Chapter 3 Resource Masters - KTL MATH CLASSES©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 3 Resource Masters The Fast FileChapter Resource system

Chapter 3Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 3 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-828006-0 Algebra 2Chapter 3 Resource Masters

2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 3-1Study Guide and Intervention . . . . . . . . 119–120Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 121Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Reading to Learn Mathematics . . . . . . . . . . 123Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 124





Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 149–150Chapter 3 Test, Form 2A . . . . . . . . . . . 151–152Chapter 3 Test, Form 2B . . . . . . . . . . . 153–154Chapter 3 Test, Form 2C . . . . . . . . . . . 155–156Chapter 3 Test, Form 2D . . . . . . . . . . . 157–158Chapter 3 Test, Form 3 . . . . . . . . . . . . 159–160Chapter 3 Open-Ended Assessment . . . . . . 161Chapter 3 Vocabulary Test/Review . . . . . . . 162Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 163Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 164Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 165Chapter 3 Cumulative Review . . . . . . . . . . . 166Chapter 3 Standardized Test Practice . . 167–168

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A26

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 3 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 3Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 150–151. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

bounded region

consistent system

constraints

kuhn·STRAYNTS

dependent system

elimination method

feasible region

FEE·zuh·buhl

inconsistent system

ihn·kuhn·SIHS·tuhnt

independent system

(continued on the next page)

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

linear programming

ordered triple

substitution method

system of equations

system of inequalities

unbounded region

vertices

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

33

Study Guide and InterventionSolving Systems of Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill 119 Glencoe Algebra 2

Less

on

3-1

Graph Systems of Equations A system of equations is a set of two or moreequations containing the same variables. You can solve a system of linear equations bygraphing the equations on the same coordinate plane. If the lines intersect, the solution isthat intersection point.

Solve the system of equations by graphing. x � 2y � 4x � y � �2

Write each equation in slope-intercept form.

x � 2y � 4 → y � � 2

x � y � �2 → y � �x � 2

The graphs appear to intersect at (0, �2).

CHECK Substitute the coordinates into each equation.x � 2y � 4 x � y � �2

0 � 2(�2) � 4 0 � (�2) � �24 � 4 ✓ �2 � �2 ✓

The solution of the system is (0, �2).

Solve each system of equations by graphing.

1. y � � � 1 2. y � 2x � 2 3. y � � � 3

y � � 4 (6, �1) y � �x � 4 (2, 2) y � (4, 1)

4. 3x � y � 0 5. 2x � � �7 6. � y � 2

x � y � �2 (1, 3) � y � 1 (�4, 3) 2x � y � �1 (�2, �3)

x

y

O

x

y

Ox

y

O

x�2

x�2

y�3

x

y

Ox

y

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x

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x�4

x�2

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(0, –2)

x�2

ExampleExample

ExercisesExercises


Classify Systems of Equations The following chart summarizes the possibilities forgraphs of two linear equations in two variables.

Graphs of Equations Slopes of Lines Classification of System Number of Solutions

Lines intersect Different slopes Consistent and independent One

Lines coincide (same line)Same slope, same

Consistent and dependent Infinitely many y-intercept

Lines are parallelSame slope, different

Inconsistent None y-intercepts

Graph the system of equations x � 3y � 6and describe it as consistent and independent, 2x � y � �3consistent and dependent, or inconsistent.

Write each equation in slope-intercept form.

x � 3y � 6 → y � x � 2

2x � y � �3 → y � 2x � 3

The graphs intersect at (�3, �3). Since there is one solution, the system is consistent and independent.

Graph the system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.

1. 3x � y � �2 2. x � 2y � 5 3. 2x � 3y � 06x � 2y � 10 3x � 15 � �6y consistent 4x � 6y � 3inconsistent and dependent inconsistent

4. 2x � y � 3 5. 4x � y � �2 6. 3x � y � 2

x � 2y � 4 consistent 2x � � �1 consistent x � y � 6 consistent

and independent and dependent and independent

x

y

O

x

y

Ox

y

O

y�2

x

y

Ox

y

Ox

y

O

1�3

x

y

O

(–3, –3)

Study Guide and Intervention (continued)

Solving Systems of Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

ExercisesExercises

ExampleExample

Skills PracticeSolving Systems of Equations By Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1


1. x � 2 2. y � �3x � 6 3. y � 4 � 3x

y � 0 (2, 0) y � 2x � 4 (2, 0) y � � x � 1 (2, �2)

4. y � 4 � x 5. y � �2x � 2 6. y � x

y � x � 2 (3, 1) y � x � 5 (3, �4) y � �3x � 4 (1, 1)

7. x � y � 3 8. x � y � 4 9. 3x � 2y � 4x � y � 1 (2, 1) 2x � 5y � 8 (4, 0) 2x � y � 1 (�2, �5)

Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.

10. y � �3x 11. y � x � 5 12. 2x � 5y � 10y � �3x � 2 �2x � 2y � �10 3x � y � 15

inconsistent consistent and consistent and dependent independent

2

2 x

y

O

x

y

O

x

y

O

x

y

Ox

y

O

x

y

O

x

y

O

x

y

O

x

y

O

1�3

x

y

Ox

y

Ox

y

O

1�2



1. x � 2y � 0 2. x � 2y � 4 3. 2x � y � 3

y � 2x � 3 (2, 1) 2x � 3y � 1 (2, 1) y � x � (3, �3)

4. y � x � 3 5. 2x � y � 6 6. 5x � y � 4y � 1 (�2, 1) x � 2y � �2 (2, �2) �2x � 6y � 4 (1, 1)

Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.

7. 2x � y � 4 8. y � �x � 2 9. 2y � 8 � x

x � y � 2 x � y � �4 y � x � 4

consistent and indep. inconsistent consistent and dep.

SOFTWARE For Exercises 10–12, use the following information.Location Mapping needs new software. Software A costs $13,000 plus $500 per additionalsite license. Software B costs $2500 plus $1200 per additional site license.

10. Write two equations that represent the cost of each software. A: y � 13,000 � 500x, B: y � 2500 � 1200x

11. Graph the equations. Estimate the break-even point of the software costs.15 additional licenses

12. If Location Mapping plans to buy 10 additional site licenses, which software will cost less? B

Software Costs

Additional Licenses

Tota

l Co

st (

$)

40 8 12 14 16 18 202 6 10

24,000

20,000

16,000

12,000

8,000

4,000

�

x

y

Ox

y

O

1�2

x

y

Ox

y

Ox

y

O

9�2

1�2

Practice (Average)

Solving Systems of Equations By Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

Reading to Learn MathematicsSolving Systems of Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


Less

on

3-1

Pre-Activity How can a system of equations be used to predict sales?

Read the introduction to Lesson 3-1 at the top of page 110 in your textbook.

• Which are growing faster, in-store sales or online sales? online sales

• In what year will the in-store and online sales be the same? 2005

Reading the Lesson

1. The Study Tip on page 110 of your textbook says that when you solve a system ofequations by graphing and find a point of intersection of the two lines, you must alwayscheck the ordered pair in both of the original equations. Why is it not good enough tocheck the ordered pair in just one of the equations?Sample answer: To be a solution of the system, the ordered pair mustmake both of the equations true.

2. Under each system graphed below, write all of the following words that apply: consistent,inconsistent, dependent, and independent.

inconsistent consistent; consistent; dependent independent

Helping You Remember

3. Look up the words consistent and inconsistent in a dictionary. How can the meaning ofthese words help you distinguish between consistent and inconsistent systems ofequations?

Sample answer: One meaning of consistent is “being in agreement,” or“compatible,” while one meaning of inconsistent is “not being inagreement” or “incompatible.” When a system is consistent, theequations are compatible because both can be true at the same time (forthe same values of x and y). When a system is inconsistent, theequations are incompatible because they can never be true at the same time.

x

y

Ox

y

Ox

y

O


InvestmentsThe following graph shows the value of two different investments over time.Line A represents an initial investment of $30,000 with a bank paying passbook savings interest. Line B represents an initial investment of $5,000 in a profitable mutual fund with dividends reinvested and capital gains accepted in shares. By deriving the linear equation y � mx � b for A and B, you can predict the value of these investments for years to come.

1. The y-intercept, b, is the initial investment. Find b for each of the following.a. line A 30,000 b. line B 5000

2. The slope of the line, m, is the rate of return. Find m for each of the following.a. line A 0.588 b. line B 2.73

3. What are the equations of each of the following lines?a. line A y � 0.588x � 30 b. line B y � 2.73x � 5

4. What will be the value of the mutual fund after 11 years of investment? $35,030

5. What will be the value of the bank account after 11 years of investment? $36,468

6. When will the mutual fund and the bank account have equal value?after 11.67 years of investment

7. Which investment has the greatest payoff after 11 years of investment? the mutual fund

A

B25

20

30

35

10

5

0

15

1 2 3 4 5 6 7 8 9

Amou

nt In

vest

ed (t

hous

ands

)

Years Invested

x

y

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

3-13-1

Study Guide and InterventionSolving Systems of Equations Algebraically

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Substitution To solve a system of linear equations by substitution, first solve for onevariable in terms of the other in one of the equations. Then substitute this expression intothe other equation and simplify.

Use substitution to solve the system of equations. 2x � y � 9x � 3y � �6

Solve the first equation for y in terms of x.2x � y � 9 First equation

�y � �2x � 9 Subtract 2x from both sides.

y � 2x � 9 Multiply both sides by �1.

Substitute the expression 2x � 9 for y into the second equation and solve for x.x � 3y � �6 Second equation

x � 3(2x � 9) � �6 Substitute 2x � 9 for y.

x � 6x � 27 � �6 Distributive Property

7x � 27 � �6 Simplify.

7x � 21 Add 27 to each side.

x � 3 Divide each side by 7.

Now, substitute the value 3 for x in either original equation and solve for y.2x � y � 9 First equation

2(3) � y � 9 Replace x with 3.

6 � y � 9 Simplify.

�y � 3 Subtract 6 from each side.

y � �3 Multiply each side by �1.

The solution of the system is (3, �3).

Solve each system of linear equations by using substitution.

1. 3x � y � 7 2. 2x � y � 5 3. 2x � 3y � �34x � 2y � 16 3x � 3y � 3 x � 2y � 2

(�1, 10) (2, 1) (�12, 7)

4. 2x � y � 7 5. 4x � 3y � 4 6. 5x � y � 66x � 3y � 14 2x � y � �8 3 � x � 0

no solution (�2, �4) (3, �9)

7. x � 8y � �2 8. 2x � y � �4 9. x � y � �2x � 3y � 20 4x � y � 1 2x � 3y � 2

(14, �2) �� , 3� (�8, �6)

10. x � 4y � 4 11. x � 3y � 2 12. 2x � 2y � 42x � 12y � 13 4x � 12 y � 8 x � 2y � 0

�5, � infinitely many � , �2�

4�

1�

1�

ExampleExample

ExercisesExercises


Elimination To solve a system of linear equations by elimination, add or subtract theequations to eliminate one of the variables. You may first need to multiply one or both of theequations by a constant so that one of the variables has the same (or opposite) coefficient inone equation as it has in the other.

Use the elimination method to solve the system of equations.2x � 4y � �263x � y � �24


Solving Systems of Equations Algebraically

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Example 1Example 1

Example 2Example 2

Multiply the second equation by 4. Then subtractthe equations to eliminate the y variable.2x � 4y � �26 2x � 4y � �263x � y � �24 Multiply by 4. 12x � 4y � �96

�10x � 70x � �7

Replace x with �7 and solve for y.2x � 4y � �26

2(�7) �4y � �26�14 � 4y � �26

�4y � �12y � 3

The solution is (�7, 3).

Multiply the first equation by 3 and the secondequation by 2. Then add the equations toeliminate the y variable.3x � 2y � 4 Multiply by 3. 9x � 6y � 125x � 3y � �25 Multiply by 2. 10x � 6y � �50

19x � �38x � �2

Replace x with �2 and solve for y.3x � 2y � 4

3(�2) � 2y � 4�6 � 2y � 4

�2y � 10y � �5

The solution is (�2, �5).

Use the elimination method to solve the system of equations.3x � 2y � 45x � 3y � �25

ExercisesExercises

Solve each system of equations by using elimination.

1. 2x � y � 7 2. x � 2y � 4 3. 3x � 4y � �10 4. 3x � y � 123x � y � 8 �x � 6y � 12 x � 4y � 2 5x � 2y � 20

(3, �1) (12, 4) (�2, �1) (4, 0)

5. 4x � y � 6 6. 5x � 2y � 12 7. 2x � y � 8 8. 7x � 2y � �1

2x � � 4 �6x � 2y � �14 3x � y � 12 4x � 3y � �13

no solution (2, 1) infinitely many (�1, 3)

9. 3x � 8y � �6 10. 5x � 4y � 12 11. �4x � y � �12 12. 5m � 2n � �8x � y � 9 7x � 6y � 40 4x � 2y � 6 4m � 3n � 2

(6, �3) (4, �2) � , �2� (�4, 6)5�

3�2

y�2

Skills PracticeSolving Systems of Equations Algebraically

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


Less

on

3-2

Solve each system of equations by using substitution.

1. m � n � 20 2. x � 3y � �3 3. w � z � 1m � n � �4 (8, 12) 4x � 3y � 6 (3, �2) 2w � 3z � 12 (3, 2)

4. 3r � s � 5 5. 2b � 3c � �4 6. x � y � 12r � s � 5 (2, �1) b � c � 3 (13, �10) 2x � 3y � 12 (3, 2)


7. 2p � q � 5 8. 2j � k � 3 9. 3c � 2d � 23p � q � 5 (2, �1) 3j � k � 2 (1, �1) 3c � 4d � 50 (6, 8)

10. 2f � 3g � 9 11. �2x � y � �1 12. 2x � y � 12f � g � 2 (3, 1) x � 2y � 3 (1, 1) 2x � y � 6 no solution

Solve each system of equations by using either substitution or elimination.

13. �r � t � 5 14. 2x � y � �5 15. x � 3y � �12�2r � t � 4 (1, 6) 4x � y � 2 �� , 4� 2x � y � 11 (3, 5)

16. 2p � 3q � 6 17. 6w � 8z � 16 18. c � d � 6�2p � 3q � �6 (3, 0) 3w � 4z � 8 c � d � 0 (3, 3)

infinitely many

19. 2u � 4v � �6 20. 3a � b � �1 21. 2x � y � 6u � 2v � 3 no solution �3a � b � 5 (�1, 2) 3x � 2y � 16 (4, �2)

22. 3y � z � �6 23. c � 2d � �2 24. 3r � 2s � 1�3y � z � 6 (�2, 0) �2c � 5d � 3 (�4, 1) 2r � 3s � 9 (�3, �5)

25. The sum of two numbers is 12. The difference of the same two numbers is �4.Find the numbers. 4, 8

26. Twice a number minus a second number is �1. Twice the second number added to three times the first number is 9. Find the two numbers. 1, 3

1�



1. 2x � y � 4 2. x � 3y � 9 3. g � 3h � 8 no3x � 2y � 1 (7, �10) x � 2y � �1 (3, �2) g � h � 9 solution

4. 2a � 4b � 6 infinitely 5. 2m � n � 6 6. 4x � 3y � �6�a � 2b � �3 many 5m � 6n � 1 (5, �4) �x � 2y � 7 (�3, �2)

7. u � 2v � 8. x � 3y � 16 9. w � 3z � 1

�u � 2v � 5 no solution 4x � y � 9 (1, �5) 3w � 5z � �4 �� , �


10. 2r � s � 5 11. 2m � n � �1 12. 6x � 3y � 63r � s � 20 (5, �5) 3m � 2n � 30 (4, 9) 8x � 5y � 12 (�1, 4)

13. 3j � k � 10 14. 2x � y � �4 no 15. 2g � h � 64j � k � 16 (6, 8) �4x � 2y � 6 solution 3g � 2h � 16 (4, �2)

16. 2t � 4v � 6 infinitely 17. 3x � 2y � 12 18. x � 3y � 11�t � 2v � �3 many

2x � y � 14 (6, 3) 8x � 5y � 17 (4, 3)

Solve each system of equations by using either substitution or elimination.

19. 8x � 3y � �5 20. 8q � 15r � �40 21. 3x � 4y � 12 infinitely10x � 6y � �13 � , �3� 4q � 2r � 56 (10, 8) x � y � many

22. 4b � 2d � 5 no 23. s � 3y � 4 24. 4m � 2p � 0�2b � d � 1 solution s � 1 (1, 1) �3m � 9p � 5 � , �

25. 5g � 4k � 10 26. 0.5x � 2y � 5 27. h � z � 3 no�3g � 5k � 7 (6, �5) x � 2y � �8 (�2, 3) �3h � 3z � 6 solution

SPORTS For Exercises 28 and 29, use the following information.Last year the volleyball team paid $5 per pair for socks and $17 per pair for shorts on atotal purchase of $315. This year they spent $342 to buy the same number of pairs of socksand shorts because the socks now cost $6 a pair and the shorts cost $18.

28. Write a system of two equations that represents the number of pairs of socks and shortsbought each year. 5x � 17y � 315, 6x � 18y � 342

29. How many pairs of socks and shorts did the team buy each year?socks: 12, shorts: 15

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Practice (Average)

Solving Systems of Equations Algebraically

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Reading to Learn MathematicsSolving Systems of Equations Algebraically

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2


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Pre-Activity How can systems of equations be used to make consumerdecisions?


• How many more minutes of long distance time did Yolanda use inFebruary than in January? 13 minutes

• How much more were the February charges than the January charges?$1.04

• Using your answers for the questions above, how can you find the rateper minute? Find $1.04 � 13.

Reading the Lesson

1. Suppose that you are asked to solve the system of equations 4x � 5y � 7at the right by the substitution method. 3x � y � �9

The first step is to solve one of the equations for one variable in terms of the other. To make your work as easy as possible,which equation would you solve for which variable? Explain.Sample answer: Solve the second equation for y because in thatequation the variable y has a coefficient of 1.

2. Suppose that you are asked to solve the system of equations 2x � 3y � �2at the right by the elimination method. 7x � y � 39

To make your work as easy as possible, which variable would you eliminate? Describe how you would do this.Sample answer: Eliminate the variable y; multiply the second equationby 3 and then add the result to the first equation.


3. The substitution method and elimination method for solving systems both have severalsteps, and it may be difficult to remember them. You may be able to remember themmore easily if you notice what the methods have in common. What step is the same inboth methods?Sample answer: After finding the value of one of the variables, you findthe value of the other variable by substituting the value you have foundin one of the original equations.


Using CoordinatesFrom one observation point, the line of sight to a downed plane is given by y � x � 1. This equation describes the distance from the observation point to the plane in a straight line. From another observation point, the line of sight is given by x � 3y � 21. What are the coordinates of the point at which the crash occurred?

Solve the system of equations � y � x � 1 .x � 3y � 21

x � 3y � 21x � 3(x � 1) � 21 Substitute x � 1 for y.

x � 3x � 3 � 214x � 24x � 6

x � 3y � 216 � 3y � 21 Substitute 6 for x.

3y � 15y � 5

The coordinates of the crash are (6, 5).

Solve the following.

1. The lines of sight to a forest fire are as follows.

From Ranger Station A: 3x � y � 9From Ranger Station B: 2x � 3y � 13Find the coordinates of the fire.

2. An airplane is traveling along the line x � y � �1 when it sees another airplane traveling along the line 5x � 3y � 19. If they continue along the same lines, at what point will their flight paths cross?

3. Two mine shafts are dug along the paths of the following equations.

x � y � 14002x � y � 1300

If the shafts meet at a depth of 200 feet, what are the coordinates of the point at which they meet?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-23-2

Study Guide and InterventionSolving Systems of Inequalities by Graphing

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3-33-3


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Graph Systems of Inequalities To solve a system of inequalities, graph the inequalitiesin the same coordinate plane. The solution set is represented by the intersection of the graphs.

Solve the system of inequalities by graphing.y � 2x � 1 and y � � 2

The solution of y � 2x � 1 is Regions 1 and 2.

The solution of y � � 2 is Regions 1 and 3.

The intersection of these regions is Region 1, which is the solution set of the system of inequalities.

Solve each system of inequalities by graphing.

1. x � y � 2 2. 3x � 2y � �1 3. y � 1x � 2y � 1 x � 4y � �12 x � 2

4. y � � 3 5. y � � 2 6. y � � � 1

y � 2x y � �2x � 1 y � 3x � 1

7. x � y � 4 8. x � 3y � 3 9. x � 2y � 62x � y � 2 x � 2y � 4 x � 4y � �4

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Region 3Region 1

Region 2

ExampleExample

ExercisesExercises


Find Vertices of a Polygonal Region Sometimes the graph of a system ofinequalities forms a bounded region. You can find the vertices of the region by a combinationof the methods used earlier in this chapter: graphing, substitution, and/or elimination.

Find the coordinates of the vertices of the figure formed by 5x � 4y � 20, y � 2x � 3, and x � 3y � 4.

Graph the boundary of each inequality. The intersections of the boundary lines are the vertices of a triangle.The vertex (4, 0) can be determined from the graph. To find thecoordinates of the second and third vertices, solve the two systems of equations

y � 2x � 3 and y � 2x � 35x � 4y � 20 x � 3y � 4

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Solving Systems of Inequalities by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

For the first system of equations, rewrite the firstequation in standard form as 2x � y � �3. Thenmultiply that equation by 4 and add to the secondequation.2x � y � �3 Multiply by 4. 8x � 4y � �125x � 4y � 20 (�) 5x � 4y � 20

13x � 8

x �

Then substitute x � in one of the original equations and solve for y.

2� � � y � �3

� y � �3

y �

The coordinates of the second vertex are � , 4 �.3�13

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For the second system ofequations, use substitution.Substitute 2x � 3 for y in thesecond equation to getx � 3(2x � 3) � 4

x � 6x � 9 � 4�5x � 13

x � �

Then substitute x � � in the

first equation to solve for y.

y � 2�� 3

y � � � 3

y � �

The coordinates of the third

vertex are ��2 , �2 �.1�5

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ExercisesExercises

Thus, the coordinates of the three vertices are (4, 0), � , 4 �, and ��2 , �2 �.

Find the coordinates of the vertices of the figure formed by each system ofinequalities.

1. y � �3x � 7 2. x � �3 3. y � � x � 3

y � x (2, 1), (�4, �2), y � � x � 3 (�3, 4), y � x � 1 (�2, 4), (2, 2),

y � �2(3, �2)

y � x � 1(�3, �4), (3, 2)

y � 3x � 10 ��3 , � �4�

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ExampleExample

Skills PracticeSolving Systems of Inequalities by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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1. x � 1 2. x � �3 3. x � 2y � �1 y � �3 x � 4 no solution

4. y � x 5. y � �4x 6. x � y � �1y � �x y � 3x � 2 3x � y � 4

7. y � 3 8. y � �2x � 3 9. x � y � 4x � 2y � 12 y � x � 2 2x � y � 4


10. y � 0 11. y � 3 � x 12. x � �2x � 0 y � 3 y � x � 2y � �x � 1 x � �5 x � y � 2 (�2, 4), (0, 0), (0, �1), (�1, 0) (0, 3), (�5, 3), (�5, 8) (�2, �4), (2, 0)

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1. y � 1 � �x 2. x � �2 3. y � 2x � 3

y � 1 2y � 3x � 6 y � � x � 2

4. x � y � �2 5. y � 1 6. 3y � 4x3x � y � �2 y � x � 1 2x � 3y � �6


7. y � 1 � x 8. x � y � 2 9. y � 2x � 2y � x � 1 x � y � 2 2x � 3y � 6x � 3 x � �2 y � 4

(1, 0), (3, 2), (3, �2) (�2, 4), (�2, �4), (2, 0) (�3, 4), � , 1�, (3, 4)

DRAMA For Exercises 10 and 11, use the following information.The drama club is selling tickets to its play. An adult ticket costs $15 and a student ticket costs $11. The auditorium willseat 300 ticket-holders. The drama club wants to collect atleast $3630 from ticket sales.

10. Write and graph a system of four inequalities that describe how many of each type of ticket the club must sell to meets its goal.x 0, y 0, x � y � 300, 15x � 11y 3630

11. List three different combinations of tickets sold thatsatisfy the inequalities. Sample answer: 250 adult and 50 student, 200 adult and 100 student, 145 adult and 148 student

Play Tickets

Adult Tickets

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1000 200 300 400

400

350

300

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100

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Practice (Average)

Solving Systems of Inequalities by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3

Reading to Learn MathematicsSolving Systems of Inequalities by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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Pre-Activity How can you determine whether your blood pressure is in a normal range?


Satish is 37 years old. He has a blood pressure reading of 135/99. Is hisblood pressure within the normal range? Explain.Sample answer: No; his systolic pressure is normal, but hisdiastolic pressure is too high. It should be between 60 and 90.

Reading the Lesson

1. Without actually drawing the graph, describe the boundary lines x � 3for the system of inequalities shown at the right. y � 5Two dashed vertical lines (x � 3 and x � �3) and two solid horizontallines (y � �5 and y � 5)

2. Think about how the graph would look for the system given above. What will be theshape of the shaded region? (It is not necessary to draw the graph. See if you canimagine it without drawing anything. If this is difficult to do, make a rough sketch tohelp you answer the question.)a rectangle

3. Which system of inequalities matches the graph shown at the right? BA. x � y � �2 B. x � y � �2

x � y � 2 x � y � 2

C. x � y � �2 D. x � y � �2x � y � 2 x � y � 2


4. To graph a system of inequalities, you must graph two or more boundary lines. Whenyou graph each of these lines, how can the inequality symbols help you rememberwhether to use a dashed or solid line?Use a dashed line if the inequality symbol is � or �, because thesesymbols do not include equality and the dashed line reminds you thatthe line itself is not included in the graph. Use a solid line if the symbolis or �, because these symbols include equality and tell you that theline itself is included in the graph.

x

y

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Tracing StrategyTry to trace over each of the figures below without tracing the same segment twice.

The figure at the left cannot be traced, but the one at the right can. The rule is that a figure is traceable if it has no more than two points where an odd number of segments meet. The figure at the left has three segments meeting at each of the four corners. However, the figure at the right has only two points, L and Q, where an odd number of segments meet.

Determine if each figure can be traced without tracing the same segment twice. If it can, then name the starting point and name the segments in the order they should be traced.

1. 2.

yes; X; X�Y�, Y�F�, F�X�, X�G�, G�F�, yes; E; E�D�, D�A�, A�E�, E�B�, B�F�, F�C�,C�K�, F�E�, E�H�, H�X�, X�W�, W�H�, H�G� K�F�, F�J�, J�H�, H�F�, F�E�, E�H�, H�G�, G�E�

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D C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionLinear Programming

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Maximum and Minimum Values When a system of linear inequalities produces abounded polygonal region, the maximum or minimum value of a related function will occurat a vertex of the region.

Graph the system of inequalities. Name the coordinates of thevertices of the feasible region. Find the maximum and minimum values of thefunction f(x, y) � 3x � 2y for this polygonal region.

y � 4y � �x � 6

y x �

y � 6x � 4First find the vertices of the bounded region. Graph the inequalities.The polygon formed is a quadrilateral with vertices at (0, 4), (2, 4), (5, 1), and (�1, �2). Use the table to find themaximum and minimum values of f(x, y) � 3x � 2y.

The maximum value is 17 at (5, 1). The minimum value is �7 at (�1, �2).

Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.

1. y � 2 2. y � �2 3. x � y � 21 � x � 5 y � 2x � 4 4y � x � 8y � x � 3 x � 2y � �1 y � 2x � 5f(x, y) � 3x � 2y f(x, y) � 4x � y f(x, y) � 4x � 3y

vertices: (1, 2), (1, 4), vertices: (�5, �2), vertices (0, 2), (4, 3),(5, 8), (5, 2); max: 11; (3, 2), (1, �2); � , � �; max: 25; min: 6 min; �5 max: 10; min: �18

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(x, y ) 3x � 2y f (x, y )

(0, 4) 3(0) � 2(4) 8

(2, 4) 3(2) � 2(4) 14

(5, 1) 3(5) � 2(1) 17

(�1, �2) 3(�1) � 2(�2) �7

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ExampleExample

ExercisesExercises


Real-World Problems When solving linear programming problems, use thefollowing procedure.

1. Define variables.2. Write a system of inequalities.3. Graph the system of inequalities.4. Find the coordinates of the vertices of the feasible region.5. Write an expression to be maximized or minimized.6. Substitute the coordinates of the vertices in the expression.7. Select the greatest or least result to answer the problem.

A painter has exactly 32 units of yellow dye and 54 units of greendye. He plans to mix as many gallons as possible of color A and color B. Eachgallon of color A requires 4 units of yellow dye and 1 unit of green dye. Eachgallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find themaximum number of gallons he can mix.

Step 1 Define the variables.x � the number of gallons of color A madey � the number of gallons of color B made

Step 2 Write a system of inequalities.Since the number of gallons made cannot benegative, x � 0 and y � 0.There are 32 units of yellow dye; each gallon of color A requires 4 units, and each gallon of color B requires 1 unit.So 4x � y � 32.Similarly for the green dye, x � 6y � 54.Steps 3 and 4 Graph the system of inequalities and find the coordinates of the vertices of the feasible region.The vertices of the feasible region are (0, 0), (0, 9), (6, 8), and (8, 0).Steps 5–7 Find the maximum number of gallons,x � y, that he can make.The maximum number of gallons the painter can make is 14,6 gallons of color A and 8 gallons of color B.

1. FOOD A delicatessen has 8 pounds of plain sausage and 10 pounds of garlic-flavoredsausage. The deli wants to make as much bratwurst as possible. Each pound of

bratwurst requires pound of plain sausage and pound of garlic-flavored sausage.

Find the maximum number of pounds of bratwurst that can be made. 10 lb

2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $16per hour. Machine B can produce 40 steering wheels per hour at a cost of $22 per hour.At least 360 steering wheels must be made in each 8-hour shift. What is the least costinvolved in making 360 steering wheels in one shift? $194

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(x, y) x � y f (x, y)

(0, 0) 0 � 0 0

(0, 9) 0 � 9 9

(6, 8) 6 � 8 14

(8, 0) 8 � 0 8

Color A (gallons)

Co

lor

B (

gal

lon

s)

5 10 15 20 25 30 35 40 45 50 550

40

35

30

25

20

15

10

5

(6, 8)

(8, 0)(0, 9)


Linear Programming

NAME ______________________________________________ DATE______________ PERIOD _____

3-43-4

ExampleExample

ExercisesExercises

Skills PracticeLinear Programming

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1. x � 2 2. x � 1 3. x � 0x � 5 y � 6 y � 0y � 1 y � x � 2 y � 7 � xy � 4 f(x, y) � x � y f(x, y) � 3x � yf(x, y) � x � y

max.: 9, min.: 3 max.: 2, min.: �5 max.: 21, min.: 0

4. x � �1 5. y � 2x 6. y � �x � 2x � y � 6 y � 6 � x y � 3x � 2f(x, y) � x � 2y y � 6 y � x � 4

f(x, y) � 4x � 3y f(x, y) � �3x � 5y

max.: 13, no min. no max., min.: 20 max.: 22, min.: �2

7. MANUFACTURING A backpack manufacturer produces an internal frame pack and anexternal frame pack. Let x represent the number of internal frame packs produced inone hour and let y represent the number of external frame packs produced in one hour.Then the inequalities x � 3y � 18, 2x � y � 16, x � 0, and y � 0 describe the constraintsfor manufacturing both packs. Use the profit function f(x) � 50x � 80y and theconstraints given to determine the maximum profit for manufacturing both backpacksfor the given constraints. $620

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1. 2x � 4 � y 2. 3x � y � 7 3. x � 0�2x � 4 � y 2x � y � 3 y � 0y � 2 y � x � 3 y � 6f(x, y) � �2x � y f(x, y) � x � 4y y � �3x � 15

f(x, y) � 3x � y

max.: 8, min.: �4 max.: 12, min.: �16 max.: 15, min.: 0

4. x � 0 5. y � 3x � 6 6. 2x � 3y � 6y � 0 4y � 3x � 3 2x � y � 24x � y � �7 x � �2 x � 0f(x, y) � �x � 4y f(x, y) � �x � 3y y � 0

f(x, y) � x � 4y � 3

max.: 28, min.: 0 max.: , no min. no max., min.:

PRODUCTION For Exercises 7–9, use the following information.A glass blower can form 8 simple vases or 2 elaborate vases in an hour. In a work shift of nomore than 8 hours, the worker must form at least 40 vases.

7. Let s represent the hours forming simple vases and e the hours forming elaborate vases.Write a system of inequalities involving the time spent on each type of vase.s 0, e 0, s � e � 8, 8s � 2e 40

8. If the glass blower makes a profit of $30 per hour worked on the simple vases and $35per hour worked on the elaborate vases, write a function for the total profit on the vases.f(s, e) � 30s � 35e

9. Find the number of hours the worker should spend on each type of vase to maximizeprofit. What is that profit? 4 h on each; $260

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Practice (Average)

Linear Programming

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

Reading to Learn MathematicsLinear Programming

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How is linear programming used in scheduling work?


Name two or more facts that indicate that you will need to use inequalitiesto model this situation.Sample answer: The buoy tender can carry up to 8 new buoys.There seems to be a limit of 24 hours on the time the crew hasat sea. The crew will want to repair or replace the maximumnumber of buoys possible.

Reading the Lesson

1. Complete each sentence.

a. When you find the feasible region for a linear programming problem, you are solving

a system of linear called . The points

in the feasible region are of the system.

b. The corner points of a polygonal region are the of the feasible region.

2. A polygonal region always takes up only a limited part of the coordinate plane. One wayto think of this is to imagine a circle or rectangle that the region would fit inside. In thecase of a polygonal region, you can always find a circle or rectangle that is large enoughto contain all the points of the polygonal region. What word is used to describe a regionthat can be enclosed in this way? What word is used to describe a region that is too largeto be enclosed in this way? bounded; unbounded

3. How do you find the corner points of the polygonal region in a linear programmingproblem? You solve a system of two linear equations.

4. What are some everyday meanings of the word feasible that remind you of themathematical meaning of the term feasible region?Sample answer: possible or achievable


5. Look up the word constraint in a dictionary. If more than one definition is given, choosethe one that seems closest to the idea of a constraint in a linear programming problem.How can this definition help you to remember the meaning of constraint as it is used inthis lesson? Sample answer: A constraint is a restriction or limitation. Theconstraints in a linear programming problem are restrictions on thevariables that translate into inequality statements.

vertices

solutions

constraintsinequalities


Computer Circuits and LogicComputers operate according to the laws of logic. The circuits of a computer can be described using logic.

1. With switch A open, no current flows. The value 0 is assigned to an open switch.

2. With switch A closed, current flows. The value 1 is assigned to a closed switch.

3. With switches A and B open, no current flows. This circuit can be described by the conjunction, A B.

4. In this circuit, current flows if either A or B is closed. This circuit can be described by the disjunction, A � B.

Truth tables are used to describe the flow of current in a circuit.The table at the left describes the circuit in diagram 4. According to the table, the only time current does not flow through the circuit is when both switches A and B are open.

Draw a circuit diagram for each of the following.

1. (A B) � C 2. (A � B) C

3. (A � B) (C � D) 4. (A B) � (C D)

5. Construct a truth table for the following circuit.

B

C

A

A

A

A

B

A B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-43-4

A B A � B

0 0 00 1 11 0 11 1 1

Study Guide and InterventionSolving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


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Systems in Three Variables Use the methods used for solving systems of linearequations in two variables to solve systems of equations in three variables. A system ofthree equations in three variables can have a unique solution, infinitely many solutions, orno solution. A solution is an ordered triple.

Solve this system of equations. 3x � y � z � �62x � y � 2z � 84x � y � 3z � �21

Step 1 Use elimination to make a system of two equations in two variables.3x � y � z � �6 First equation 2x � y � 2z � 8 Second equation

(�) 2x � y � 2z � 8 Second equation (�) 4x � y � 3z � �21 Third equation

5x � z � 2 Add to eliminate y. 6x � z � �13 Add to eliminate y.

Step 2 Solve the system of two equations.5x � z � 2

(�) 6x � z � �1311x � �11 Add to eliminate z.

x � �1 Divide both sides by 11.

Substitute �1 for x in one of the equations with two variables and solve for z.5x � z � 2 Equation with two variables

5(�1) � z � 2 Replace x with �1.

�5 � z � 2 Multiply.

z � 7 Add 5 to both sides.

The result so far is x � �1 and z � 7.

Step 3 Substitute �1 for x and 7 for z in one of the original equations with three variables.3x � y � z � �6 Original equation with three variables

3(�1) � y � 7 � �6 Replace x with �1 and z with 7.

�3 � y � 7 � �6 Multiply.

y � 4 Simplify.

The solution is (�1, 4, 7).

Solve each system of equations.

1. 2x � 3y � z � 0 2. 2x � y � 4z � 11 3. x � 2y � z � 8x � 2y � 4z � 14 x � 2y � 6z � �11 2x � y � z � 03x � y � 8z � 17 3x � 2y �10z � 11 3x � 6y � 3z � 24

(4, �3, �1) �2, �5, � infinitely manysolutions

4. 3x � y � z � 5 5. 2x � 4y � z � 10 6. x � 6y � 4z � 23x � 2y � z � 11 4x � 8y � 2z � 16 2x � 4y � 8z � 166x � 3y � 2z � �12 3x � y � z � 12 x � 2y � 5

� , 2, �5� no solution �6, , � �1�

1�

2�

1�

ExampleExample

ExercisesExercises


Real-World Problems

The Laredo Sports Shop sold 10 balls, 3 bats, and 2 bases for $99 onMonday. On Tuesday they sold 4 balls, 8 bats, and 2 bases for $78. On Wednesdaythey sold 2 balls, 3 bats, and 1 base for $33.60. What are the prices of 1 ball, 1 bat,and 1 base?

First define the variables.x � price of 1 bally � price of 1 batz � price of 1 base

Translate the information in the problem into three equations.

10x � 3y � 2z � 994x � 8y � 2z � 782x � 3y �z � 33.60


Solving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

ExampleExample

Subtract the second equation from the firstequation to eliminate z.

10x � 3y � 2z � 99(�) 4x � 8y � 2z � 78

6x � 5y � 21

Multiply the third equation by 2 andsubtract from the second equation.

4x � 8y � 2z � 78(�) 4x � 6y � 2z � 67.20

2y � 10.80y � 5.40

Substitute 5.40 for y in the equation 6x � 5y � 21.

6x �5(5.40) � 216x � 48x � 8

Substitute 8 for x and 5.40 for y in one ofthe original equations to solve for z.

10x � 3y � 2z � 9910(8) � 3(5.40) � 2z � 99

80 � 16.20 � 2z � 992z � 2.80z � 1.40

So a ball costs $8, a bat $5.40, and a base $1.40.

1. FITNESS TRAINING Carly is training for a triathlon. In her training routine each week,she runs 7 times as far as she swims, and she bikes 3 times as far as she runs. One weekshe trained a total of 232 miles. How far did she run that week? 56 miles

2. ENTERTAINMENT At the arcade, Ryan, Sara, and Tim played video racing games, pinball,and air hockey. Ryan spent $6 for 6 racing games, 2 pinball games, and 1 game of airhockey. Sara spent $12 for 3 racing games, 4 pinball games, and 5 games of air hockey. Timspent $12.25 for 2 racing games, 7 pinball games, and 4 games of air hockey. How much dideach of the games cost? Racing game: $0.50; pinball: $0.75; air hockey: $1.50

3. FOOD A natural food store makes its own brand of trail mix out of dried apples, raisins,and peanuts. One pound of the mixture costs $3.18. It contains twice as much peanutsby weight as apples. One pound of dried apples costs $4.48, a pound of raisins $2.40, anda pound of peanuts $3.44. How many ounces of each ingredient are contained in 1 poundof the trail mix? 3 oz of apples, 7 oz of raisins, 6 oz of peanuts

ExercisesExercises

Skills PracticeSolving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. 2a � c � �10 (5, �5, �20) 2. x � y � z � 3 (0, 2, 1)b � c � 15 13x � 2z � 2a � 2b � c � �5 �x � 5z � �5

3. 2x � 5y � 2z � 6 (�3, 2, 1) 4. x � 4y � z � 1 no solution5x � 7y � �29 3x � y � 8z � 0z � 1 x � 4y � z � 10

5. �2z � �6 (2, �1, 3) 6. 3x � 2y � 2z � �2 (�2, 1, 3)2x � 3y � z � �2 x � 6y � 2z � �2x � 2y � 3z � 9 x � 2y � 0

7. �x � 5z � �5 (0, 0, 1) 8. �3r � 2t � 1 (1, �6, 2)y � 3x � 0 4r � s � 2t � �613x � 2z � 2 r � s � 4t � 3

9. x � y � 3z � 3 no solution 10. 5m � 3n � p � 4 (�2, 3, 5)�2x � 2y � 6z � 6 3m � 2n � 0y � 5z � �3 2m � n � 3p � 8

11. 2x � 2y � 2z � �2 infinitely many 12. x � 2y � z � 4 (1, 2, 1)2x � 3y � 2z � 4 3x � y � 2z � 3x � y � z � �1 �x � 3y � z � 6

13. 3x � 2y � z � 1 (5, 7, 0) 14. 3x � 5y � 2z � �12 infinitely many�x � y � z � 2 x � 4y � 2z � 85x � 2y � 10z � 39 �3x � 5y � 2z � 12

15. 2x � y � 3z � �2 (�1, 3, �1) 16. 2x � 4y � 3z � 0 (3, 0, �2)x � y � z � �3 x � 2y � 5z � 133x � 2y � 3z � �12 5x � 3y � 2z � 19

17. �2x � y � 2z � 2 (1, �2, 3) 18. x � 2y � 2z � �1 infinitely many3x � 3y � z � 0 x � 2y � z � 6x � y � z � 2 �3x � 6y � 6z � 3

19. The sum of three numbers is 18. The sum of the first and second numbers is 15,and the first number is 3 times the third number. Find the numbers. 9, 6, 3



1. 2x � y � 2z � 15 2. x � 4y � 3z � �27 3. a � b � 3�x � y � z � 3 2x � 2y � 3z � 22 �b � c � 33x � y � 2z � 18 4z � �16 a � 2c � 10(3, 1, 5) (1, 4, �4) (2, 1, 4)

4. 3m � 2n � 4p � 15 5. 2g � 3h � 8j � 10 6. 2x � y � z � �8m � n � p � 3 g � 4h � 1 4x � y � 2z � �3m � 4n � 5p � 0 �2g � 3h � 8j � 5 �3x � y � 2z � 5(3, 3, 3) no solution (�2, �3, 1)

7. 2x � 5y � z � 5 8. 2x � 3y � 4z � 2 9. p � 4r � �73x � 2y � z � 17 5x � 2y � 3z � 0 p � 3q � �84x � 3y � 2z � 17 x � 5y � 2z � �4 q � r � 1(5, 1, 0) (2, 2, �2) (1, 3, �2)

10. 4x � 4y � 2z � 8 11. d � 3e � f � 0 12. 4x � y � 5z � �93x � 5y � 3z � 0 �d � 2e � f � �1 x � 4y � 2z � �2 2x � 2y � z � 4 4d � e � f � 1 2x � 3y � 2z � 21infinitely many (1, �1, 2) (2, 3, �4)

13. 5x � 9y � z � 20 14. 2x � y � 3z � �3 15. 3x � 3y � z � 102x � y � z � �21 3x � 2y � 4z � 5 5x � 2y � 2z � 75x � 2y � 2z � �21 �6x � 3y � 9z � 9 3x � 2y � 3z � �9(�7, 6, 1) infinitely many (1, 3, �2)

16. 2u � v � w � 2 17. x � 5y � 3z � �18 18. x � 2y � z � �1�3u � 2v � 3w � 7 3x � 2y � 5z � 22 �x � 2y � z � 6 �u � v � 2w � 7 �2x � 3y � 8z � 28 �4y � 2z � 1(0, �1, 3) (1, �2, 3) no solution

19. 2x � 2y � 4z � �2 20. x � y � 9z � �27 21. 2x � 5y � 3z � 73x � 3y � 6z � �3 2x � 4y � z � �1 �4x � 10y � 2z � 6�2x � 3y � z � 7 3x � 6y � 3z � 27 6x � 15y � z � �19(4, 5, 0) (2, 2, �3) (1, 2, �5)

22. The sum of three numbers is 6. The third number is the sum of the first and secondnumbers. The first number is one more than the third number. Find the numbers.4, �1, 3

23. The sum of three numbers is �4. The second number decreased by the third is equal tothe first. The sum of the first and second numbers is �5. Find the numbers. �3, �2, 1

24. SPORTS Alexandria High School scored 37 points in a football game. Six points areawarded for each touchdown. After each touchdown, the team can earn one point for theextra kick or two points for a 2-point conversion. The team scored one fewer 2-pointconversions than extra kicks. The team scored 10 times during the game. How manytouchdowns were made during the game? 5

Practice (Average)

Solving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

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Reading to Learn MathematicsSolving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

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Pre-Activity How can you determine the number and type of medals U.S.Olympians won?


At the 1996 Summer Olympics in Atlanta, Georgia, the United States won101 medals. The U.S. team won 12 more gold medals than silver and 7fewer bronze medals than silver. Using the same variables as those in theintroduction, write a system of equations that describes the medals won forthe 1996 Summer Olympics.g � s � b � 101; g � s � 12; b � s � 7

Reading the Lesson

1. The planes for the equations in a system of three linear equations in three variablesdetermine the number of solutions. Match each graph description below with thedescription of the number of solutions of the system. (Some of the items on the right maybe used more than once, and not all possible types of graphs are listed.)

a. three parallel planes I. one solution

b. three planes that intersect in a line II. no solutions

c. three planes that intersect in one point III. infinite solutions

d. one plane that represents all three equations

2. Suppose that three classmates, Monique, Josh, and Lilly, are studying for a quiz on thislesson. They work together on solving a system of equations in three variables, x, y, andz, following the algebraic method shown in your textbook. They first find that z � 3, thenthat y � �2, and finally that x � �1. The students agree on these values, but disagreeon how to write the solution. Here are their answers:

Monique: (3, �2, �1) Josh: (�2, �1, 3) Lilly: (�1, �2, 3)

a. How do you think each student decided on the order of the numbers in the orderedtriple? Sample answer: Monique arranged the values in the order inwhich she found them. Josh arranged them from smallest to largest.Lilly arranged them in alphabetical order of the variables.

b. Which student is correct? Lilly


3. How can you remember that obtaining the equation 0 � 0 indicates a system withinfinitely many solutions, while obtaining an equation such as 0 � 8 indicates a systemwith no solutions? 0 � 0 is always true, while 0 � 8 is never true.

III

I

III

II


BilliardsThe figure at the right shows a billiard table.The object is to use a cue stick to strike the ball at point C so that the ball will hit the sides (or cushions) of the table at least once before hitting the ball located at point A. In playing the game, you need to locate point P.

Step 1 Find point B so that � and � . B is called the reflected

image of C in .

Step 2 Draw .

Step 3 intersects at the desired point P.

For each billiards problem, the cue ball at point C must strike the indicatedcushion(s) and then strike the ball at point A. Draw and label the correct path for the cue ball using the process described above.

1. cushion 2. cushion

3. cushion , then cushion 4. cushion , then cushion

C

A

RK

T S

C

ARK

T S

RSKTRSTS

C

A

RK

T S

C

A

RK

T S

RSKR

STAB

AB

STCHBH

STBC

K R

C

A

B

P

HT S

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

Chapter 3 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

entWrite the letter for the correct answer in the blank at the right of each question.

1. A system of equations may not have A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.

Choose the correct description of each system of equations.A. consistent and independent B. consistent and dependentC. inconsistent D. inconsistent and dependent 2.

2. 4x � 2y � �6 3. 3x � y � 32x � y � 8 x � 2y � 4 3.

To solve each system of equations, which expression could be substituted for x into the first equation?

4. 5x � 2y � 8 A. y � 1 B. y � 1x � y � 1

C. ��25�x � �

58� D. x � 1 4.

5. 4x � 3y � 12 A. 3y � 5 B. y � 35x � 3y � �5

C. �3y � 5 D. ��13�x � �

53� 5.

The first equation of each system is multiplied by 4. By what number would you multiply the second equation in order to eliminate the x variable by adding?

6. 3x � 2y � 4 A. 3 B. �34x � 5y � 28 C. 4 D. �4 6.

7. �6x � 3y � 12 A. 3 B. �38x � 2y � 16 C. 6 D. �6 7.

Solve each system of equations.8. 3x � 2y � 5 A. (1, 1) B. (2, 0)

x � y � 2 C. (0, �2) D. (1, �1) 8.

9. 2x � 3y � 5 A. (3, 4) B. (�2, 3)3x � 2y � 1 C. (1, 1) D. (4, �1) 9.

10. Which system of equations is graphed?

A. y � �13�x � 0 B. y � 3x � 0

x � y � �2 x � y � �2

C. y � 3x � 0 D. y � �13�x � 0 10.

x � y � 2 x � y � 2

11. Which system of inequalities is graphed?A. y � �1 B. y � �1

y � �2x�1 y � �2x � 1C. y � �1 D. y � �1 11.

y � �2x � 1 y � �2x � 1

33

y

xO

(1, 3)

y

xO


Chapter 3 Test, Form 1 (continued)

12. Find the coordinates of the vertices A. (0, 0), (3, 0), (3, 2), (0, 2)of the figure formed by the system y � 0, x � 0, y � 2, and x � 3.

12.Use the system of inequalities y � 0, x � 0, and y � �2x � 4.13. Find the coordinates of the vertices

of the feasible region.

13.

14. Find the maximum value of f(x, y) � 3x � y for the feasible region.A. 2 B. 4 C. 6 D. 12 14.

15. Find the minimum value of f(x, y) � 3x � y for the feasible region.A. 6 B. 4 C. 2 D. 0 15.

For Questions 16–18, use the following information. A college arena sells tickets to students and to the public. Student tickets are $8 each and general public tickets are $32 each. The college reserves at least 5000 tickets for students. The arena seats 18,000.16. Let s represent the number of student tickets and p represent the number

of general public tickets. Which system of inequalities represents the number of tickets sold? A. s � 0, p � 0, s � p � 18,000 B. s � 5000, p � 0, s � p � 18,000C. s � 8, p � 32, s � p � 40 D. s � 0, p � 0, s � p � 18,000 16.

17. How many general public tickets should the college sell to maximize revenue (amount collected)?

A. 18,000 B. 0 C. 13,000 D. 5000 17.

18. What is the maximum revenue?A. $456,000 B. $416,000 C. $40,000 D. $576,000 18.

19. What is the value of y in the solution of the system of equations?

A. 1 B. 2 C. 3 D. 4 19.

20. The 300 students at Holmes School work a total of 5000 hours each month.Each student in group A works 10 hours, each in group B works 15 hours,and each in group C works 20 hours each month. There are twice as many students in group B as in group A. Which equation would not be included in the system used to solve this problem?A. A � 2B B. 10A � 15B � 20C � 5000C. A � B � C � 300 D. B � 2A 20.

Bonus Find the area of the region defined by the system of inequalities x � 0, y � 0, and x � 2y � 4. B:

2x � y � z � 132x � y � 3z � �3x � 2y � 4z � 20

NAME DATE PERIOD

33

A. (0, 0), (3, 0), (3, 2), (0, 2)B. (0, 0), (2, 0), (2, 3), (0, 3)C. (0, 0), (�3, 0), (�3, �2), (0, �2)D. (0, 0), (�2, 0), (�2, �3), (0, �3)

A. (0, 0), (�2, 0), (0, �4)B. (0, 0), (2, 0), (0, 4)C. (0, 0), (4, 0), (0, 2)D. (0, 0), (�4, 0), (0, 2)

Chapter 3 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm


1. The system of equations y � �3x � 5 and y � �3x � 7 has A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.


2. 2x � y � 4 3. 9x � 3y � 15 3.4x � 2y � 6 6x � 2y � 10

To solve each system of equations, which expression could be substituted for y into the first equation?

4. 5x � 3y � 9 A. 12x � 3y B. ��35�x � 3

4x � y � 8 C. 4x � 8 D. 8 � 4x 4.

5. 3x � 6y � 12 A. �12�y � �

52� B. 2x � 5

2x � y � 5 C. 2x � 5 D. 12y � 5 5.

6. The first equation of the system is multiplied by 3. 4x � 3y � 6By what number would you multiply the second 6x � 1y � 10equation to eliminate the x variable by adding?A. �2 B. 2 C. 9 D. �9 6.

7. The first equation of the system is multiplied by 2. 4x � 3y � 6By what number would you multiply the second 6x � 1y � 10equation to eliminate the y variable by adding?A. �2 B. �1 C. 6 D. �3 7.

For Questions 8 and 9, solve each system of equations.8. 5x � 2y � 1 A. (1, �2) B. (1, 2)

y � 1 � 3x C. �0, �12�� D. (�2, 1) 8.

9. 3x � 4y � 12 A. (3, 0) B. (4, 0)2x � 3y � �9 C. (�1, 4) D. (0, 3) 9.

10. Which system of equations is graphed?A. x � y � 5 B. x � y � 5

x � 2y � 2 x � 2y � 2

C. x � y � 5 D. x � y � 5 10.x � 2y � 2 x � 2y � 2

11. Which system of inequalities is graphed?A. 2x � y � 2 B. 2x � y � 2

x � 3y � 6 x � 3y � 6C. 2x � y � 2 D. 2x � y � 2 11.

x � 3y � 6 x � 3y � 6

33

y

xO(4, �1)

y

xO


Chapter 3 Test, Form 2A (continued)

12. Find the coordinates of the vertices A. (0, 0), (3, 0), (3, 2), (0, 2)of the figure formed by the system x � �1, y � �2, and 2x � y � 6.

12.For Questions 13–15, use the system of inequalities x � 2, y � x � �3,and x � y � 5.13. Find the coordinates of the vertices of the feasible region.

A. (2, �1), (2, 3), (4, 1) B. (0, �3), (0, 5), (4, 1)C. (2, 0), (3, 0), (4, 1), (2, 3) D. (0, 0), (0, 5), (3, 0), (4, 1) 13.

14. Find the maximum value of f(x, y) � x � 4y for the feasible region.A. 14 B. 0 C. 8 D. 6 14.

15. Find the minimum value of f(x, y) � x � 4y for the feasible region.A. �2 B. 0 C. �10 D. �4 15.

16. What is the value of z in the solution of the system of equations?

A. 4 B. 1 C. �2 D. �34� 16.

An office building containing 96,000 square feet of space is to be made into apartments. There will be at most 15 one-bedroom units, each with 800 square feet of space. The remaining units, each with 1200 square feet of space, will have two bedrooms. Rent for each one-bedroom unit will be $650 and for each two-bedroom unit will be $900.17. Let x represent the number of one-bedroom apartments and y represent

the number of two-bedroom apartments. Which system of inequalities represents the number of apartments to be built? A. x � 15, y � 0, 650x � 900y � 96,000B. x � 15, y � 0, 800x � 1200y � 96,000C. x � 650, y � 900, 800x � 1200y � 96,000D. x � 15, y � 0, 800x � 1200y � 96,000 17.

18. How many two-bedroom apartments should be built to maximize revenue? A. 70 B. 15 C. 80 D. 120 18.

At a university, 1200 students are enrolled in engineering. There are twice as many in electrical engineering as in mechanical engineering,and three times as many in chemical engineering as in mechanical engineering.19. Which system of equations represents the number of students in each program?

A. c � m � e � 1200, 2m � e, 3m � c B. c � m � e � 1200, 3m � e, 2m � cC. c � m � e � 1200, 2e � m, 3c � m D. c � m � e � 1200, 2m � e, 3m � 2e 19.

20. How many students are enrolled in the mechanical engineering program?A. 200 B. 400 C. 600 D. 1200 20.

Bonus Find the value of x in the solution of the system of

equations x � y � �98� and x � 2y � �

98�. B:

2x � 3y � z � 9x � 2y � z � 4x � 3y � 2z � �3

NAME DATE PERIOD

33

A. (0, 0), (3, 0), (0, 6)B. (�1, 8), (�1, �2), (4, �2)C. (0, 0), (0, 3), (6, 0)D. (�1, �2), (�1, 6), (4, 0)

Chapter 3 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm


1. The system of equations y � 2x � 3 and y � 4x � 3 has A. exactly one solution. B. no solution.C. infinitely many solutions. D. exactly two solutions. 1.


2. x � 2y � 7 3. 2x � 3y � 10 3.3x � 2y � 5 4x � 6y � 20

To solve each system of equations, which expression could be substituted for x into the first equation?

4. 3x � 5y � 14 A. 10 � 4y B. 4y � 10x � 4y � 10 C. �

14�x � �

52� D. ��

14�x � �

52� 4.

5. 2x � 7y � 10 A. �12�x � 15 B. �

12�x � 15

x � 2y � 15 C. 2y � 15 D. 2y � 15 5.

6. The first equation of the system is multiplied by 2.By what number would you multiply the second equation to eliminate the x variable by adding? A. 3 B. �3 C. 2 D. �2 6.

7. The first equation of the system is multiplied by 4.By what number would you multiply the second equation to eliminate the y variable by adding?A. 5 B. �5 C. 2 D. �2 7.

For Questions 8 and 9, solve each system of equations.8. 4x � 3y � 14 A. (1, 1) B. (5, 2)

y � �3x � 4 C. (�4, �10) D. (2, �2) 8.

9. 4x � 3y � 8 A. (�2, 1) B. (2, 0)2x � 5y � �9 C. (0, �83) D. ��

12�, �2� 9.

10. Which system of equations is graphed?A. 2x � y � 1 B. 2x � y � �1

�3x � y � 3 3x � y � 3C. 2x � y � 1 D. 2x � y � �1 10.

3x � y � 3 �3x � y � 3

11. Which system of inequalities is graphed?A. 2x � y � 5 B. 2x � y � 5

3x � 2y � 9 3x � 2y � 9C. 2x � y � �5 D. �2x � y � 5 11.

3x � 2y � 9 3x � 2y � 9

2x � 5y � 168x � 4y � 10

6x � 5y � 214x � 7y � 15

33

y

xO

(�2, 3)

yxO


Chapter 3 Test, Form 2B (continued)

12. Find the coordinates of the vertices of the figure formed by the system x � 0, y � �2, and 2x � y � 4.A. (3, �2), (0, 4), (0, �2) B. (�2, 0), (4, 0), (�2, 3)C. (0, 0), (0, 4), (2, 0) D. (�2, 3), (0, 4), (0, �2) 12.

For Questions 13–15, use the system of inequalities y � 1, y � x � 6,and x � 2y � 6.13. Find the coordinates of the vertices of the feasible region.

A. (�6, 0), (�2, 4), (6, 0) B. (�5, 1), (�2, 4), (4, 1)C. (0, 1), (0, 3), (4, 1) D. (�5, 1), (�2, 4), (0, 3), (0, 1) 13.

14. Find the maximum value of f(x, y) � 2x � y for the feasible region.A. 0 B. 11 C. 9 D. 8 14.

15. Find the minimum value of f(x, y) � 2x � y for the feasible region.A. �10 B. 0 C. �9 D. �4 15.

16. What is the value of z in the solution of the system of equations?

A. �1 B. 12 C. 3 D. �2 16.

Tickets to a golf tournament are sold in advance for $40 each, and on the day of the event for $50 each. For the tournament to occur, at least 2000 of the 8000 tickets must be sold in advance.17. Let a represent the number of advance tickets sold and d represent the

number sold on the day of the tournament. Which system of inequalities represents the number of tickets sold? A. a � 2000, d � 0, a � d � 8000 B. a � 0, d � 0, a � d � 8000C. a � 0, d � 0, a � d � 2000 D. a � 40, d � 50, a � d � 2000 17.

18. How many advance tickets should be sold to maximize revenue?A. 6000 B. 2000 C. 4000 D. 8000 18.

A local gas station sells low-grade (�), mid-grade (m), and premium (p)gasoline. Mid-grade gasoline costs $0.10 per gallon more than low-grade,and premium gasoline costs $0.10 per gallon more than mid-grade gasoline. Five gallons of low-grade gas cost $9.19. Which system of equations represents the cost of each type of gasoline?

A. 5� � m � 9, m � � � 0.10, p � m � 0.10B. 5� � 9, m � � � 0.10, p � m � 0.10C. 5� � 9, m � ��0.10, p � m � 0.10D. 0.10� � 0.10m � 5p � 9, 0.10� � m � 0, 0.10m � p � 0 19.

20. What is the cost of one gallon of premium gasoline?A. $1.80 B. $1.90 C. $2.00 D. $2.10 20.

Bonus Solve the system of equations. B:a � b � 6 c � d � 4 f � a � 2b � c � 5 d � f � 3

2x � 3y � z � 124x � y � z � �3�2x � 2y � z � 3

NAME DATE PERIOD

33

Chapter 3 Test, Form 2C



1. 3x � 2y � 62x � y � 4

2. 3x � y � 13y � 9x � 6

Describe each system of equations as consistent andindependent, consistent and dependent, or inconsistent.

3. y � 2x � 5 4. 2x � y � 5y � �3x � 4 6x � 3y � 15


5. 3x � 7y � 19 6. x � 3y � 12x � y � 5 5x � y � 4


7. 3x � 2y � 4 8. 4x � y � 102x � 3y � 7 5x � 2y � 6


9. 4x � 3y � 92x � y � 5

10. y � �32�x � 2

2y � x � 4

Find the coordinates of the vertices of the figure formed by each system of inequalities.

11. y � �3 12. x � 3y � 2x � 1 y � 2x � 4x � 2 x � y � �2

3y � �2x � 12

NAME DATE PERIOD

SCORE 33

Ass

essm

ent1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

y

xO

y

xO

y

xO

y

xO


Chapter 3 Test, Form 2C (continued)

Use the system of inequalities x � �2, x � y � 7, and y � 2x � 1.

13. Find the coordinates of the vertices of the feasible region.

14. Find the maximum and minimum values of the function f(x, y) � 3x � y for the feasible region.

The area of a parking lot is 600 square meters. A car requires 6 square meters and a bus requires 30 square meters of space. The lot can handle a maximum of 60 vehicles.

15. Let c represent the number of cars and b represent the number of buses. Write a system of inequalities to represent the number of vehicles that can be parked.

16. If a car costs $3 and a bus costs $8 to park in the lot, determine the number of each vehicle to maximize the amount collected.


17. x � 2y � 3z � 5x � y � 2z � �3x � y � z � 2

18. 3x � y � 2z � 12x � y � z � �3x � y � 4z � �3

A printing company sells small packages of personalized stationery for $7 each, medium packages for $12 each,and large packages for $15 each. Yesterday, the company sold 9 packages of stationery, collecting a total of $86.Three times as many medium packages were sold as large packages.

19. Let s represent the number of small packages, m the number of medium packages, and � the number of large packages. Write a system of three equations that represents the number of packages sold.

20. Find the number of each size package sold.

Bonus Find the perimeter of the region defined by the system of inequalities:�2 � x � 5�4 � y � �1 B:

NAME DATE PERIOD

33

13.

14.

15.

16.

17.

18.

19.

20.

Chapter 3 Test, Form 2D



1. x � y � 52y � x � 2

2. y � �23�x � 1

2x � y � �1


3. 3x � 4y � 5 4. 2x � 7y � 146x � 8y � �5 x � 3y � 6


5. 4x � y � 10 6. x � y � 6y � 3x � 6 3x � 2y � �22


7. 5x � 2y � 1 8. 5x � 3y � 162x � 3y � 7 2x � 7y � �10


9. 2x � 3y � �33y � �2x � 6

10. x � 3y � 6

y � �32�x � 2

Find the coordinates of the vertices of the figure formed by the solution of each system of inequalities.

11. x � �3 12. y � 3y � �2 4y � 3x � 122x � y � �2 x � y � �4

y � 2x � 1

NAME DATE PERIOD

SCORE 33

Ass

essm

ent1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

y

xO

y

xO

y

xO

y

xO


Chapter 3 Test, Form 2D (continued)

For Questions 13 and 14, use the system of inequalities y � 7, x � y � 2, y � 2x � 5.

13. Find the coordinates of the vertices of the feasible region. 13.

14. Find the maximum and minimum values of the function 14.f(x, y) � 3x � y for the feasible region.

Kristin earns $7 per hour at a video store and $10 per hour at a landscaping company. She must work at least 4 hours per week at the video store, but the total number of hours she works at both jobs cannot be greater than 15.

15. Let v represent the number of hours working at the video 15.store and � represent the number of hours working at the landscaping company. Write a system of inequalities to represent the number of hours worked in one week.

16. Determine Kristin’s maximum weekly earnings (before 16.deductions).


17. 2x � y � z � 4 18. 3x � y � z � 12 17.3x � y � 4z � 11 2x � 3y � z � 5x � y � 5z � 20 x � 2y � z � 9 18.

The price of a sweatshirt at a local shop is twice the price of a pair of shorts. The price of a T-shirt at the shop is $4 less than the price of a pair of shorts. Brad purchased 3 sweatshirts, 2 pairs of shorts, and 5 T-shirts for a total cost of $136.

19. Let w represent the price of one sweatshirt, t represent the 19.price of one T-shirt, and h represent the price of one pair of shorts. Write a system of three equations that represents the prices of the clothing.

20. Find the cost of one sweatshirt. 20.

Bonus Solve the system of equations. B:

�23�x � �

12�y � 4

��16�x � �

18�y � �3

NAME DATE PERIOD

33

Chapter 3 Test, Form 3



1. �13�x � �

12�y � 1

�12�x � �

14�y � ��

12�

2. x � 0.5y � 0.52.5 � 1.5x � y


3. 2x � �23�y � �

130� 4. �1

34�

x � �114�

y � �12�

9x � 3y � 15 6x � 2( y � 5)


5. �12�x � �

13�y � 1 6. �

45�a � b � �

34� � 0

�23�y � x � 6 2a � �

54� � �

52�b


7. �23�x � 4y � �1 8. 0.2c � 1.5d � �2.7

�12�x � 3y � ��

94�

1.2c � 0.5d � 2.8


9. x � y � 5x � y � �3x � 2y � 6x � 2y � �2

10. � x � � 1� y � 1 � � 3x � 3y � �6

Find the coordinates of the vertices of the figure formed by the solution of each system of inequalities.

11. x � 2 12. �2 � x � 5�4 � y � 3

�12�x � �

12�y � 1

x � y � �32x � y � � 2y � �4

NAME DATE PERIOD

SCORE 33

Ass

essm

ent1.

2.

3.

4.

5.

6.

7.

8.9.

10.

11.

12.

y

xO

y

xO

y

xO

y

xO


Chapter 3 Test, Form 3 (continued)

For Questions 13 and 14, use the system of inequalities

x � �3, x � y � �4, x � y � 1, and �13�x � �

12�y � 2.

13. Find the coordinates of the vertices of the feasible region. 13.

14. Find the maximum and minimum values of the function

f(x, y) � 3x � �12�y for the feasible region. 14.

A dog food manufacturer wants to advertise its products.A magazine charges $60 per ad and requires the purchase of at least three ads. A radio station charges $150 per commercial minute and requires the purchase of at least four minutes. Each magazine ad reaches 12,000 people while each commercial minute reaches 16,000 people. At most $900 can be spent on advertising.

15. Let a represent the number of magazine ads and m represent 15.the number of commercial minutes. Write a system of inequalities that represents the advertising plan for the company.

16. How many ads and commercial minutes should be purchased 16.to reach the most people? How many people would this be?


17. 4x � 6y � z � 1 18. 10a � b � �13�c � �6

17.3x � �

12�y � �

23�z � �

92� 2a � 3b � �

17�c � �

352�

5x � 3y � 2z � ��121� ��

52�a � �

14�c � �

149� 18.

An electronic repair shop offers three types of service:on-site, at-store, and by-mail. On-site service costs 3 times as much as at-store service. By-mail service costs $10 less than at-store service. Last week, the shop completed 15 services on-site, 40 services at-store, and 5 services by mail for total sales of $2650.

19. Let s represent the cost of one on-site repair, a represent 19.the cost of one at-store repair, and m represent the cost of one by-mail repair service. Write a system of three equations that represents the cost of each repair service.

20. Determine the cost of one on-site repair service. 20.

Bonus Solve the system of equations. B:

�3x� � �

4y� � �

52�

�6x� � �

1y� � �

12�

NAME DATE PERIOD

33

Chapter 3 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solution in more thanone way or investigate beyond the requirements of the problem.

1. The square of Janet’s age is 400 more than the square of the sumof Kim’s and Sue’s ages. Kim’s and Sue’s ages total 10 less thanJanet’s age. Find the square of the sum of the ages of Janet, Kim,and Sue. Explain your reasoning.

2. A feasible region has vertices at (0, 8), (6, 0), and (0, 0).a. Write a system of inequalities whose graph forms this feasible

region.b. Explain how to find the maximum and minimum values of

f(x, y) � x � y for the region.

3. Explain what the algebraic solution of a system of two linearequations is and what that means in terms of the graphs of theequations of the system.

4. When describing a system of two linear equations, a studentindicated that the system was inconsistent and dependent.Discuss the meaning of the student’s description. Then assess thestudent’s understanding of these terms.

5. A business owner asks the company finance manager to developa formula, or function, for the cost incurred in the production oftheir products, and another function for the revenue (moneycollected) that the company earns when the products are sold. Inpreparing the report for the owner, the finance manager preparesa graph which shows both the cost function and the revenuefunction. The business owner, on seeing the graph of two parallellines, makes a major business decision. What might that decisionbe and why would it be made?

6. Explain what it means for a system of two linear inequalities tohave no solution. Sketch a graph of such a system and write thesystem of inequalities represented by your graph.

NAME DATE PERIOD

SCORE 33

Ass

essm

ent


Chapter 3 Vocabulary Test/Review

Choose from the terms above to complete each sentence.

1. A system of equations with no solutions is called a(n)

.

2. A(n) is a system of equations withan infinite number of solutions.

3. (3, �2, 7) is an example of a(n) .

4. If your first step in solving a system of equations is to solve one of the equations for one variable in terms of the others, you are using the

.

5. In a linear programming problem, the inequalities are called

.

6. A set of two or more inequalities that are considered together is called

a(n) .

7. If you are solving a system of equations and one of your steps is to add

the equations, you are using the .

8. If you are solving a system of equations by graphing and your graph shows two intersecting lines, the system can be described both as a(n)

and as a(n) .

9. Graphing, the substitution method, and the elimination method are all

methods for solving a(n) .

10. The process of finding the maximum or minimum values of a function

defined by a feasible region is called .

In your own words—Define each term.

11. feasible region

12. unbounded region

bounded regionconsistent systemconstraintsdependent system

elimination methodfeasible regioninconsistent systemindependent system

linear programmingordered triplesubstitution methodsystem of equations

system of inequalitiesunbounded regionvertices

NAME DATE PERIOD

SCORE 33

Chapter 3 Quiz (Lessons 3–1 and 3–2)

33


1. Solve the system x � y � 4 and x � 2y � 1 by graphing.

2. Graph the system y � x � 2 and 2y � 2x � 4. Describe it as consistent and independent, consistent and dependent,or inconsistent.

3. Solve the system 3y � 2x, y � �23�x � 2, by using substitution.

4. Solve the system of equations by 4x � 5y � �2, and 3x � 2y � �13 using elimination.

5. Standardized Test Practice Compare the quantity in Column A and the quantity in Column B. Then determine whether:A. the quantity in Column A is greater,B. the quantity in Column B is greater,C. the two quantities are equal, or D. the relationship cannot be determined from the

information given.2x � 3y � 25x � y � 22

Column A Column B

NAME DATE PERIOD

SCORE

Chapter 3 Quiz (Lesson 3–3)


1. x � y � �3 2. y � �13�x � 1

2x � y � 6y � �

13�x � 2

Find the coordinates of the vertices of the figure formed by each system of inequalities.

3. x � 0 4. y � �2y � 0 x � 32x � y � 4 x � y � 2

y � 2x � 4

NAME DATE PERIOD

SCORE 33

Ass

essm

ent1.

2.

3.

4.

5.

y

xO

y

xO

1.

2.

3.

4.

y

xO

y

xO

yx


1. A feasible region has vertices at (�2, 3), (1, 6), (1, �1), and (�3, �2). Find the maximum and minimum of the function f(x, y) � �x � 2y over this region.

2. Graph the system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region.4y � x � 12�4y � 3x � 45x � 4y � 4f(x, y) � x � 5y

A clothing company makes jackets and pants. Each jacket requires 1 hour of cutting and 4 hours of sewing. Each pair of pants requires 2 hours of cutting and 2 hours of sewing.The total time per day available for cutting is 20 hours and for sewing is 32 hours.

3. Let j represent the number of jackets and let p represent the number of pairs of pants. Write a system of inequalities to represent the number of items that can be produced.

4. If the profit on a jacket is $14 and the profit on a pair of pants is $8, determine the number of each that should be made each day to maximize profit. What is the maximum profit?


Solve each system of equations.1. 4x � 6y � 3z � 20 2. 3x � y � 2z � 1 1.

x � 5y � z � �15 �2x � �4�7x � y � 2z � 1 x � 3y � 11 2.

3. x � 2y � z � �73.3x � 3y � z � 13

2x � 5y � 2z � 0

During one month, a rental car agency rented a total of 155 cars, vans, and trucks. Nine times as many cars were rented as vans, and three times as many vans were rented as trucks.

4. Let x represent the number of cars, let y represent the 4.number of vans, and let z represent the number of trucks.Write a system of three equations that represents the number of vehicles rented. 5.

5. Find the number of each type of vehicle rented.

NAME DATE PERIOD

SCORE


33

NAME DATE PERIOD

SCORE

33

1.

2.

3.

4.

y

xO

Chapter 3 Mid-Chapter Test (Lessons 3–1 through 3–3)


Write the letter for the correct answer in the blank at the right of each question.

1. Choose the correct description 3x � y � 5of the system of equations. 6x � 2y � 5 A. consistent and independentB. consistent and dependentC. inconsistentD. inconsistent and dependent 1.

2. Solve 3x � 2y � 7 and x � 4y � �21 by using substitution.

A. (3, �1) B. ��73�, �

72�� C. (�1, 5) D. (1, 5) 2.


3. 2x � 5y � 18 A. (�1, 4) B. �9, �158��

3x � 2y � �11 C. (1, 4) D. (�3, 1) 3.

4. 3x � 5y � 14 A. (3, �1) B. (8, 2)2x � 3y � 3 C. (0, 1) D. (6, �3) 4.

5. Solve 3x � y � �4 and x � 2y � �6 by graphing.

6. Solve 2x � y � �1 and x � y � 4 by graphing.

7. Classify the system as consistent 3x � y � 5and independent, consistent and 6x � 2( y � 5)dependent, or inconsistent.

8. Solve the system of inequalities 4x � y � 4by graphing. 3y � �x � 6

9. Find the coordinates of the x � �2vertices of the figure formed y � 6by the system of inequalities. y � ��

32�x � 6

x � 2y � 4

Part II

Part I

NAME DATE PERIOD

SCORE 33

Ass

essm

ent

5.

6.

7.

8.

9.

y

xO

y

xO

y

xO


Chapter 3 Cumulative Review (Chapters 1–3)

1. Simplify �13�(6x � 21) � 4(x � 5). (Lesson 1-2) 1.

2. Solve 7 � 2(m � 3) � 4 � m. (Lesson 1-3) 2.

3. Solve the inequality � 3 � 2x � � 7. Then graph the solution 3.set. (Lesson 1-6)

4. Find f(2a) if f(x) � � x3 � 2x � 5. (Lesson 2-1) 4.

For Questions 5 and 6, state whether each equation or function is linear. If not, explain. (Lesson 2-2)

5. f(x) � �1x� � 5 6. y � x2 � 2

7. Write an equation for the line that passes through (2, �3) 7.and is parallel to the line whose equation is y � �4x � 3.(Lesson 2-4)

8. Evaluate f��18�� if f(x) � 5 � 3x. (Lesson 2-6) 8.

9. Describe the system of equations as consistent and 9.independent, consistent and dependent, or inconsistent.2x � 3y � 114x � 6y � 22 (Lesson 3-1)

10. Solve the system of equations y � 2x � 5 10.by using substitution. 4x � 5y � �1(Lesson 3-2)

11. Solve the system of equations y � 3x � 5 11.by using elimination. 4x � 9y � �22(Lesson 3-2)

For Questions 12 and 13, use the system of inequalities x � 1, y � �2, and x � y � 4.

12. Find the coordinates of the vertices of the figure formed by 12.the system of inequalities. (Lesson 3-3)

13. Find the maximum and minimum values of the function 13.f(x, y) � y � 3x for the feasible region. (Lesson 3-4)

14. Solve the system of equations. 2x � y � 3z � 9 14.(Lesson 3-5) x � 2y � z � �8

x � 3y � 2z � 11

�1�2�3�4�5�6 0 1 2

NAME DATE PERIOD

33

5.

6.

Standardized Test Practice (Chapters 1–3)


1. A number is 8 more than the ratio of q and 2. What is three-fourths of the number in terms of q?

A. 6 B. 5 � �38q� C. �

38q� D. 6 � �

38q� 1.

2. Carlisle’s salary was raised from $19,000 to $23,750. Find the percent of increase.E. 20% F. 25% G. 30% H. 47.5% 2.

3. Which of the following fractions is more than one-fourth but less than three-eights?

A. �27� B. �

28� C. �

34� D. �

12� 3.

4. If the area of a square is 49, the difference between the length of a diagonal and the length of a side is _____.E. 7�2� � 7 F. 0 G. �2� H. 1 � �2� 4.

5. Which is not equal to �2�?

A. 2�12�

B. �4

4� C. ��2��2 D. �6

8� 5.

6. Find the area of the triangle at the right.E. 32 F. 48G. 60 H. 120 6.

7. If m is an integer greater than 5, then which of the following must represent an odd integer?A. m2 B. m � 1 C. 2m � 3 D. m � 2 7.

8. Desiree and Mario together have $30. Mario and Scott together have $26. Desiree and Scott together have $34. What is the least amount of money any of them has?E. $11 F. $15 G. $19 H. $26 8.

9. If x, y, and z are negative integers, which of the following must be true?A. x(y � z) � 0 B. xyz � 0C. x � (y � z) � 0 D. x � y � z 9.

10. If �r �r

2� � �

43�, then �2

r� � _____.

E. �14� F. �

23� G. 1 H. 3 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

33

Ass

essm

ent

10

12

10

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.


Standardized Test Practice (continued)

11. If the operation � is defined by the equation 11. 12.a � b � ab � b2, what is the value of n in theequation 4 � n � 4?

12. In a survey of high school sophomores,60 students named Science as their favorite subject. If 70% of the students surveyed had a favorite subject other than Science, how many sophomores were surveyed?

13. If figure RSTV is 13. 14.a parallelogram,what is the value of x?

14. What number is in the hundreds place in the product of 540 and 9?

Column A Column B

15. p � r3 � 4, q � r3 � 4 15.

16. 16.

17. 17.

18. x � 0 18.

19. 2x � 1 � 13 and 5y � 14 19.

5y2x

DCBA

�xx

1

6

0��

xx

6

2�

DCBA

DCBA0.09�0.09�

DCBA0.4 2c�25

� of 6c

qp

DCBA

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

33

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

R U

S T3y˚

5x˚

y˚

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 150–151 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

33

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 14–18, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 14 16 18

12

13

15 17

Select the best answer from the choices given and fill in the corresponding oval.

19 21

20 22 DCBADCBA

DCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

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0 0 0

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.

99 9 987654321

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0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

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0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 3-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g S

yste

ms

of

Eq

uat

ion

s by

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1

©G

lenc

oe/M

cGra

w-H

ill11

9G

lenc

oe A

lgeb

ra 2

Lesson 3-1

Gra

ph

Sys

tem

s o

f Eq

uat

ion

sA

sys

tem

of

equ

atio

ns

is a

set

of

two

or m

ore

equ

atio

ns

con

tain

ing

the

sam

e va

riab

les.

You

can

sol

ve a

sys

tem

of

lin

ear

equ

atio

ns

bygr

aph

ing

the

equ

atio

ns

on t

he

sam

e co

ordi

nat

e pl

ane.

If t

he

lin

es i

nte

rsec

t,th

e so

luti

on i

sth

at i

nte

rsec

tion

poi

nt.

Sol

ve t

he

syst

em o

f eq

uat

ion

s b

y gr

aph

ing.

x �

2y�

4x

�y

��

2W

rite

eac

h e

quat

ion

in

slo

pe-i

nte

rcep

t fo

rm.

x �

2y�

4→

y�

�2

x�

y�

�2

→y

��

x�

2

Th

e gr

aph

s ap

pear

to

inte

rsec

t at

(0,

�2)

.

CH

ECK

Su

bsti

tute

th

e co

ordi

nat

es i

nto

eac

h e

quat

ion

.x

�2y

�4

x�

y�

�2

0 �

2(�

2) �

40

�(�

2)�

�2

4 �

4✓

�2

��

2✓

Th

e so

luti

on o

f th

e sy

stem

is

(0,�

2).

Sol

ve e

ach

sys

tem

of

equ

atio

ns

by

grap

hin

g.

1.y

��

�1

2.y

�2x

�2

3.y

��

�3

y�

�4

(6,�

1)y

��

x�

4(2

,2)

y�

(4,1

)

4.3x

�y

�0

5.2x

��

�7

6.�

y�

2

x�

y�

�2

(1,3

)�

y�

1(�

4,3)

2x�

y�

�1

(�2,

�3)

( –2,

–3)

x

y

O

( –4,

3)

x

y

O

( 1, 3

)

x

y

O

x � 2

x � 2y � 3

( 4, 1

)

x

y

O

( 2, 2

)

x

y

O

( 6, –

1)

x

y

O

x � 4x � 2

x � 2x � 3

x

y

O

( 0, –

2)

x � 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill12

0G

lenc

oe A

lgeb

ra 2

Cla

ssif

y Sy

stem

s o

f Eq

uat

ion

sT

he

foll

owin

g ch

art

sum

mar

izes

th

e po

ssib

ilit

ies

for

grap

hs

of t

wo

lin

ear

equ

atio

ns

in t

wo

vari

able

s.

Gra

ph

s o

f E

qu

atio

ns

Slo

pes

of

Lin

esC

lass

ific

atio

n o

f S

yste

mN

um

ber

of

So

luti

on

s

Line

s in

ters

ect

Diff

eren

t sl

opes

Con

sist

ent

and

inde

pend

ent

One

Line

s co

inci

de (

sam

e lin

e)S

ame

slop

e, s

ame

Con

sist

ent

and

depe

nden

tIn

finite

ly m

any

y-in

terc

ept

Line

s ar

e pa

ralle

lS

ame

slop

e, d

iffer

ent

Inco

nsis

tent

Non

e y-

inte

rcep

ts

Gra

ph

th

e sy

stem

of

equ

atio

ns

x�

3y�

6an

d d

escr

ibe

it a

s co

nsi

sten

t a

nd

in

dep

end

ent,

2x�

y�

�3

con

sist

ent

an

d d

epen

den

t,or

in

con

sist

ent.

Wri

te e

ach

equ

atio

n i

n s

lope

-in

terc

ept

form

.

x�

3y�

6→

y�

x�

2

2x�

y�

�3

→y

�2x

�3

Th

e gr

aph

s in

ters

ect

at (

�3,

�3)

.Sin

ce t

her

e is

on

e so

luti

on,t

he

syst

em i

s co

nsi

sten

t an

d in

depe

nde

nt.

Gra

ph

th

e sy

stem

of

equ

atio

ns

and

des

crib

e it

as

con

sist

ent

an

d i

nd

epen

den

t,co

nsi

sten

t a

nd

dep

end

ent,

or i

nco

nsi

sten

t.

1.3x

�y

��

2 2.

x�

2y�

5 3.

2x�

3y�

06x

�2y

�10

3x�

15 �

�6y

co

nsi

sten

t4x

�6y

�3

inco

nsi

sten

tan

d d

epen

den

tin

con

sist

ent

4.2x

�y

�3

5.4x

�y

��

26.

3x�

y�

2

x�

2y�

4co

nsi

sten

t 2x

��

�1

con

sist

ent

x�

y�

6 co

nsi

sten

t

and

ind

epen

den

t an

d d

epen

den

t an

d in

dep

end

ent

x

y

O

x

y

Ox

y

O

y � 2

x

y

Ox

y

Ox

y

O

1 � 3

x

y O

( –3,

–3)

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g S

yste

ms

of

Eq

uat

ion

s by

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1

Exer

cises

Exer

cises

Exam

ple

Exam

ple


An

swer

s


Skil

ls P

ract

ice

So

lvin

g S

yste

ms

of

Eq

uat

ion

s B

y G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1

©G

lenc

oe/M

cGra

w-H

ill12

1G

lenc

oe A

lgeb

ra 2

Lesson 3-1

Sol

ve e

ach

sys

tem

of

equ

atio

ns

by

grap

hin

g.

1.x

�2

2.y

��

3x�

63.

y�

4 �

3x

y�

0(2

,0)

y�

2x�

4(2

,0)

y�

�x

�1

(2,�

2)

4.y

�4

�x

5.y

��

2x�

26.

y�

x

y�

x�

2(3

,1)

y�

x�

5(3

,�4)

y�

�3x

�4

(1,1

)

7.x

�y

�3

8.x

�y

�4

9.3x

�2y

�4

x�

y�

1(2

,1)

2x�

5y�

8(4

,0)

2x�

y�

1(�

2,�

5)

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s an

d d

escr

ibe

it a

s co

nsi

sten

t a

nd

ind

epen

den

t,co

nsi

sten

t a

nd

dep

end

ent,

or i

nco

nsi

sten

t.

10.y

��

3x11

.y�

x�

512

.2x

�5y

�10

y�

�3x

�2

�2x

�2y

��

103x

�y

�15

inco

nsi

sten

tco

nsi

sten

t an

d

con

sist

ent

and

d

epen

den

tin

dep

end

ent

( 5, 0

)

2

2x

y

O

x

y

O

x

y

O

( –2,

–5)

x

y

O( 4

, 0)

x

y

O( 2

, 1)

x

y

O

( 1, 1

)

x

y

O( 3

, –4)

x

y

O

( 3, 1

)

x

y

O

1 � 3

( 2, –

2)x

y

O( 2

, 0)

x

y

O( 2

, 0)

x

y

O

1 � 2

©G

lenc

oe/M

cGra

w-H

ill12

2G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

sys

tem

of

equ

atio

ns

by

grap

hin

g.

1.x

�2y

�0

2.x

�2y

�4

3.2x

�y

�3

y�

2x�

3(2

,1)

2x�

3y�

1(2

,1)

y�

x�

(3,�

3)

4.y

�x

�3

5.2x

�y

�6

6.5x

�y

�4

y�

1(�

2,1)

x�

2y�

�2

(2,�

2)�

2x�

6y�

4(1

,1)

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s an

d d

escr

ibe

it a

s co

nsi

sten

t a

nd

ind

epen

den

t,co

nsi

sten

ta

nd

dep

end

ent,

or i

nco

nsi

sten

t.

7.2x

�y

�4

8.y

��

x�

29.

2y�

8 �

x

x�

y�

2x

�y

��

4y

�x

�4

con

sist

ent

and

ind

ep.

inco

nsi

sten

tco

nsi

sten

t an

d d

ep.

SOFT

WA

RE

For

Exe

rcis

es 1

0–12

,use

th

e fo

llow

ing

info

rmat

ion

.L

ocat

ion

Map

pin

g n

eeds

new

sof

twar

e.S

oftw

are

A c

osts

$13

,000

plu

s $5

00 p

er a

ddit

ion

alsi

te l

icen

se.S

oftw

are

B c

osts

$25

00 p

lus

$120

0 pe

r ad

diti

onal

sit

e li

cen

se.

10.W

rite

tw

o eq

uat

ion

s th

at r

epre

sen

t th

e co

st o

f ea

ch s

oftw

are.

A:

y�

13,0

00 �

500x

,B

:y

�25

00 �

1200

x

11.G

raph

th

e eq

uat

ion

s.E

stim

ate

the

brea

k-ev

en

poin

t of

th

e so

ftw

are

cost

s.15

ad

dit

ion

al li

cen

ses

12.I

f L

ocat

ion

Map

pin

g pl

ans

to b

uy

10 a

ddit

ion

al

site

lic

ense

s,w

hic

h s

oftw

are

wil

l co

st l

ess?

B

( 15,

20,

500)

A

B

So

ftw

are

Co

sts

Ad

dit

ion

al L

icen

ses

Total Cost ($)

40

812

1416

1820

26

10

24,0

00

20,0

00

16,0

00

12,0

00

8,00

0

4,00

0

x

y

O

x

y

O( 2

, 0)

x

y

O

1 � 2

y

( 1, 1

)x

O( 2

, –2)

x

y O( –

2, 1

)

x

y

O

( 3, –

3)

x

y

O( 2

, 1) x

y

O

( 2, 1

)

x

y

O

9 � 21 � 2

Pra

ctic

e (

Ave

rag

e)

So

lvin

g S

yste

ms

of

Eq

uat

ion

s B

y G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Sys

tem

s o

f E

qu

atio

ns

by G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1

©G

lenc

oe/M

cGra

w-H

ill12

3G

lenc

oe A

lgeb

ra 2

Lesson 3-1

Pre-

Act

ivit

yH

ow c

an a

sys

tem

of

equ

atio

ns

be

use

d t

o p

red

ict

sale

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-1

at

the

top

of p

age

110

in y

our

text

book

.

•W

hic

h a

re g

row

ing

fast

er,i

n-s

tore

sal

es o

r on

lin

e sa

les?

on

line

sale

s

•In

wh

at y

ear

wil

l th

e in

-sto

re a

nd

onli

ne

sale

s be

th

e sa

me?

2005

Rea

din

g t

he

Less

on

1.T

he

Stu

dy T

ip o

n p

age

110

of y

our

text

book

say

s th

at w

hen

you

sol

ve a

sys

tem

of

equ

atio

ns

by g

raph

ing

and

fin

d a

poin

t of

in

ters

ecti

on o

f th

e tw

o li

nes

,you

mu

st a

lway

sch

eck

the

orde

red

pair

in

bot

hof

th

e or

igin

al e

quat

ion

s.W

hy

is i

t n

ot g

ood

enou

gh t

och

eck

the

orde

red

pair

in

just

on

e of

th

e eq

uat

ion

s?S

amp

le a

nsw

er:T

o b

e a

solu

tio

n o

f th

e sy

stem

,th

e o

rder

ed p

air

mu

stm

ake

bo

th o

f th

e eq

uat

ion

s tr

ue.

2.U

nde

r ea

ch s

yste

m g

raph

ed b

elow

,wri

te a

ll o

f th

e fo

llow

ing

wor

ds t

hat

app

ly:c

onsi

sten

t,in

con

sist

ent,

dep

end

ent,

and

ind

epen

den

t.

inco

nsi

sten

tco

nsi

sten

t;co

nsi

sten

t;d

epen

den

tin

dep

end

ent

Hel

pin

g Y

ou

Rem

emb

er

3.L

ook

up

the

wor

ds c

onsi

sten

tan

d in

con

sist

ent

in a

dic

tion

ary.

How

can

th

e m

ean

ing

ofth

ese

wor

ds h

elp

you

dis

tin

guis

h b

etw

een

con

sist

ent

and

inco

nsi

sten

t sy

stem

s of

equ

atio

ns?

Sam

ple

an

swer

:O

ne

mea

nin

g o

f co

nsi

sten

tis

“b

ein

g in

ag

reem

ent,”

or

“co

mp

atib

le,”

wh

ile o

ne

mea

nin

g o

f in

con

sist

ent

is “

no

t b

ein

g in

agre

emen

t”o

r “i

nco

mp

atib

le.”

Wh

en a

sys

tem

is c

on

sist

ent,

the

equ

atio

ns

are

com

pat

ible

bec

ause

bo

th c

an b

e tr

ue

at t

he

sam

e ti

me

(fo

rth

e sa

me

valu

es o

f x

and

y).

Wh

en a

sys

tem

is in

con

sist

ent,

the

equ

atio

ns

are

inco

mp

atib

le b

ecau

se t

hey

can

nev

er b

e tr

ue

at t

he

sam

e ti

me.

x

y

Ox

y

Ox

y

O

©G

lenc

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w-H

ill12

4G

lenc

oe A

lgeb

ra 2

Inve

stm

ents

The

fol

low

ing

grap

h sh

ows

the

valu

e of

tw

o di

ffer

ent

inve

stm

ents

ove

r ti

me.

Lin

e A

repr

esen

ts a

n i

nit

ial

inve

stm

ent

of $

30,0

00 w

ith

a b

ank

payi

ng

pass

book

sav

ings

inte

rest

.Lin

e B

rep

rese

nts

an in

itia

l inv

estm

ent

of $

5,00

0 in

a

prof

itab

le m

utua

l fun

d w

ith

divi

dend

s re

inve

sted

and

capi

tal g

ains

acc

epte

d in

sh

ares

.By

deri

ving

the

line

ar e

quat

ion

y�

mx

�b

for

A a

nd

B,y

ou c

an p

redi

ct

the

valu

e of

th

ese

inve

stm

ents

for

yea

rs t

o co

me.

1.T

he

y-in

terc

ept,

b,is

th

e in

itia

l in

vest

men

t.F

ind

bfo

r ea

ch o

f th

e fo

llow

ing.

a.li

ne

A30

,000

b.

lin

e B

5000

2.T

he

slop

e of

th

e li

ne,

m,i

s th

e ra

te o

f re

turn

.Fin

d m

for

each

of

the

foll

owin

g.a.

lin

e A

0.58

8b

.li

ne

B2.

73

3.W

hat

are

th

e eq

uat

ion

s of

eac

h o

f th

e fo

llow

ing

lin

es?

a.li

ne

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hat

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he

mu

tual

fu

nd

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r 11

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rs o

f in

vest

men

t?$3

5,03

0

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hat

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l be

th

e va

lue

of t

he

ban

k ac

cou

nt

afte

r 11

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rs o

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vest

men

t?$3

6,46

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6.W

hen

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l th

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utu

al f

un

d an

d th

e ba

nk

acco

un

t h

ave

equ

al v

alu

e?af

ter

11.6

7 ye

ars

of

inve

stm

ent

7.W

hich

inve

stm

ent

has

the

grea

test

pay

off a

fter

11

year

s of

inve

stm

ent?

the

mu

tual

fu

nd

A B25 203035 10 5 0

15

12

34

56

78

9

Amount Invested (thousands)

Year

s In

vest

ed

x

y

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-1

3-1


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g S

yste

ms

of

Eq

uat

ion

s A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-2

3-2

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lenc

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cGra

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ill12

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lenc

oe A

lgeb

ra 2

Lesson 3-2

Sub

stit

uti

on

To

solv

e a

syst

em o

f li

nea

r eq

uat

ion

s by

su

bst

itu

tion

,fir

st s

olve

for

on

eva

riab

le i

n t

erm

s of

th

e ot

her

in

on

e of

th

e eq

uat

ion

s.T

hen

su

bsti

tute

th

is e

xpre

ssio

n i

nto

the

oth

er e

quat

ion

an

d si

mpl

ify.

Use

su

bst

itu

tion

to

solv

e th

e sy

stem

of

equ

atio

ns.

2x�

y�

9x

�3y

��

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olve

th

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rst

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or y

in t

erm

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x.

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irst

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act

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oth

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ultip

ly b

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�1.

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pres

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2x

�9

for

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to t

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r x.

x�

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trib

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plify

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dd 2

7 to

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ivid

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ide

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.

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3 fo

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ith

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irst

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lace

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.

6 �

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ify.

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1.

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stem

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(3,�

3).

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ve e

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ear

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ng

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142x

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83

�x

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no

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9)

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2 � 34 � 3

1 � 4

1 � 2

Exam

ple

Exam

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Exer

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Exer

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ill12

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lenc

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lgeb

ra 2

Elim

inat

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To

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nea

r eq

uat

ion

s by

eli

min

atio

n,a

dd o

r su

btra

ct t

he

equ

atio

ns

to e

lim

inat

e on

e of

th

e va

riab

les.

You

may

fir

st n

eed

to m

ult

iply

on

e or

bot

h o

f th

eeq

uat

ion

s by

a c

onst

ant

so t

hat

on

e of

th

e va

riab

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has

th

e sa

me

(or

oppo

site

) co

effi

cien

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one

equ

atio

n a

s it

has

in

th

e ot

her

.

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th

e el

imin

atio

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eth

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o so

lve

the

syst

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f eq

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s.2x

�4y

��

263x

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24

Stu

dy G

uid

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onti

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ms

of

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s A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

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ER

IOD

____

_

3-2

3-2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Mu

ltip

ly t

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seco

nd

equ

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en s

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equ

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to e

lim

inat

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.2x

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��

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�y

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24M

ultip

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96�

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x�

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Rep

lace

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7) �

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26�

4y�

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y�

3T

he

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ltip

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3 an

d th

e se

con

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2.T

hen

add

th

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s to

elim

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5x�

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��

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2

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th

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imin

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n m

eth

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o so

lve

the

syst

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5x�

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cises

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ve e

ach

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ng

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.

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2.x

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3.3x

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104.

3x�

y�

123x

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25x

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no

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610

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11.�

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64m

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4,6)

5 � 2

3 � 2y � 2



Skil

ls P

ract

ice

So

lvin

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yste

ms

of

Eq

uat

ion

s A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-2

3-2

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ill12

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lgeb

ra 2

Lesson 3-2

Sol

ve e

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sys

tem

of

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by

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ng

sub

stit

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1.m

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w�

z�

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,12)

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5.2b

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46.

x�

y�

12r

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)

Sol

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by

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elim

inat

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.

7.2p

�q

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8.2j

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911

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ach

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ub

stit

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13.�

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.2x

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)

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x�

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no

so

luti

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(�1,

2)3x

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25.T

he

sum

of

two

nu

mbe

rs i

s 12

.Th

e di

ffer

ence

of

the

sam

e tw

o n

um

bers

is

�4.

Fin

d th

e n

um

bers

.4,

8

26.T

wic

e a

nu

mbe

r m

inu

s a

seco

nd

nu

mbe

r is

�1.

Tw

ice

the

seco

nd

nu

mbe

r ad

ded

to t

hre

e ti

mes

th

e fi

rst

nu

mbe

r is

9.F

ind

the

two

nu

mbe

rs.

1,3

1 � 2

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lenc

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w-H

ill12

8G

lenc

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lgeb

ra 2

Sol

ve e

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tem

of

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ns

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ng

sub

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on.

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no

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,�5)

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.

10.2

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)

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infi

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luti

on

SPO

RTS

For

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rcis

es 2

8 an

d 2

9,u

se t

he

foll

owin

g in

form

atio

n.

Las

t ye

ar t

he

voll

eyba

ll t

eam

pai

d $5

per

pai

r fo

r so

cks

and

$17

per

pair

for

sh

orts

on

ato

tal

purc

has

e of

$31

5.T

his

yea

r th

ey s

pen

t $3

42 t

o bu

y th

e sa

me

nu

mbe

r of

pai

rs o

f so

cks

and

shor

ts b

ecau

se t

he

sock

s n

ow c

ost

$6 a

pai

r an

d th

e sh

orts

cos

t $1

8.

28.W

rite

a s

yste

m o

f tw

o eq

uat

ion

s th

at r

epre

sen

ts t

he

nu

mbe

r of

pai

rs o

f so

cks

and

shor

tsbo

ugh

t ea

ch y

ear.

5x�

17y

�31

5,6x

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y�

342

29.H

ow m

any

pair

s of

soc

ks a

nd

shor

ts d

id t

he

team

bu

y ea

ch y

ear?

sock

s:12

,sh

ort

s:15

2 � 31 � 3

4 � 34 � 9

1 � 31 � 2

2 � 3

1 � 2

1 � 21 � 2

1 � 2

1 � 3

Pra

ctic

e (

Ave

rag

e)

So

lvin

g S

yste

ms

of

Eq

uat

ion

s A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-2

3-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Sys

tem

s o

f E

qu

atio

ns

Alg

ebra

ical

ly

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-2

3-2

©G

lenc

oe/M

cGra

w-H

ill12

9G

lenc

oe A

lgeb

ra 2

Lesson 3-2

Pre-

Act

ivit

yH

ow c

an s

yste

ms

of e

qu

atio

ns

be

use

d t

o m

ake

con

sum

erd

ecis

ion

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-2

at

the

top

of p

age

116

in y

our

text

book

.

•H

ow m

any

mor

e m

inu

tes

of l

ong

dist

ance

tim

e di

d Yo

lan

da u

se i

nF

ebru

ary

than

in

Jan

uar

y?13

min

ute

s

•H

ow m

uch

mor

e w

ere

the

Feb

ruar

y ch

arge

s th

an t

he

Jan

uar

y ch

arge

s?$1

.04

•U

sin

g yo

ur

answ

ers

for

the

ques

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s ab

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how

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you

fin

d th

e ra

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inu

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Fin

d $

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din

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on

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righ

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th

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bsti

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eth

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3x�

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e fi

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olve

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in t

erm

s of

th

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mak

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ur

wor

k as

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y as

pos

sibl

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quat

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wou

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ou s

olve

for

wh

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var

iabl

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xpla

in.

Sam

ple

an

swer

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olv

e th

e se

con

d e

qu

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or

yb

ecau

se in

th

ateq

uat

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th

e va

riab

le y

has

a c

oef

fici

ent

of

1.

2.S

upp

ose

that

you

are

ask

ed t

o so

lve

the

syst

em o

f eq

uat

ion

s 2x

�3y

��

2at

th

e ri

ght

by t

he

elim

inat

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�39

To

mak

e yo

ur

wor

k as

eas

y as

pos

sibl

e,w

hic

h v

aria

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wou

ld

you

eli

min

ate?

Des

crib

e h

ow y

ou w

ould

do

this

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amp

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nsw

er:

Elim

inat

e th

e va

riab

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;m

ult

iply

th

e se

con

d e

qu

atio

n b

y3

and

th

en a

dd

th

e re

sult

to

th

e fi

rst

equ

atio

n.

Hel

pin

g Y

ou R

emem

ber

3.T

he

subs

titu

tion

met

hod

an

d el

imin

atio

n m

eth

od f

or s

olvi

ng

syst

ems

both

hav

e se

vera

lst

eps,

and

it m

ay b

e di

ffic

ult

to

rem

embe

r th

em.Y

ou m

ay b

e ab

le t

o re

mem

ber

them

mor

e ea

sily

if

you

not

ice

wh

at t

he

met

hod

s h

ave

in c

omm

on.W

hat

ste

p is

th

e sa

me

inbo

th m

eth

ods?

Sam

ple

an

swer

:A

fter

fin

din

g t

he

valu

e o

f o

ne

of

the

vari

able

s,yo

u f

ind

the

valu

e o

f th

e o

ther

var

iab

le b

y su

bst

itu

tin

g t

he

valu

e yo

u h

ave

fou

nd

in o

ne

of

the

ori

gin

al e

qu

atio

ns.

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Usi

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Co

ord

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rom

on

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serv

atio

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oin

t,th

e li

ne

of s

igh

t to

a d

own

ed p

lan

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giv

en b

y y

�x

�1.

Th

is e

quat

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des

crib

es t

he

dist

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fro

m t

he

obse

rvat

ion

poi

nt

to t

he

plan

e in

a s

trai

ght

lin

e.F

rom

an

oth

er o

bser

vati

on p

oin

t,th

e li

ne

of

sigh

t is

giv

en b

y x

�3y

�21

.Wh

at a

re t

he

coor

din

ates

of

the

poin

t at

wh

ich

th

e cr

ash

occ

urr

ed?

Sol

ve t

he

syst

em o

f eq

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s�y

�x

�1

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�3y

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x�

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x�

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for

x.

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15y

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e co

ordi

nat

es o

f th

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ash

are

(6,

5).

Sol

ve t

he

foll

owin

g.

1.T

he

lin

es o

f si

ght

to a

for

est

fire

are

as

foll

ows.

Fro

m R

ange

r S

tati

on A

:3x

�y

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Fro

m R

ange

r S

tati

on B

:2x

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d th

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2.A

n a

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ong

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lin

e x

�y

��

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it

sees

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avel

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5x�

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19.I

f th

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onti

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e al

ong

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wh

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ill

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r fl

igh

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ths

cros

s?(2

,3)

3.T

wo

min

e sh

afts

are

du

g al

ong

the

path

s of

th

e fo

llow

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equ

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x�

y�

1400

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shaf

ts m

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hat

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th

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mee

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rich

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____

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____

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3-2

3-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g S

yste

ms

of

Ineq

ual

itie

s by

Gra

ph

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NA

ME

____

____

____

____

____

____

____

____

____

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3-3

3-3

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Lesson 3-3

Gra

ph

Sys

tem

s o

f In

equ

alit

ies

To s

olve

a s

yste

m o

f in

equa

litie

s,gr

aph

the

ineq

ualit

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in t

he s

ame

coor

dina

te p

lane

.The

sol

utio

n se

t is

rep

rese

nted

by

the

inte

rsec

tion

of

the

grap

hs.

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ve t

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syst

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f in

equ

alit

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by

grap

hin

g.y

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�1

and

y�

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f y

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1 an

d 2.

Th

e so

luti

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f y

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2 is

Reg

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s 1

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3.

Th

e in

ters

ecti

on o

f th

ese

regi

ons

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egio

n 1

,wh

ich

is

the

solu

tion

set

of

the

syst

em o

f in

equ

alit

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Sol

ve e

ach

sys

tem

of

ineq

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s b

y gr

aph

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1.x

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2.3x

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y

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y

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x � 3

x � 3

x

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on 3

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Regi

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ple

Exam

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Exer

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Exer

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Fin

d V

erti

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of

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lyg

on

al R

egio

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omet

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th

e gr

aph

of

a sy

stem

of

ineq

ual

itie

s fo

rms

a bo

un

ded

regi

on.Y

ou c

an f

ind

the

vert

ices

of

the

regi

on b

y a

com

bin

atio

nof

th

e m

eth

ods

use

d ea

rlie

r in

th

is c

hap

ter:

grap

hin

g,su

bsti

tuti

on,a

nd/

or e

lim

inat

ion

.

Fin

d t

he

coor

din

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of

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vert

ices

of

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figu

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orm

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y 5x

�4y

�20

,y�

2x�

3,an

d x

�3y

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ph t

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bou

nda

ry o

f ea

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neq

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ity.

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e in

ters

ecti

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he

bou

nda

ry l

ines

are

th

e ve

rtic

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tria

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e.T

he

vert

ex (

4,0)

can

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th

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aph

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fin

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nat

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f th

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con

d an

d th

ird

vert

ices

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ve t

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two

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ems

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quat

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s

y�

2x�

3an

dy

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5x�

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x

y

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nte

rven

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)

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lvin

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ms

of

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ual

itie

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Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

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3-3

3-3

For

th

e fi

rst

syst

em o

f eq

uat

ion

s,re

wri

te t

he

firs

teq

uat

ion

in

sta

nda

rd f

orm

as

2x�

y�

�3.

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enm

ult

iply

th

at e

quat

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4 an

d ad

d to

th

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con

deq

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�y

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3M

ultip

ly b

y 4.

8x�

4y�

�12

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)5x

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en s

ubs

titu

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�in

on

e of

th

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igin

al e

quat

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s an

d so

lve

for

y.

2 ��

y�

�3

�y

��

3

y�

Th

e co

ordi

nat

es o

f th

e se

con

d ve

rtex

are

�,4

�.3 � 13

8 � 13

55 � 13

16 � 138 � 13

8 � 13

8 � 13

For

th

e se

con

d sy

stem

of

equ

atio

ns,

use

su

bsti

tuti

on.

Su

bsti

tute

2x

�3

for

yin

th

ese

con

d eq

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get

x�

3(2x

�3)

�4

x�

6x�

9 �

4�

5x�

13x

��

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ubs

titu

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��

in t

he

firs

t eq

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solv

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r y.

y�

2 ��

3

y�

��

3

y�

�

Th

e co

ordi

nat

es o

f th

e th

ird

vert

ex a

re ��

2,�

2�.

1 � 53 � 5

11 � 526 � 5

13 � 5

13 � 5

13 � 5

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cises

Exer

cises

Th

us,

the

coor

din

ates

of

the

thre

e ve

rtic

es a

re (

4,0)

, �,4

�,an

d ��2

,�2

�.

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d t

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coor

din

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ices

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re f

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y ea

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m o

fin

equ

alit

ies.

1.y

��

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72.

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�3

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4 � 53 � 5

1 � 21 � 3

1 � 2

1 � 2

1 � 53 � 5

3 � 138 � 13

Exam

ple

Exam

ple


An

swer

s


Skil

ls P

ract

ice

So

lvin

g S

yste

ms

of

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ual

itie

s by

Gra

ph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

3-3

3-3

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3G

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Lesson 3-3

Sol

ve e

ach

sys

tem

of

ineq

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s b

y gr

aph

ing.

1.x

�1

2.x

��

33.

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��

1y

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no

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vert

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of

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figu

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y ea

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m o

fin

equ

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DR

AM

AF

or E

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10

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th

e fo

llow

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dram

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ts p

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seat

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om t

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250

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100

stu

den

t,14

5 ad

ult

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tud

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Pla

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Ad

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Student Tickets

100

020

030

040

0

400

350

300

250

200

150

100 50

3 � 2

x

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y

xO

y

xO

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So

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ph

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NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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____

____

__P

ER

IOD

____

_

3-3

3-3



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Sys

tem

s o

f In

equ

alit

ies

by G

rap

hin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3

©G

lenc

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cGra

w-H

ill13

5G

lenc

oe A

lgeb

ra 2

Lesson 3-3

Pre-

Act

ivit

yH

ow c

an y

ou d

eter

min

e w

het

her

you

r b

lood

pre

ssu

re i

s in

a

nor

mal

ran

ge?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-3

at

the

top

of p

age

123

in y

our

text

book

.

Sat

ish

is

37 y

ears

old

.He

has

a b

lood

pre

ssu

re r

eadi

ng

of 1

35/9

9.Is

his

bloo

d pr

essu

re w

ith

in t

he

nor

mal

ran

ge?

Exp

lain

.S

amp

le a

nsw

er:

No

;h

is s

ysto

lic p

ress

ure

is n

orm

al,b

ut

his

dia

sto

lic p

ress

ure

is t

oo

hig

h.I

t sh

ou

ld b

e b

etw

een

60

and

90.

Rea

din

g t

he

Less

on

1.W

ith

out

actu

ally

dra

win

g th

e gr

aph

,des

crib

e th

e bo

un

dary

lin

es

x

�3

for

the

syst

em o

f in

equ

alit

ies

show

n a

t th

e ri

ght.

y

�5

Two

das

hed

ver

tica

l lin

es (

x�

3 an

d x

��

3) a

nd

tw

o s

olid

ho

rizo

nta

llin

es (

y�

�5

and

y�

5)

2.T

hin

k ab

out

how

th

e gr

aph

wou

ld l

ook

for

the

syst

em g

iven

abo

ve.W

hat

wil

l be

th

esh

ape

of t

he

shad

ed r

egio

n?

(It

is n

ot n

eces

sary

to

draw

th

e gr

aph

.See

if

you

can

imag

ine

it w

ith

out

draw

ing

anyt

hin

g.If

th

is i

s di

ffic

ult

to

do,m

ake

a ro

ugh

ske

tch

to

hel

p yo

u a

nsw

er t

he

ques

tion

.)a

rect

ang

le

3.W

hic

h s

yste

m o

f in

equ

alit

ies

mat

ches

th

e gr

aph

sh

own

at

th

e ri

ght?

BA

.x

�y

��

2B

.x

�y

��

2x

�y

�2

x�

y�

2

C.

x�

y�

�2

D.

x�

y�

�2

x�

y�

2x

�y

�2

Hel

pin

g Y

ou

Rem

emb

er

4.T

o gr

aph

a s

yste

m o

f in

equ

alit

ies,

you

mu

st g

raph

tw

o or

mor

e bo

un

dary

lin

es.W

hen

you

gra

ph e

ach

of

thes

e li

nes

,how

can

th

e in

equ

alit

y sy

mbo

ls h

elp

you

rem

embe

rw

het

her

to

use

a d

ash

ed o

r so

lid

lin

e?U

se a

das

hed

lin

e if

th

e in

equ

alit

y sy

mb

ol i

s �

or

�,b

ecau

se t

hes

esy

mb

ols

do

no

t in

clu

de

equ

alit

y an

d t

he

das

hed

lin

e re

min

ds

you

th

atth

e lin

e it

self

is n

ot

incl

ud

ed in

th

e g

rap

h.U

se a

so

lid li

ne

if t

he

sym

bo

lis

o

r �

,bec

ause

th

ese

sym

bo

ls in

clu

de

equ

alit

y an

d t

ell y

ou

th

at t

he

line

itse

lf is

incl

ud

ed in

th

e g

rap

h.

x

y

O

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Trac

ing

Str

ateg

yT

ry t

o tr

ace

over

eac

h o

f th

e fi

gure

s be

low

wit

hou

t tr

acin

g th

e sa

me

segm

ent

twic

e.

Th

e fi

gure

at

the

left

can

not

be

trac

ed,b

ut

the

one

at t

he

righ

t ca

n.T

he

rule

is

that

a

figu

re i

s tr

acea

ble

if i

t h

as n

o m

ore

than

tw

o po

ints

wh

ere

an o

dd n

um

ber

of

segm

ents

mee

t.T

he

figu

re a

t th

e le

ft h

as t

hre

e se

gmen

ts m

eeti

ng

at e

ach

of

the

fou

r co

rner

s.H

owev

er,t

he

figu

re a

t th

e ri

ght

has

on

ly t

wo

poin

ts,L

and

Q,w

her

e an

odd

nu

mbe

r of

seg

men

ts m

eet.

Det

erm

ine

if e

ach

fig

ure

can

be

trac

ed w

ith

out

trac

ing

the

sam

e se

gmen

t tw

ice.

If i

t ca

n,t

hen

nam

e th

e st

arti

ng

poi

nt

and

nam

e th

e se

gmen

ts i

n

the

ord

er t

hey

sh

ould

be

trac

ed.

1.2.

yes;

X;

X�Y�

,Y�F�,

F�X�,X�

G�,G�

F�,ye

s;E

;E�

D�,D�

A�,A�

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B�,B�

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,C�K�

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,E�H�

,H�X�

,X�W�

,W�H�

,H�G�

K�F�,

F�J�,J�

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F�,F�E�

,E�H�

,H�G�

,G�E�

3.

no

t tr

acea

ble

T SRU

PV

A

EF

DK

BC

GJ

H

EF

HG

XW

Y

K M

JL

PQ

AB

DC

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-3

3-3


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Lin

ear

Pro

gra

mm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

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Lesson 3-4

Max

imu

m a

nd

Min

imu

m V

alu

esW

hen

a s

yste

m o

f li

nea

r in

equ

alit

ies

prod

uce

s a

bou

nde

d po

lygo

nal

reg

ion

,th

e m

axim

um

or m

inim

um

valu

e of

a r

elat

ed f

un

ctio

n w

ill

occu

rat

a v

erte

x of

th

e re

gion

.

Gra

ph

th

e sy

stem

of

ineq

ual

itie

s.N

ame

the

coor

din

ates

of

the

vert

ices

of

the

feas

ible

reg

ion

.Fin

d t

he

max

imu

m a

nd

min

imu

m v

alu

es o

f th

efu

nct

ion

f(x

,y)

�3x

�2y

for

this

pol

ygon

al r

egio

n.

y�

4y

��

x�

6

y�

x�

y�

6x�

4F

irst

fin

d th

e ve

rtic

es o

f th

e bo

un

ded

regi

on.G

raph

th

e in

equ

alit

ies.

Th

e po

lygo

n f

orm

ed i

s a

quad

rila

tera

l w

ith

ver

tice

s at

(0

,4),

(2,4

),(5

,1),

and

(�1,

�2)

.Use

th

e ta

ble

to f

ind

the

max

imu

m a

nd

min

imu

m v

alu

es o

f f(

x,y)

�3x

�2y

.

Th

e m

axim

um

val

ue

is 1

7 at

(5,

1).T

he

min

imu

m v

alu

e is

�7

at (

�1,

�2)

.

Gra

ph

eac

h s

yste

m o

f in

equ

alit

ies.

Nam

e th

e co

ord

inat

es o

f th

e ve

rtic

es o

f th

efe

asib

le r

egio

n.F

ind

th

e m

axim

um

an

d m

inim

um

val

ues

of

the

give

n f

un

ctio

n f

orth

is r

egio

n.

1.y

�2

2.y

��

23.

x�

y�

21

�x

�5

y�

2x�

44y

�x

�8

y�

x�

3x

�2y

��

1y

�2x

�5

f(x,

y) �

3x�

2yf(

x,y)

�4x

�y

f(x,

y) �

4x�

3y

vert

ices

:(1

,2),

(1,4

),ve

rtic

es:

(�5,

�2)

,ve

rtic

es (

0,2)

,(4,

3),

(5,8

),(5

,2);

max

:11

;(3

,2),

(1,�

2);

�,�

�;max

:25

;m

in:

6

min

;�

5m

ax:

10;

min

:�

181 � 3

7 � 3

x

y

Ox

y

O

x

y

O

(x,y

)3x

�2y

f(x,

y)

(0,

4)3(

0) �

2(4)

8

(2,

4)3(

2) �

2(4)

14

(5,

1)3(

5) �

2(1)

17

(�1,

�2)

3(�

1) �

2(�

2)�

7

x

y O

3 � 21 � 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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Rea

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orl

d P

rob

lem

sW

hen

sol

vin

g li

nea

r p

rogr

amm

ing

prob

lem

s,u

se t

he

foll

owin

g pr

oced

ure

.

1.D

efin

e va

riab

les.

2.W

rite

a s

yste

m o

f in

equ

alit

ies.

3.G

raph

th

e sy

stem

of

ineq

ual

itie

s.4.

Fin

d th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f th

e fe

asib

le r

egio

n.

5.W

rite

an

exp

ress

ion

to

be m

axim

ized

or

min

imiz

ed.

6.S

ubs

titu

te t

he

coor

din

ates

of

the

vert

ices

in

th

e ex

pres

sion

.7.

Sel

ect

the

grea

test

or

leas

t re

sult

to

answ

er t

he

prob

lem

.

A p

ain

ter

has

exa

ctly

32

un

its

of y

ello

w d

ye a

nd

54

un

its

of g

reen

dye

.He

pla

ns

to m

ix a

s m

any

gall

ons

as p

ossi

ble

of

colo

r A

an

d c

olor

B.E

ach

gall

on o

f co

lor

A r

equ

ires

4 u

nit

s of

yel

low

dye

an

d 1

un

it o

f gr

een

dye

.Eac

hga

llon

of

colo

r B

req

uir

es 1

un

it o

f ye

llow

dye

an

d 6

un

its

of g

reen

dye

.Fin

d t

he

max

imu

m n

um

ber

of

gall

ons

he

can

mix

.

Ste

p 1

Def

ine

the

vari

able

s.x

�th

e n

um

ber

of g

allo

ns

of c

olor

A m

ade

y�

the

nu

mbe

r of

gal

lon

s of

col

or B

mad

e

Ste

p 2

Wri

te a

sys

tem

of

ineq

ual

itie

s.S

ince

th

e n

um

ber

of g

allo

ns

mad

e ca

nn

ot b

en

egat

ive,

x�

0 an

d y

�0.

Th

ere

are

32 u

nit

s of

yel

low

dye

;eac

h g

allo

n o

f co

lor

A r

equ

ires

4 u

nit

s,an

d ea

ch g

allo

n o

f co

lor

B r

equ

ires

1 u

nit

.S

o 4x

�y

�32

.S

imil

arly

for

th

e gr

een

dye

,x�

6y�

54.

Ste

ps

3 an

d 4

Gra

ph t

he

syst

em o

f in

equ

alit

ies

and

fin

d th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f th

e fe

asib

le r

egio

n.

Th

e ve

rtic

es o

f th

e fe

asib

le r

egio

n a

re (

0,0)

,(0,

9),(

6,8)

,an

d (8

,0).

Ste

ps

5–7

Fin

d th

e m

axim

um

nu

mbe

r of

gal

lon

s,x

�y,

that

he

can

mak

e.T

he

max

imu

m n

um

ber

of g

allo

ns

the

pain

ter

can

mak

e is

14,

6 ga

llon

s of

col

or A

an

d 8

gall

ons

of c

olor

B.

1.FO

OD

A d

elic

ates

sen

has

8 p

oun

ds o

f pl

ain

sau

sage

an

d 10

pou

nds

of

garl

ic-f

lavo

red

sau

sage

.Th

e de

li w

ants

to

mak

e as

mu

ch b

ratw

urs

t as

pos

sibl

e.E

ach

pou

nd

of

brat

wu

rst

requ

ires

po

un

d of

pla

in s

ausa

ge a

nd

pou

nd

of g

arli

c-fl

avor

ed s

ausa

ge.

Fin

d th

e m

axim

um

nu

mbe

r of

pou

nds

of

brat

wu

rst

that

can

be

mad

e.10

lb

2.M

AN

UFA

CTU

RIN

GM

ach

ine

A c

an p

rodu

ce 3

0 st

eeri

ng

wh

eels

per

hou

r at

a c

ost

of $

16pe

r h

our.

Mac

hin

e B

can

pro

duce

40

stee

rin

g w

hee

ls p

er h

our

at a

cos

t of

$22

per

hou

r.A

t le

ast

360

stee

rin

g w

hee

ls m

ust

be

mad

e in

eac

h 8

-hou

r sh

ift.

Wh

at i

s th

e le

ast

cost

invo

lved

in

mak

ing

360

stee

rin

g w

hee

ls i

n o

ne

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t?$1

94

2 � 3

1 � 43 � 4

(x,y

)x

�y

f(x,

y)

(0,

0)0

�0

0

(0,

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�9

9

(6,

8)6

�8

14

(8,

0)8

�0

8

Co

lor

A (

gal

lon

s)

Color B (gallons)

510

1520

2530

3540

4550

550

40 35 30 25 20 15 10 5

( 6, 8

)

( 8, 0

)( 0

, 9)

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Lin

ear

Pro

gra

mm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Lin

ear

Pro

gra

mm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

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Lesson 3-4

Gra

ph

eac

h s

yste

m o

f in

equ

alit

ies.

Nam

e th

e co

ord

inat

es o

f th

e ve

rtic

es o

f th

efe

asib

le r

egio

n.F

ind

th

e m

axim

um

an

d m

inim

um

val

ues

of

the

give

n f

un

ctio

n f

orth

is r

egio

n.

1.x

�2

2.x

�1

3.x

�0

x�

5y

�6

y�

0y

�1

y�

x�

2y

�7

�x

y�

4f(

x,y)

�x

�y

f(x,

y) �

3x�

yf(

x,y)

�x

�y

max

.:9,

min

.:3

max

.:2,

min

.:�

5m

ax.:

21,m

in.:

0

4.x

��

15.

y�

2x6.

y�

�x

�2

x�

y�

6y

�6

�x

y�

3x�

2f(

x,y)

�x

�2y

y�

6y

�x

�4

f(x,

y) �

4x�

3yf(

x,y)

��

3x�

5y

max

.:13

,no

min

.n

o m

ax.,

min

.:20

max

.:22

,min

.:�

2

7.M

AN

UFA

CTU

RIN

GA

bac

kpac

k m

anu

fact

ure

r pr

odu

ces

an i

nte

rnal

fra

me

pack

an

d an

exte

rnal

fra

me

pack

.Let

xre

pres

ent

the

nu

mbe

r of

in

tern

al f

ram

e pa

cks

prod

uce

d in

one

hou

r an

d le

t y

repr

esen

t th

e n

um

ber

of e

xter

nal

fra

me

pack

s pr

odu

ced

in o

ne

hou

r.T

hen

th

e in

equ

alit

ies

x�

3y�

18,2

x�

y�

16,x

�0,

and

y�

0 de

scri

be t

he

con

stra

ints

for

man

ufa

ctu

rin

g bo

th p

acks

.Use

th

e pr

ofit

fu

nct

ion

f(x

) �

50x

�80

yan

d th

eco

nst

rain

ts g

iven

to

dete

rmin

e th

e m

axim

um

pro

fit

for

man

ufa

ctu

rin

g bo

th b

ackp

acks

for

the

give

n c

onst

rain

ts.

$620

( 1, 5

)

( –1,

–1)

( –3,

1)

x

y

O

( 3, 6

)

( 2, 4

)

x

y

O

( –1,

7)

x

y

O

x

y

O

( 0, 7

)

( 0, 0

)( 7

, 0)

x

y

O

( 1, 6

)

( 1, –

1)

( 8, 6

)

x

y

O

( 2, 4

)

( 2, 1

)( 5

, 1)

( 5, 4

)

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ph

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um

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PRO

DU

CTI

ON

For

Exe

rcis

es 7

–9,u

se t

he

foll

owin

g in

form

atio

n.

A g

lass

blo

wer

can

for

m 8

sim

ple

vase

s or

2 e

labo

rate

vas

es i

n a

n h

our.

In a

wor

k sh

ift

of n

om

ore

than

8 h

ours

,th

e w

orke

r m

ust

for

m a

t le

ast

40 v

ases

.

7.L

et s

repr

esen

t th

e h

ours

for

min

g si

mpl

e va

ses

and

eth

e h

ours

for

min

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abor

ate

vase

s.W

rite

a s

yste

m o

f in

equ

alit

ies

invo

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g th

e ti

me

spen

t on

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h t

ype

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ase.

s

0,e

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th

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mpl

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$35

per

hou

r w

orke

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abor

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vase

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rite

a f

un

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or t

he

tota

l pr

ofit

on

th

e va

ses.

f(s,

e) �

30s

�35

e

9.F

ind

the

nu

mbe

r of

hou

rs t

he

wor

ker

shou

ld s

pen

d on

eac

h t

ype

of v

ase

to m

axim

ize

prof

it.W

hat

is

that

pro

fit?

4 h

on

eac

h;

$260

17 � 234 � 5

x

y

(3 – 2, 1)

( 0, 2

) O

(–7 – 5, 9 – 5)

( –2,

0)

x

y O

x

y ( 0, –

7)

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( 0, 0

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( 3, 6

)

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)

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)

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4)

x

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Pra

ctic

e (

Ave

rag

e)

Lin

ear

Pro

gra

mm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csL

inea

r P

rog

ram

min

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill14

1G

lenc

oe A

lgeb

ra 2

Lesson 3-4

Pre-

Act

ivit

yH

ow i

s li

nea

r p

rogr

amm

ing

use

d i

n s

ched

uli

ng

wor

k?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-4

at

the

top

of p

age

129

in y

our

text

book

.

Nam

e tw

o or

mor

e fa

cts

that

in

dica

te t

hat

you

wil

l n

eed

to u

se i

neq

ual

itie

sto

mod

el t

his

sit

uat

ion

.S

amp

le a

nsw

er:T

he

buoy

ten

der

can

car

ry u

p t

o 8

new

bu

oys.

Th

ere

seem

s to

be

a lim

it o

f 24

ho

urs

on

th

e ti

me

the

crew

has

at s

ea.T

he

crew

will

wan

t to

rep

air

or

rep

lace

th

e m

axim

um

nu

mb

er o

f bu

oys

po

ssib

le.

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sen

ten

ce.

a.W

hen

you

fin

d th

e fe

asib

le r

egio

n f

or a

lin

ear

prog

ram

min

g pr

oble

m,y

ou a

re s

olvi

ng

a sy

stem

of

lin

ear

call

ed

.Th

e po

ints

in t

he

feas

ible

reg

ion

are

of

th

e sy

stem

.

b.

Th

e co

rner

poi

nts

of

a po

lygo

nal

reg

ion

are

th

e of

th

e fe

asib

le r

egio

n.

2.A

pol

ygon

al r

egio

n a

lway

s ta

kes

up

only

a l

imit

ed p

art

of t

he

coor

din

ate

plan

e.O

ne

way

to t

hin

k of

th

is i

s to

im

agin

e a

circ

le o

r re

ctan

gle

that

th

e re

gion

wou

ld f

it i

nsi

de.I

n t

he

case

of

a po

lygo

nal

reg

ion

,you

can

alw

ays

fin

d a

circ

le o

r re

ctan

gle

that

is

larg

e en

ough

to c

onta

in a

ll t

he

poin

ts o

f th

e po

lygo

nal

reg

ion

.Wh

at w

ord

is u

sed

to d

escr

ibe

a re

gion

that

can

be

encl

osed

in

th

is w

ay?

Wh

at w

ord

is u

sed

to d

escr

ibe

a re

gion

th

at i

s to

o la

rge

to b

e en

clos

ed i

n t

his

way

?b

ou

nd

ed;

un

bo

un

ded

3.H

ow d

o yo

u f

ind

the

corn

er p

oin

ts o

f th

e po

lygo

nal

reg

ion

in

a l

inea

r pr

ogra

mm

ing

prob

lem

?Yo

u s

olv

e a

syst

em o

f tw

o li

nea

r eq

uat

ion

s.

4.W

hat

are

som

e ev

eryd

ay m

ean

ings

of

the

wor

d fe

asib

leth

at r

emin

d yo

u o

f th

em

ath

emat

ical

mea

nin

g of

th

e te

rm f

easi

ble

regi

on?

Sam

ple

an

swer

:p

oss

ible

or

ach

ieva

ble

Hel

pin

g Y

ou

Rem

emb

er

5.L

ook

up

the

wor

d co

nst

rain

tin

a d

icti

onar

y.If

mor

e th

an o

ne

defi

nit

ion

is

give

n,c

hoo

seth

e on

e th

at s

eem

s cl

oses

t to

th

e id

ea o

f a

con

stra

int

in a

lin

ear

prog

ram

min

g pr

oble

m.

How

can

th

is d

efin

itio

n h

elp

you

to

rem

embe

r th

e m

ean

ing

of c

onst

rain

tas

it

is u

sed

inth

is l

esso

n?

Sam

ple

an

swer

:A

co

nst

rain

tis

a r

estr

icti

on

or

limit

atio

n.T

he

con

stra

ints

in a

lin

ear

pro

gra

mm

ing

pro

ble

m a

re r

estr

icti

on

s o

n t

he

vari

able

s th

at t

ran

slat

e in

to in

equ

alit

y st

atem

ents

.

vert

ices

solu

tio

ns

con

stra

ints

ineq

ual

itie

s

©G

lenc

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cGra

w-H

ill14

2G

lenc

oe A

lgeb

ra 2

Co

mp

ute

r C

ircu

its

and

Lo

gic

Com

pute

rs o

pera

te a

ccor

din

g to

th

e la

ws

of l

ogic

.Th

e ci

rcu

its

of a

com

pute

r ca

n b

e de

scri

bed

usi

ng

logi

c.

1.W

ith s

witc

h A

open

, no

cur

rent

flo

ws.

The

val

ue 0

is a

ssig

ned

to a

n op

en s

witc

h.

2.W

ith s

witc

h A

clos

ed,

curr

ent

flow

s. T

he v

alue

1 is

ass

igne

d to

a c

lose

d sw

itch.

3.W

ith s

witc

hes

Aan

d B

ope

n, n

o cu

rren

t flo

ws.

Thi

s ci

rcui

t ca

n be

des

crib

ed b

y th

e co

njun

ctio

n, A

B

.

4.In

thi

s ci

rcui

t, cu

rren

t flo

ws

if ei

ther

Aor

B is

clo

sed.

Thi

s ci

rcui

t ca

n be

des

crib

ed b

y th

e di

sjun

ctio

n, A

�B

.

Tru

th t

able

s ar

e u

sed

to d

escr

ibe

the

flow

of

curr

ent

in a

cir

cuit

.T

he

tabl

e at

th

e le

ft d

escr

ibes

th

e ci

rcu

it i

n d

iagr

am 4

.Acc

ordi

ng

to t

he

tabl

e,th

e on

ly t

ime

curr

ent

does

not

flo

w t

hro

ugh

th

e ci

rcu

it i

s w

hen

bot

h s

wit

ches

A a

nd

B a

re o

pen

.

Dra

w a

cir

cuit

dia

gram

for

eac

h o

f th

e fo

llow

ing.

1.(A

B

) �

C2.

(A �

B)

C

3.(A

�B

)

(C �

D)

4.(A

B

) �

(C

D)

5.C

onst

ruct

a t

ruth

tab

le f

or t

he

foll

owin

g ci

rcu

it.

B C

A

A CDB

A BD

C

A

BC

AB

C

A A A B

AB

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-4

3-4

AB

C(B

�C

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(B �

C)

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Stu

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of

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s in

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ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

3-5

3-5

©G

lenc

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cGra

w-H

ill14

3G

lenc

oe A

lgeb

ra 2

Lesson 3-5

Syst

ems

in T

hre

e V

aria

ble

sU

se t

he

met

hod

s u

sed

for

solv

ing

syst

ems

of l

inea

req

uat

ion

s in

tw

o va

riab

les

to s

olve

sys

tem

s of

equ

atio

ns

in t

hre

e va

riab

les.

A s

yste

m o

fth

ree

equ

atio

ns

in t

hre

e va

riab

les

can

hav

e a

un

iqu

e so

luti

on,i

nfi

nit

ely

man

y so

luti

ons,

orn

o so

luti

on.A

sol

uti

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s an

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tri

ple

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ve t

his

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tem

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3x�

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ple

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Th

e L

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por

ts S

hop

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day

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60.W

hat

are

th

e p

rice

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1 b

all,

1 b

at,

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1 b

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st d

efin

e th

e va

riab

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x�

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rmat

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nto

th

ree

equ

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dy G

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lvin

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ms

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uat

ion

s in

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ree

Var

iab

les

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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ER

IOD

____

_

3-5

3-5

Exam

ple

Exam

ple

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btra

ct t

he

seco

nd

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rom

th

e fi

rst

equ

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olve

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a ba

t $5

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1.40

.

1.FI

TNES

S TR

AIN

ING

Car

ly i

s tr

ain

ing

for

a tr

iath

lon

.In

her

tra

inin

g ro

uti

ne

each

wee

k,sh

e ru

ns

7 ti

mes

as

far

as s

he

swim

s,an

d sh

e bi

kes

3 ti

mes

as

far

as s

he

run

s.O

ne

wee

ksh

e tr

ain

ed a

tot

al o

f 23

2 m

iles

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far

did

sh

e ru

n t

hat

wee

k?56

mile

s

2.EN

TERT

AIN

MEN

TA

t th

e ar

cade

,Rya

n,S

ara,

and

Tim

pla

yed

vide

o ra

cing

gam

es,p

inba

ll,

and

air

hoc

key.

Rya

n s

pen

t $6

for

6 r

acin

g ga

mes

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inba

ll g

ames

,an

d 1

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e of

air

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ey.S

ara

spen

t $1

2 fo

r 3

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ng g

ames

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ll g

ames

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ames

of

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ey.T

imsp

ent

$12.

25 f

or 2

rac

ing

gam

es,7

pin

ball

gam

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nd 4

gam

es o

f ai

r ho

ckey

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muc

h di

dea

ch o

f th

e ga

mes

cos

t?R

acin

g g

ame:

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0;p

inb

all:

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r h

ock

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0

3.FO

OD

A n

atu

ral

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re m

akes

its

ow

n b

ran

d of

tra

il m

ix o

ut

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uts

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e po

un

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re c

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18.I

t co

nta

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twic

e as

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ch p

ean

uts

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eigh

t as

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les.

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e po

un

d of

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pple

s co

sts

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8,a

pou

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aisi

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0,an

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44.H

ow m

any

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ces

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ach

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ined

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1 p

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th

e tr

ail

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les,

7 o

z o

f ra

isin

s,6

oz

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pea

nu

ts

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

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yste

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ree

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NA

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____

____

____

____

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on

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ore

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ind

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e th

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qual

to

the

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t.T

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sum

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con

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um

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is

�5.

Fin

d th

e n

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3,�

2,1

24.S

POR

TSA

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igh

Sch

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e.S

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reaw

arde

d fo

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ch t

ouch

dow

n.A

fter

eac

h t

ouch

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n,t

he

team

can

ear

n o

ne

poin

t fo

r th

eex

tra

kick

or

two

poin

ts f

or a

2-p

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t co

nve

rsio

n.T

he

team

sco

red

one

few

er 2

-poi

nt

con

vers

ion

s th

an e

xtra

kic

ks.T

he

team

sco

red

10 t

imes

du

rin

g th

e ga

me.

How

man

yto

uch

dow

ns

wer

e m

ade

duri

ng

the

gam

e?5

Pra

ctic

e (

Ave

rag

e)

So

lvin

g S

yste

ms

of

Eq

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s in

Th

ree

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iab

les

NA

ME

____

____

____

____

____

____

____

____

____

____

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____

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ER

IOD

____

_

3-5

3-5



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Sys

tem

s o

f E

qu

atio

ns

in T

hre

e V

aria

ble

s

NA

ME

____

____

____

____

____

____

____

____

____

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ER

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____

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3-5

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Lesson 3-5

Pre-

Act

ivit

yH

ow c

an y

ou d

eter

min

e th

e n

um

ber

an

d t

ype

of m

edal

s U

.S.

Oly

mp

ian

s w

on?

Rea

d th

e in

trod

uct

ion

to

Les

son

3-5

at

the

top

of p

age

138

in y

our

text

book

.

At

the

1996

Su

mm

er O

lym

pics

in

Atl

anta

,Geo

rgia

,th

e U

nit

ed S

tate

s w

on10

1 m

edal

s.T

he

U.S

.tea

m w

on 1

2 m

ore

gold

med

als

than

sil

ver

and

7fe

wer

bro

nze

med

als

than

sil

ver.

Usi

ng

the

sam

e va

riab

les

as t

hos

e in

th

ein

trod

uct

ion

,wri

te a

sys

tem

of

equ

atio

ns

that

des

crib

es t

he

med

als

won

for

the

1996

Su

mm

er O

lym

pics

.g

�s

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Rea

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g t

he

Less

on

1.T

he

plan

es f

or t

he

equ

atio

ns

in a

sys

tem

of

thre

e li

nea

r eq

uat

ion

s in

th

ree

vari

able

sde

term

ine

the

nu

mbe

r of

sol

uti

ons.

Mat

ch e

ach

gra

ph d

escr

ipti

on b

elow

wit

h t

he

desc

ript

ion

of

the

nu

mbe

r of

sol

uti

ons

of t

he

syst

em.(

Som

e of

th

e it

ems

on t

he

righ

t m

aybe

use

d m

ore

than

on

ce,a

nd

not

all

pos

sibl

e ty

pes

of g

raph

s ar

e li

sted

.)

a.th

ree

para

llel

pla

nes

I.

one

solu

tion

b.

thre

e pl

anes

th

at i

nte

rsec

t in

a l

ine

II.

no

solu

tion

s

c.th

ree

plan

es t

hat

in

ters

ect

in o

ne

poin

t II

I.in

fin

ite

solu

tion

s

d.

one

plan

e th

at r

epre

sen

ts a

ll t

hre

e eq

uat

ion

s

2.S

upp

ose

that

th

ree

clas

smat

es,M

oniq

ue,

Josh

,an

d L

illy

,are

stu

dyin

g fo

r a

quiz

on

th

isle

sson

.Th

ey w

ork

toge

ther

on

sol

vin

g a

syst

em o

f eq

uat

ion

s in

th

ree

vari

able

s,x,

y,an

dz,

foll

owin

g th

e al

gebr

aic

met

hod

sh

own

in

you

r te

xtbo

ok.T

hey

fir

st f

ind

that

z�

3,th

enth

at y

��

2,an

d fi

nal

ly t

hat

x�

�1.

Th

e st

ude

nts

agr

ee o

n t

hes

e va

lues

,bu

t di

sagr

eeon

how

to

wri

te t

he

solu

tion

.Her

e ar

e th

eir

answ

ers:

Mon

iqu

e:(3

,�2,

�1)

Josh

:(�

2,�

1,3)

Lil

ly:(

�1,

�2,

3)

a.H

ow d

o yo

u t

hin

k ea

ch s

tude

nt

deci

ded

on t

he

orde

r of

th

e n

um

bers

in

th

e or

dere

dtr

iple

?S

amp

le a

nsw

er:

Mo

niq

ue

arra

ng

ed t

he

valu

es in

th

e o

rder

inw

hic

h s

he

fou

nd

th

em.J

osh

arr

ang

ed t

hem

fro

m s

mal

lest

to

larg

est.

Lill

y ar

ran

ged

th

em in

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hab

etic

al o

rder

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the

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able

s.

b.

Wh

ich

stu

den

t is

cor

rect

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illy

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

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ou r

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that

obt

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ing

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n 0

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cate

s a

syst

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ith

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ely

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y so

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wh

ile

obta

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g an

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n s

uch

as

0 �

8 in

dica

tes

a sy

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h n

o so

luti

ons?

0 �

0 is

alw

ays

tru

e,w

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0 �

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er t

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III

I

III

II

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Bill

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igur

e at

the

rig

ht s

how

s a

bill

iard

tab

le.

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obj

ect

is t

o us

e a

cue

stic

k to

str

ike

the

ball

at

poi

nt C

so t

hat

the

ball

wil

l hi

t th

e si

des

(or

cush

ions

) of

the

tab

le a

t le

ast

once

bef

ore

hitt

ing

the

ball

loc

ated

at

poin

t A

.In

play

ing

the

gam

e,yo

u ne

ed t

o lo

cate

poi

nt P

.

Ste

p 1

Fin

d po

int

Bso

tha

t �

and

�

.Bis

cal

led

the

refl

ecte

d im

age

of C

in

.

Ste

p 2

Dra

w

.

Ste

p 3

inte

rsec

ts

at t

he d

esir

ed p

oint

P.

For

eac

h b

illi

ard

s p

rob

lem

,th

e cu

e b

all

at p

oin

t C

mu

st s

trik

e th

e in

dic

ated

cush

ion

(s)

and

th

en s

trik

e th

e b

all

at p

oin

t A

.Dra

w a

nd

lab

el t

he

corr

ect

pat

h

for

the

cue

bal

l u

sin

g th

e p

roce

ss d

escr

ibed

ab

ove.

1.cu

shio

n 2.

cush

ion

3.cu

shio

n ,t

hen

cush

ion

4.cu

shio

n ,t

hen

cush

ion C

A

B1

B2

P1

P2

RK T

S

C

A

B1

B2

P1

P2R

K TS

RS

KT

RS

TS

C

A

B

R P

K TS

C

A

B

RK T

S

RS

KR

ST

AB

AB

ST

CH

BH

ST

BC

KR

C

A

B

P

HT

S

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

3-5

3-5


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. B

D

D

A

C

A

B

D

B

C

B

4 units2

A

B

A

C

B

D

C

B

A

B

B

C

D

A

B

C

A

A

C

D

Chapter 3 Assessment Answer KeyForm 1 Form 2APage 149 Page 150 Page 151

An

swer

s

(continued on the next page)


12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

C

B

A

D

C

C

B

A

C

D

D

D

A

B

C

A

B

A

A

�98

�

A

A

C

D

C

C

D

A

B

Chapter 3 Assessment Answer KeyForm 2A (continued) Form 2BPage 152 Page 153 Page 154

a � 2, b � 4, c � 1,d � 3, f � 0


1.

2.

no solution

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: 20 units

(�1, 2, 1)

(0, 1, �1)

50 cars, 10 busses

y

xO

y

xO

(2, �2)

(2, 1)

(0, 4)

(4, 1)

y

xO

y

xO(2, 0)

Chapter 3 Assessment Answer KeyForm 2CPage 155 Page 156

An

swer

s

consistent andindependent

consistent anddependent

(�2, �3), (2, �3),(2, 5)

(�2, 0), (3, �5),(3, 2), (0, 4)

(�2, �3), (�2, 9),(2, 5)

max: f(2, 5) � 1;min: f(�2, 9) � �15

c � 0, b � 0,6c � 30b � 600,c � b � 60

s � m � � � 9 7s � 12m � 15� � 86m � 3�

5 small packages,3 medium packages,1 large package


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: (12, �8)

$24

(3, 1, �4)

(�2, 3, 5)

$138

(�5, 7), (1, 7), (�1, 3)

y

xO

y

xO

(2, �2)

(�1, 3)

(�2, �8)

(4, 6)

inconsistent

y

xO(0, �1)

y

xO

(4, 1)

Chapter 3 Assessment Answer KeyForm 2DPage 157 Page 158

consistent and independent

(�3, �2), (0, �2),(�3, 4)

(�1, �3), (�4, 0),(0, 3), (2, 3)

max: f(1, 7) � 10;min: f(�5, 7) � �8

� � 0, v � 4,v � � � 15

w � 2h, t � h � 4,3w � 2h � 5t � 136


1.

2.

3.

4.

5.

6.

7.

8.9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: (6, 2)

$90

��15

�, �1, 21�

��12

�, ��23

�, �5�

y

xO

y

xO

(1.5, �2)

��3, �14

��

no solution

(�2, 6)

inconsistent

y

xO (1, �1)

y

xO

(�3, �4)

Chapter 3 Assessment Answer KeyForm 3Page 159 Page 160

An

swer

s

(�6, 3), (2, 3),(2, �4), (1, �4)

(�2, 4), (5, �3),(5, �4), (1, �4),(�2, 2)

(�3, �4), (�3, 1),(0, 4), (3, 2)

max: f(3, 2) � 8;

min: f(�3, 1) � ��129�

60a � 150m � 900,a � 3, m � 4

5 ads and 4 commercial minutes;124,000 people

s � 3a, m � a � 10,15s � 40a � 5m � 2650



Chapter 3 Assessment Answer KeyPage 161, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs are mostly accurate and appropriate.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofsolving systems of linear equations by graphing, bysubstitution, and by elimination; solving systems ofinequalities by graphing; solving problems using linearprogramming; and solving systems in three variables.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs are mostly accurate.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Graphs may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of solvingsystems of linear equations by graphing, by substitution,and by elimination; solving systems of inequalities bygraphing; solving problems using linear programming; andsolving systems in three variables.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

Chapter 3 Assessment Answer KeyPage 161, Open-Ended Assessment

Sample Answers


1. Let J � Janet’s age, K � Kim’s age, andS � Sue’s age.

J2 � (K � S)2 � 400 and K � S � J � 10J2 � (J � 10)2 � 400J2 � J2 � 20J � 100 � 400

20J � 500J � 25

Thus, K � S � 25 � 10 or 15 and J � (K � S) � 25 � 15 or 40. Therefore,(J � K � S)2 � 402 or 1600.

2a. x � 0y � 0

y � ��43�x � 8

2b. Sample answer: Evaluate f(x, y) at eachof the vertices of the region.

3. Student responses should indicate thatthe solution of a system of equations isan ordered pair representing the pointat which the graphs of the equationsintersect. Solutions should include thefact that a system may have no solution,indicating that the graphs of the linesare parallel and that a system may havean infinite number of solutions,indicating that the equations representthe same line.

4. Student responses should indicate thatit is impossible for a system to be both inconsistent (a system of non-intersecting lines) and dependent (a system of two representations of the same line).

5. Students should demonstrate anunderstanding that, if the graphs of thecost and revenue functions are parallel,the cost and revenue have the same rateof change. This means that the companywill never break even on the productionand sale of these products and, therefore,never make a profit. The owner maydecide to increase the price charged forthe product, may look for ways to cutcosts, may put the company resourcesinto the production of other products, ormay even decide to close up shop.

6. Students should indicate that a systemof two linear inequalities has nosolution when there are no orderedpairs which satisfy both inequalities.Graphically, this means that the shadedregions which represent the graphs ofthe inequalities do not intersect.Students’ graphs should show twoparallel lines with shading everywherebut between them.Sample answer: x � �3, x � 2

In addition to the scoring rubric found on page A22, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


Chapter 3 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 3–1 and 3–2) Quiz (Lesson 3–4)

Page 162 Page 163 Page 164

1. inconsistent system

2. dependent system

3. ordered triple

4. substitution method

5. constraints

6. system ofinequalities

7. elimination method

8. consistent systemindependent system

9. system of equations

10. linear programming

11. Sample answer: In alinear programmingproblem, the regionthat is theintersection of thegraphs of theconstraints is calledthe feasible region.

12. Sample answer:When a system oflinear inequalities isgraphed, if thesolutions form aregion that is not apolygonal region,then we say theregion is anunbounded region.

1.

2.

inconsistent

3.

4.

5.

Quiz (Lesson 3–3)

Page 163

1.

2.

3.

4.

1.

2.

vertices:(�4, 2), (4, 4), (0, �1);max: f(0, �1) � 5;min: f(4, 4) � �16

3.

4.

Quiz (Lesson 3–5)

Page 164

1.

2.

3.

4.

5.

(1, �2, 4)

(2, 3, �1)

(�1, 2, �4)

y

xO

(4, 4)

(�4, 2)

(0, �1)

(0, 0), (0, 4), (2, 0)

y

xO

y

xO

A

(�3, �2)

y

xO

y

xO (3, 1)

no solution;inconsistent system

(1, �2), (2, 0), (3, �1),(3, �2)

max: f(1, 6) � 11;min: f(1, �1) � �3

j � 0, p � 0,j � 2p � 20,4j � 2p � 32

4 jackets and 8 pairs of pants;

$120

x � y � z � 155,x � 9y, y � 3z

135 cars, 15 vans,5 trucks


1.

2.

3.

4.

5.

6.

7.

8.

9.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

max: f(1, 3) � 0;min: f(6, �2) � �20

(1, 3), (6, �2), (1, �2)

consistent andindependent

�387�

No, because x has anexponent other than 1.

No, because the variableappears in adenominator.

�8a3 � 4a � 5

�1�2�3�4�5�6 0 1 2

{x � x � �5 or x � 2} or (�, �5) � (2, �)

y

xO

y

xO

(1, 3)

y

xO

(�2, 2)

A

A

C

C

Chapter 3 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 165 Page 166

An

swer

s


(�2, 3), (�2, 6),(0, 6), (4, 0)


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17.

18.

19. DCBA

DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

8

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

9

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 0 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 3 Assessment Answer KeyStandardized Test Practice

Page 167 Page 168

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Chapter 3 Resource Masters - KTL MATH CLASSES©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter 3 Resource Masters The Fast FileChapter Resource system