Chapter 3 Resource Masters - Math Classmrsstuckeysmathclass.weebly.com/uploads/6/1/1/2/61126813/...©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter

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

O

x

y

O

x$4

x$2

x$2

x$3

x

y

O

(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 " 2x # 2y " 4 consistent 2x # " !1 consistent x # y " 6 consistent and independent and dependent and independent

x

y

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y

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y$2

x

y

Ox

y

Ox

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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

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O

x

y

Ox

y

O

x

y

O

x

y

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y

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x

y

O

1$3

x

y

Ox

y

Ox

y

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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)


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

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y

O

1$2

x

y

Ox

y

Ox

y

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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.


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$

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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)

Solve each system of equations by using elimination.

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$


Solve each system of equations by using substitution.

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 !! , "Solve each system of equations by using elimination.

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


Less

on

3-2

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

NAME ______________________________________________ DATE ____________ PERIOD _____

3-33-3


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on

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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

x

y

Ox

y

O

x

y

O

x

y

Ox

y

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x

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x$4

x$3

x$2

x

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y

Ox

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x$3

x$3

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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

8$13

55$13

16$13

8$13

8$13

8$13

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

3$5

11$5

26$5

13$5

13$5

13$5

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$

3$

1$2

1$3

1$2

1$2

1$5

3$5

3$13

8$13

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)

x

y

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Ox

y

O 2

2

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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

Stu

den

t Ti

cket

s

1000 200 300 400

400

350

300

250

200

150

100

50

3$

x

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1$2

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 _____

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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.

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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#

3. T

S R

U

P V

A

E FD K

B C

G JH

E F

H G

XW

Y

K

M

J LPQ

A B

D C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

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Study Guide and InterventionLinear Programming

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

1$

7$

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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

x

y

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3$2

1$2

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 lb2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $16

per 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

2!3

1%4

3%4

(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

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

17$

34$

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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 _____

3-53-5


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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 _____

3-53-5

Reading to Learn MathematicsSolving Systems of Equations in Three Variables

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5


Less

on

3-5

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

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Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

3-53-5

©G

lencoe/McG

raw-H

illA

2G

lencoe Algebra 2

Answ

ers(Lesson 3-1)

Study Guide and InterventionSolving Systems of Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1


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 ✓



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)

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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 " 2x # 2y " 4 consistent 2x # " !1 consistent x # y " 6 consistent and independent and dependent and independent

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Solving Systems of Equations by Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

ExercisesExercises

ExampleExample

©G

lencoe/McG

raw-H

illA

3G

lencoe Algebra 2

Answers

Answ

ers(Lesson 3-1)

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)


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

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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)


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

(15, 20,500)

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Practice (Average)

Solving Systems of Equations By Graphing

NAME ______________________________________________ DATE ____________ PERIOD _____

3-13-1

© Glencoe/McGraw-Hill A4 Glencoe Algebra 2

Answers (Lesson 3-1)

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Sub

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#12

y"

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infin

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2 $ 34 $ 3

1 $ 4

1 $ 2

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eth

od t

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lve

the

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uat

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s.2x

!4y

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263x

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____

____

____

____

____

____

____

____

____

____

____

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____

____

____

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3-2

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Exam

ple1

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Mul

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ly b

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x"

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lace

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!7

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solv

e fo

r y.

2x!

4y"

!26

2(!

7) !

4y"

!26

!14

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26!

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y"

3T

he s

olut

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7,3)

.

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3 an

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the

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64m

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3)(4

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2 "(!

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5 $ 2

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Skil

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Sol

ving

Sys

tem

s of

Equ

atio

ns A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

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____

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____

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solu

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solu

tion

SPO

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For

Exe

rcis

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8 an

d 2

9,u

se t

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foll

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form

atio

n.

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2

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ach

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cks:

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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 (A

vera

ge)

Sol

ving

Sys

tem

s of

Equ

atio

ns A

lgeb

raic

ally

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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3-2

3-2


An

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Rea

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____

____

____

____

____

____

____

____

____

____

____

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____

____

____

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3-2

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trod

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the

top

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page

116

in y

our

text

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min

ute?

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13.

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din

g t

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Less

on

1.Su

ppos

e th

at y

ou a

re a

sked

to

solv

e th

e sy

stem

of

equa

tion

s 4x

!5y

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at t

he r

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by

the

subs

titu

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3x#

y"

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The

fir

st s

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quat

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var

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ter

ms

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mak

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asy

as p

ossi

ble,

whi

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quat

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wou

ld y

ou s

olve

for

whi

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aria

ble?

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ampl

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in t

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pos

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atio

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Hel

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u R

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3.T

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hav

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not

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val

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the

var

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ou f

ind

the

valu

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the

oth

er v

aria

ble

by s

ubst

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lue

you

have

foun

din

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orig

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atio

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Usi

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ght

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Thi

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the

dis

tanc

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om t

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the

pla

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trai

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m a

noth

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line

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y x

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.Wha

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e th

e co

ordi

nate

s of

the

poi

nt a

t w

hich

th

e cr

ash

occu

rred

?

Solv

e th

e sy

stem

of

equa

tion

s!y

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!1

.x

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3y"

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x!

1) "

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itute

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1 fo

r y.

x#

3x!

3 "

214x

"24

x"

6

x#

3y"

216

#3y

"21

Sub

stitu

te 6

for

x.

3y"

15y

"5

The

coo

rdin

ates

of

the

cras

h ar

e (6

,5).

Sol

ve t

he

foll

owin

g.

1.T

he li

nes

of s

ight

to

a fo

rest

fir

e ar

e as

fol

low

s.

Fro

m R

ange

r St

atio

n A

:3x

#y

"9

Fro

m R

ange

r St

atio

n B

:2x

#3y

"13

Fin

d th

e co

ordi

nate

s of

the

fir

e.(2

,3)

2.A

n ai

rpla

ne is

tra

velin

g al

ong

the

line

x!

y"

!1

whe

n it

see

s an

othe

r ai

rpla

ne t

rave

ling

alon

g th

e lin

e 5x

#3y

"19

.If

they

con

tinu

e al

ong

the

sam

e lin

es,a

t w

hat

poin

t w

ill t

heir

flig

ht p

aths

cro

ss?

(2,3

)

3.T

wo

min

e sh

afts

are

dug

alo

ng t

he p

aths

of

the

follo

win

g eq

uati

ons.

x!

y"

1400

2x#

y"

1300

If t

he s

haft

s m

eet

at a

dep

th o

f 20

0 fe

et,w

hat

are

the

coor

dina

tes

of t

he

poin

t at

whi

ch t

hey

mee

t?(9

00,!

500)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-2

3-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Sys

tem

s of

Ineq

ualit

ies

by G

raph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

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lenc

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Lesson 3-3

Gra

ph S

yste

ms

of In

equa

litie

sTo

sol

ve a

sys

tem

of i

nequ

alit

ies,

grap

h th

e in

equa

litie

sin

the

sam

e co

ordi

nate

pla

ne.T

he s

olut

ion

set

is r

epre

sent

ed b

y th

e in

ters

ecti

on o

f the

gra

phs.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by g

rap

hin

g.y

&2x

!1

and

y'

#2

The

sol

utio

n of

y%

2x!

1 is

Reg

ions

1 a

nd 2

.

The

sol

utio

n of

y&

#2

is R

egio

ns 1

and

3.

The

inte

rsec

tion

of

thes

e re

gion

s is

Reg

ion

1,w

hich

is

the

solu

tion

set

of

the

syst

em o

f in

equa

litie

s.

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.x

!y

%2

2.3x

!2y

%!

13.

y

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x#

2y'

1x

#4y

'!

12x

&2

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35.

y(

#2

6.y

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y(

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(!

2x#

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(3x

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7.x

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2x!

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2x

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x#

4y(

!4

x

y

Ox

y

O

x

y

O

x

y

Ox

y

O

x

y

O

x $ 4x $ 3

x $ 2

x

y

Ox

y

Ox

y

O

x $ 3x $ 3

x

y

O

Regi

on 3

Regi

on 1

Regi

on 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

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Fin

d V

erti

ces

of

a Po

lyg

on

al R

egio

nSo

met

imes

the

gra

ph o

f a

syst

em o

fin

equa

litie

s fo

rms

a bo

unde

d re

gion

.You

can

fin

d th

e ve

rtic

es o

f th

e re

gion

by

a co

mbi

nati

onof

the

met

hods

use

d ea

rlie

r in

thi

s ch

apte

r:gr

aphi

ng,s

ubst

itut

ion,

and/

or e

limin

atio

n.

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

figu

re f

orm

ed b

y 5x

#4y

(20

,y(

2x#

3,an

d x

!3y

(4.

Gra

ph t

he b

ound

ary

of e

ach

ineq

ualit

y.T

he in

ters

ecti

ons

of t

he

boun

dary

line

s ar

e th

e ve

rtic

es o

f a

tria

ngle

.T

he v

erte

x (4

,0)

can

be d

eter

min

ed f

rom

the

gra

ph.T

o fi

nd t

heco

ordi

nate

s of

the

sec

ond

and

thir

d ve

rtic

es,s

olve

the

tw

o sy

stem

s of

equ

atio

ns

y"

2x#

3an

dy

"2x

#3

5x#

4y"

20x

!3y

"4

x

y

O

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Sys

tem

s of

Ineq

ualit

ies

by G

raph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

For

the

firs

t sy

stem

of

equa

tion

s,re

wri

te t

he f

irst

equa

tion

in s

tand

ard

form

as

2x!

y"

!3.

The

nm

ulti

ply

that

equ

atio

n by

4 a

nd a

dd t

o th

e se

cond

equa

tion

.2x

!y

"!

3M

ultip

ly b

y 4.

8x!

4y"

!12

5x#

4y"

20(#

)5x

#4y

"20

13x

"8

x"

The

n su

bsti

tute

x"

in o

ne o

f th

e or

igin

al e

quat

ions

an

d so

lve

for

y.

2 "#!

y"

!3

!y

"!

3

y"

The

coo

rdin

ates

of

the

seco

nd v

erte

x ar

e "

,4#.

3 $ 138 $ 13

55 $ 13

16 $ 138 $ 13

8 $ 13

8 $ 13

For

the

seco

nd s

yste

m o

feq

uati

ons,

use

subs

titu

tion

.Su

bsti

tute

2x

#3

for

yin

the

seco

nd e

quat

ion

to g

etx

!3(

2x#

3) "

4x

!6x

!9

"4

!5x

"13

x"

!

The

n su

bsti

tute

x"

!in

the

firs

t eq

uati

on t

o so

lve

for

y.

y"

2 "!##

3

y"

!#

3

y"

!

The

coo

rdin

ates

of

the

thir

d

vert

ex a

re "!

2,!

2#.

1 $ 53 $ 5

11 $ 526 $ 5

13 $ 5

13 $ 5

13 $ 5

Exer

cises

Exer

cises

Thu

s,th

e co

ordi

nate

s of

the

thr

ee v

erti

ces

are

(4,0

),"

,4#,a

nd "!

2,!

2#.

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

figu

re f

orm

ed b

y ea

ch s

yste

m o

fin

equ

alit

ies.

1.y

%!

3x#

72.

x&

!3

3.y

(!

x#

3

y(

x(2

,1),

(!4,

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3,4)

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2,2)

,

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(3,!

2)y

&x

!1

(!3,

!4)

,(3,

2)y

(3x

#10

!!3

,!"

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

Sol

ving

Sys

tem

s of

Ineq

ualit

ies

by G

raph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

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ra 2

Lesson 3-3

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.x

(1

2.x

'!

33.

x%

2y

'!

1y

'!

3x

&4

no s

olut

ion

4.y

'x

5.y

(!

4x6.

x!

y'

!1

y'

!x

y'

3x!

23x

!y

%4

7.y

(3

8.y

(!

2x#

39.

x!

y%

4x

#2y

(12

y%

x!

22x

#y

(4

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

figu

re f

orm

ed b

y ea

ch s

yste

m o

fin

equ

alit

ies.

10.y

(0

11.y

(3

!x

12.x

'!

2x

(0

y'

3y

&x

!2

y'

!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)

x

y

Ox

y

Ox

y

O2

2

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O

©G

lenc

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4G

lenc

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Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.y

#1

(!

x2.

x&

!2

3.y

%2x

!3

y'

12y

'3x

#6

y%

!x

#2

4.x

#y

&!

25.

y

%1

6.3y

&4x

3x!

y'

!2

y(

x!

12x

!3y

&!

6

Fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

figu

re f

orm

ed b

y ea

ch s

yste

m o

fin

equ

alit

ies.

7.y

'1

!x

8.x

!y

%2

9.y

'2x

!2

y%

x!

1x

#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)

DR

AM

AF

or E

xerc

ises

10

and

11,

use

th

e fo

llow

ing

info

rmat

ion

.T

he d

ram

a cl

ub is

sel

ling

tick

ets

to it

s pl

ay.A

n ad

ult

tick

et

cost

s $1

5 an

d a

stud

ent

tick

et c

osts

$11

.The

aud

itor

ium

will

seat

300

tic

ket-

hold

ers.

The

dra

ma

club

wan

ts t

o co

llect

at

leas

t $3

630

from

tic

ket

sale

s.

10.W

rite

and

gra

ph a

sys

tem

of

four

ineq

ualit

ies

that

de

scri

be h

ow m

any

of e

ach

type

of

tick

et t

he c

lub

mus

t se

ll to

mee

ts it

s go

al.

x)

0,y

)0,

x#

y&

300,

15x

#11

y)

3630

11.L

ist

thre

e di

ffer

ent

com

bina

tion

s of

tic

kets

sol

d th

atsa

tisf

y th

e in

equa

litie

s.S

ampl

e an

swer

:25

0 ad

ult

and

50 s

tude

nt,2

00 a

dult

and

100

stud

ent,

145

adul

t an

d 14

8 st

uden

t

Play

Tic

kets

Ad

ult

Tic

kets

Student Tickets

100

020

030

040

0

400

350

300

250

200

150

100 50

3 $ 2

x

y O

y

xO

y

xO

x

y

Ox

y

Ox

y

O

1 $ 2

Pra

ctic

e (A

vera

ge)

Sol

ving

Sys

tem

s of

Ineq

ualit

ies

by G

raph

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3



Rea

din

g t

o L

earn

Math

emati

csS

olvi

ng S

yste

ms

of In

equa

litie

s by

Gra

phin

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3

©G

lenc

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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 bl

ood

pre

ssu

re i

s in

a

nor

mal

ran

ge?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 3-

3 at

the

top

of

page

123

in y

our

text

book

.

Sati

sh is

37

year

s ol

d.H

e ha

s a

bloo

d pr

essu

re r

eadi

ng o

f 13

5/99

.Is

his

bloo

d pr

essu

re w

ithi

n th

e no

rmal

ran

ge?

Exp

lain

.S

ampl

e an

swer

:No;

his

syst

olic

pre

ssur

e is

nor

mal

,but

his

dias

tolic

pre

ssur

e is

too

hig

h.It

shou

ld b

e be

twee

n 60

and

90.

Rea

din

g t

he

Less

on

1.W

itho

ut a

ctua

lly d

raw

ing

the

grap

h,de

scri

be t

he b

ound

ary

lines

x

(

3fo

r th

e sy

stem

of

ineq

ualit

ies

show

n at

the

rig

ht.

y

%5

Two

dash

ed v

ertic

al li

nes

(x"

3 an

d x

"!

3) a

nd t

wo

solid

hor

izon

tal

lines

(y

"!

5 an

d y

"5)

2.T

hink

abo

ut h

ow t

he g

raph

wou

ld lo

ok f

or t

he s

yste

m g

iven

abo

ve.W

hat

will

be

the

shap

e of

the

sha

ded

regi

on?

(It

is n

ot n

eces

sary

to

draw

the

gra

ph.S

ee if

you

can

imag

ine

it w

itho

ut d

raw

ing

anyt

hing

.If

this

is d

iffi

cult

to

do,m

ake

a ro

ugh

sket

ch t

ohe

lp y

ou a

nsw

er t

he q

uest

ion.

)a

rect

angl

e

3.W

hich

sys

tem

of

ineq

ualit

ies

mat

ches

the

gra

ph s

how

n at

the

rig

ht?

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.To

gra

ph a

sys

tem

of

ineq

ualit

ies,

you

mus

t gr

aph

two

or m

ore

boun

dary

line

s.W

hen

you

grap

h ea

ch o

f th

ese

lines

,how

can

the

ineq

ualit

y sy

mbo

ls h

elp

you

rem

embe

rw

heth

er t

o us

e a

dash

ed o

r so

lid li

ne?

Use

a d

ashe

d lin

e if

the

ineq

ualit

y sy

mbo

l is

'or

(,b

ecau

se t

hese

sym

bols

do

not

incl

ude

equa

lity

and

the

dash

ed li

ne r

emin

ds y

ou t

hat

the

line

itsel

f is

not

incl

uded

in t

he g

raph

.Use

a s

olid

line

if t

he s

ymbo

lis

)or

&,b

ecau

se t

hese

sym

bols

incl

ude

equa

lity

and

tell

you

that

the

line

itsel

f is

incl

uded

in t

he g

raph

.

x

y

O

©G

lenc

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cGra

w-H

ill13

6G

lenc

oe A

lgeb

ra 2

Trac

ing

Str

ateg

yT

ry t

o tr

ace

over

eac

h of

the

fig

ures

bel

ow w

itho

ut t

raci

ng t

he s

ame

segm

ent

twic

e.

The

fig

ure

at t

he le

ft c

anno

t be

tra

ced,

but

the

one

at t

he r

ight

can

.The

rul

e is

tha

t a

figu

re is

tra

ceab

le if

it h

as n

o m

ore

than

tw

o po

ints

whe

re a

n od

d nu

mbe

r of

se

gmen

ts m

eet.

The

fig

ure

at t

he le

ft h

as t

hree

seg

men

ts m

eeti

ng a

t ea

ch o

f th

e fo

ur c

orne

rs.H

owev

er,t

he f

igur

e at

the

rig

ht h

as o

nly

two

poin

ts,L

and

Q,w

here

an

odd

num

ber

of s

egm

ents

mee

t.

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#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#

3.

not

trac

eabl

e

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

ATE

____

____

____

PE

RIO

D__

___

3-3

3-3


An

swer

s


Stu

dy

Gu

ide

an

d I

nte

rven

tion

Line

ar P

rogr

amm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill13

7G

lenc

oe A

lgeb

ra 2

Lesson 3-4

Max

imu

m a

nd

Min

imu

m V

alu

esW

hen

a sy

stem

of

linea

r in

equa

litie

s pr

oduc

es a

boun

ded

poly

gona

l reg

ion,

the

max

imum

or m

inim

umva

lue

of a

rel

ated

fun

ctio

n w

ill o

ccur

at a

ver

tex

of t

he r

egio

n.

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

unde

d re

gion

.Gra

ph

the

ineq

ualit

ies.

The

pol

ygon

for

med

is a

qua

drila

tera

l wit

h ve

rtic

es a

t (0

,4),

(2,4

),(5

,1),

and

(!1,

!2)

.Use

the

tab

le t

o fi

nd t

hem

axim

um a

nd m

inim

um v

alue

s of

f(x

,y) "

3x#

2y.

The

max

imum

val

ue is

17

at (

5,1)

.The

min

imum

val

ue 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)

;!

,$";m

ax:2

5;m

in:6

m

in;$

5m

ax:1

0;m

in:$

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

©G

lenc

oe/M

cGra

w-H

ill13

8G

lenc

oe A

lgeb

ra 2

Rea

l-W

orl

d P

rob

lem

sW

hen

solv

ing

lin

ear

pro

gram

min

gpr

oble

ms,

use

the

follo

win

g pr

oced

ure.

1.D

efin

e va

riab

les.

2.W

rite

a s

yste

m o

f in

equa

litie

s.3.

Gra

ph t

he s

yste

m o

f in

equa

litie

s.4.

Fin

d th

e co

ordi

nate

s of

the

ver

tice

s of

the

fea

sibl

e re

gion

.5.

Wri

te a

n ex

pres

sion

to

be m

axim

ized

or

min

imiz

ed.

6.Su

bsti

tute

the

coo

rdin

ates

of

the

vert

ices

in t

he e

xpre

ssio

n.7.

Sele

ct t

he g

reat

est

or le

ast

resu

lt t

o an

swer

the

pro

blem

.

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 c

olor

A a

nd

col

or B

.Eac

hga

llon

of

colo

r A

req

uir

es 4

un

its

of y

ello

w d

ye a

nd

1 u

nit

of

gree

n d

ye.E

ach

gall

on o

f co

lor

B r

equ

ires

1 u

nit

of

yell

ow d

ye a

nd

6 u

nit

s of

gre

en d

ye.F

ind

th

em

axim

um

nu

mbe

r of

gal

lon

s h

e ca

n m

ix.

Step

1D

efin

e th

e va

riab

les.

x"

the

num

ber

of g

allo

ns o

f co

lor

A m

ade

y"

the

num

ber

of g

allo

ns o

f co

lor

B m

ade

Step

2W

rite

a s

yste

m o

f in

equa

litie

s.Si

nce

the

num

ber

of g

allo

ns m

ade

cann

ot b

ene

gati

ve,x

$0

and

y$

0.T

here

are

32

unit

s of

yel

low

dye

;eac

h ga

llon

of

colo

r A

req

uire

s 4

unit

s,an

d ea

ch g

allo

n of

co

lor

B r

equi

res

1 un

it.

So 4

x#

y%

32.

Sim

ilarl

y fo

r th

e gr

een

dye,

x#

6y%

54.

Step

s 3

and

4G

raph

the

sys

tem

of

ineq

ualit

ies

and

find

the

coo

rdin

ates

of

the

vert

ices

of

the

feas

ible

reg

ion.

The

ver

tice

s of

the

fea

sibl

e re

gion

are

(0,

0),(

0,9)

,(6,

8),a

nd (

8,0)

.St

eps

5–7

Fin

d th

e m

axim

um n

umbe

r of

gal

lons

,x

#y,

that

he

can

mak

e.T

he m

axim

um n

umbe

r of

gal

lons

the

pai

nter

can

mak

e is

14,

6 ga

llons

of

colo

r A

and

8 g

allo

ns o

f co

lor

B.

1.FO

OD

A d

elic

ates

sen

has

8 po

unds

of

plai

n sa

usag

e an

d 10

pou

nds

of g

arlic

-fla

vore

dsa

usag

e.T

he d

eli w

ants

to

mak

e as

muc

h br

atw

urst

as

poss

ible

.Eac

h po

und

of

brat

wur

st r

equi

res

poun

d of

pla

in s

ausa

ge a

nd

poun

d of

gar

lic-f

lavo

red

saus

age.

Fin

d th

e m

axim

um n

umbe

r of

pou

nds

of b

ratw

urst

tha

t ca

n be

mad

e.10

lb2.

MA

NU

FAC

TUR

ING

Mac

hine

A c

an p

rodu

ce 3

0 st

eeri

ng w

heel

s pe

r ho

ur a

t a

cost

of

$16

per

hour

.Mac

hine

B c

an p

rodu

ce 4

0 st

eeri

ng w

heel

s pe

r ho

ur a

t a

cost

of

$22

per

hour

.A

t le

ast

360

stee

ring

whe

els

mus

t be

mad

e in

eac

h 8-

hour

shi

ft.W

hat

is t

he le

ast

cost

invo

lved

in m

akin

g 36

0 st

eeri

ng w

heel

s in

one

shi

ft?

$194

2 & 3

1 & 43 & 4

(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

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Line

ar P

rogr

amm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Line

ar P

rogr

amm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

©G

lenc

oe/M

cGra

w-H

ill13

9G

lenc

oe A

lgeb

ra 2

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

.no

max

.,m

in.:

20m

ax.:

22,m

in.:

!2

7.M

AN

UFA

CTU

RIN

GA

bac

kpac

k m

anuf

actu

rer

prod

uces

an

inte

rnal

fra

me

pack

and

an

exte

rnal

fra

me

pack

.Let

xre

pres

ent

the

num

ber

of in

tern

al f

ram

e pa

cks

prod

uced

inon

e ho

ur a

nd le

t y

repr

esen

t th

e nu

mbe

r of

ext

erna

l fra

me

pack

s pr

oduc

ed in

one

hou

r.T

hen

the

ineq

ualit

ies

x#

3y%

18,2

x#

y%

16,x

'0,

and

y'

0 de

scri

be t

he c

onst

rain

tsfo

r m

anuf

actu

ring

bot

h pa

cks.

Use

the

pro

fit

func

tion

f(x)

"50

x#

80y

and

the

cons

trai

nts

give

n to

det

erm

ine

the

max

imum

pro

fit

for

man

ufac

turi

ng b

oth

back

pack

sfo

r th

e gi

ven

cons

trai

nts.

$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

)

©G

lenc

oe/M

cGra

w-H

ill14

0G

lenc

oe A

lgeb

ra 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.

1.2x

!4

%y

2.3x

!y

%7

3.x

'0

!2x

!4

%y

2x!

y'

3y

'0

y%

2y

'x

!3

y%

6f(

x,y)

"!

2x#

yf(

x,y)

"x

!4y

y%

!3x

#15

f(x,

y) "

3x#

y

max

.:8,

min

.:!

4m

ax.:

12,m

in.:

!16

max

.:15

,min

.:0

4.x

%0

5.y

%3x

#6

6.2x

#3y

'6

y%

04y

#3x

%3

2x!

y%

24x

#y

'!

7x

'!

2x

'0

f(x,

y) "

!x

!4y

f(x,

y) "

!x

#3y

y'

0f(

x,y)

"x

#4y

#3

max

.:28

,min

.:0

max

.:,n

o m

in.

no m

ax.,

min

.:

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 in

an

hour

.In

a w

ork

shif

t of

no

mor

e th

an 8

hou

rs,t

he w

orke

r m

ust

form

at

leas

t 40

vas

es.

7.L

et s

repr

esen

t th

e ho

urs

form

ing

sim

ple

vase

s an

d e

the

hour

s fo

rmin

g el

abor

ate

vase

s.W

rite

a s

yste

m o

f in

equa

litie

s in

volv

ing

the

tim

e sp

ent

on e

ach

type

of

vase

.s

)0,

e)

0,s

#e

&8,

8s#

2e)

408.

If t

he g

lass

blo

wer

mak

es a

pro

fit

of $

30 p

er h

our

wor

ked

on t

he s

impl

e va

ses

and

$35

per

hour

wor

ked

on t

he e

labo

rate

vas

es,w

rite

a f

unct

ion

for

the

tota

l pro

fit

on t

he v

ases

.f(

s,e)

"30

s#

35e

9.F

ind

the

num

ber

of h

ours

the

wor

ker

shou

ld s

pend

on

each

typ

e of

vas

e to

max

imiz

epr

ofit

.Wha

t is

tha

t pr

ofit

?4

h on

eac

h;$2

60

17 $ 234 $ 5

x

y

(3 – 2, 1)

( 0, 2

) O

(–7 – 5, 9 – 5)

( –2,

0)

x

y O

x

y ( 0, –

7)

(–7 – 4, 0)

( 0, 0

)O

( 3, 6

) ( 5, 0

)

( 0, 6

)

( 0, 0

)x

y

O

( 2, –

1)

( 4, 5

)

( 0, –

3)

x

y

O

( –3,

2)

( 3, 2

)

( 0, –

4)

x

y

O

Pra

ctic

e (A

vera

ge)

Line

ar P

rogr

amm

ing

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csLi

near

Pro

gram

min

g

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

ucti

on t

o L

esso

n 3-

4 at

the

top

of

page

129

in y

our

text

book

.

Nam

e tw

o or

mor

e fa

cts

that

indi

cate

tha

t yo

u w

ill n

eed

to u

se in

equa

litie

sto

mod

el t

his

situ

atio

n.S

ampl

e an

swer

:The

buo

y te

nder

can

car

ry u

p to

8 n

ew b

uoys

.Th

ere

seem

s to

be

a lim

it of

24

hour

s on

the

tim

e th

e cr

ew h

asat

sea

.The

cre

w w

ill w

ant

to r

epai

r or

rep

lace

the

max

imum

num

ber

of b

uoys

pos

sibl

e.

Rea

din

g t

he

Less

on

1.C

ompl

ete

each

sen

tenc

e.

a.W

hen

you

find

the

fea

sibl

e re

gion

for

a li

near

pro

gram

min

g pr

oble

m,y

ou a

re s

olvi

ng

a sy

stem

of

linea

r ca

lled

.The

poi

nts

in t

he f

easi

ble

regi

on a

re

of t

he s

yste

m.

b.T

he c

orne

r po

ints

of

a po

lygo

nal r

egio

n ar

e th

e of

the

fe

asib

le r

egio

n.

2.A

pol

ygon

al r

egio

n al

way

s ta

kes

up o

nly

a lim

ited

par

t of

the

coo

rdin

ate

plan

e.O

ne w

ayto

thi

nk o

f th

is is

to

imag

ine

a ci

rcle

or

rect

angl

e th

at t

he r

egio

n w

ould

fit

insi

de.I

n th

eca

se o

f a

poly

gona

l reg

ion,

you

can

alw

ays

find

a c

ircl

e or

rec

tang

le t

hat

is la

rge

enou

ghto

con

tain

all

the

poin

ts o

f th

e po

lygo

nal r

egio

n.W

hat

wor

d is

use

d to

des

crib

e a

regi

onth

at c

an b

e en

clos

ed in

thi

s w

ay?

Wha

t w

ord

is u

sed

to d

escr

ibe

a re

gion

tha

t is

too

larg

eto

be

encl

osed

in t

his

way

?bo

unde

d;un

boun

ded

3.H

ow d

o yo

u fi

nd t

he c

orne

r po

ints

of

the

poly

gona

l reg

ion

in a

line

ar p

rogr

amm

ing

prob

lem

?Yo

u so

lve

a sy

stem

of

two

linea

r eq

uatio

ns.

4.W

hat

are

som

e ev

eryd

ay m

eani

ngs

of t

he w

ord

feas

ible

that

rem

ind

you

of t

hem

athe

mat

ical

mea

ning

of

the

term

feas

ible

reg

ion?

Sam

ple

answ

er:p

ossi

ble

or a

chie

vabl

e

Hel

pin

g Y

ou

Rem

emb

er

5.L

ook

up t

he w

ord

cons

trai

ntin

a d

icti

onar

y.If

mor

e th

an o

ne d

efin

itio

n is

giv

en,c

hoos

eth

e on

e th

at s

eem

s cl

oses

t to

the

idea

of

a co

nstr

aint

in a

line

ar p

rogr

amm

ing

prob

lem

.H

ow c

an t

his

defi

niti

on h

elp

you

to r

emem

ber

the

mea

ning

of c

onst

rain

tas

it is

use

d in

this

less

on?

Sam

ple

answ

er:A

con

stra

int

is a

res

tric

tion

or li

mita

tion.

The

cons

trai

nts

in a

line

ar p

rogr

amm

ing

prob

lem

are

res

tric

tions

on

the

vari

able

s th

at t

rans

late

into

ineq

ualit

y st

atem

ents

.

vert

ices

solu

tions

cons

trai

nts

ineq

ualit

ies

©G

lenc

oe/M

cGra

w-H

ill14

2G

lenc

oe A

lgeb

ra 2

Com

pute

r C

ircu

its a

nd L

ogic

Com

pute

rs o

pera

te a

ccor

ding

to

the

law

s of

logi

c.T

he c

ircu

its

of a

com

pute

r ca

n be

des

crib

ed u

sing

logi

c.

1.W

ith s

witc

h A

open

, no

curr

ent f

low

s. T

he v

alue

0 is

ass

igne

d to

an

open

sw

itch.

2.W

ith s

witc

h A

clos

ed, c

urre

nt fl

ows.

The

val

ue 1

is a

ssig

ned

to a

clo

sed

switc

h.

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

this

circ

uit,

curr

ent f

low

s if

eith

er A

or B

is c

lose

d. T

his

circ

uit

can

be d

escr

ibed

by

the

disj

unct

ion,

A#

B.

Tru

th t

able

s ar

e us

ed t

o de

scri

be t

he f

low

of

curr

ent

in a

cir

cuit

.T

he t

able

at

the

left

des

crib

es t

he c

ircu

it in

dia

gram

4.A

ccor

ding

to

the

tab

le,t

he o

nly

tim

e cu

rren

t do

es n

ot f

low

thr

ough

the

ci

rcui

t is

whe

n bo

th s

wit

ches

A a

nd B

are

ope

n.

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

follo

win

g ci

rcui

t.

B C

A

A CDB

A BDC

A

BC

AB

C

A A A B

AB

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-4

3-4

AB

C(B

#C

)A

(B #

C)

00

00

00

01

10

01

11

00

10

10

10

11

11

10

11

11

11

11

00

00

AB

A#

B

00

00

11

10

11

11



Stu

dy

Gu

ide

and I

nte

rven

tion

Sol

ving

Sys

tem

s of

Equ

atio

ns in

Thr

ee V

aria

bles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

©G

lenc

oe/M

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 m

etho

ds u

sed

for

solv

ing

syst

ems

of li

near

equa

tion

s in

tw

o va

riab

les

to s

olve

sys

tem

s of

equ

atio

ns in

thr

ee v

aria

bles

.A s

yste

m o

fth

ree

equa

tion

s in

thr

ee v

aria

bles

can

hav

e a

uniq

ue s

olut

ion,

infi

nite

ly m

any

solu

tion

s,or

no s

olut

ion.

A s

olut

ion

is a

n or

der

ed t

rip

le.

Sol

ve t

his

sys

tem

of

equ

atio

ns.

3x#

y!

z"

!6

2x!

y#

2z"

84x

#y

!3z

"!

21

Step

1U

se e

limin

atio

n to

mak

e a

syst

em o

f tw

o eq

uati

ons

in t

wo

vari

able

s.3x

#y

!z

"!

6F

irst e

quat

ion

2x!

y#

2z"

8S

econ

d eq

uatio

n(#

) 2x

!y

#2z

"8

Sec

ond

equa

tion

(#) 4

x#

y!

3z"

!21

Thi

rd e

quat

ion

5x#

z"

2A

dd to

elim

inat

e y.

6x!

z

"!

13A

dd to

elim

inat

e y.

Step

2So

lve

the

syst

em o

f tw

o eq

uati

ons.

5x#

z"

2(#

) 6x

!z

"!

1311

x"

!11

Add

to e

limin

ate

z.x

"!

1D

ivid

e bo

th s

ides

by

11.

Subs

titu

te !

1 fo

r x

in o

ne o

f th

e eq

uati

ons

wit

h tw

o va

riab

les

and

solv

e fo

r z.

5x#

z"

2E

quat

ion

with

two

varia

bles

5(!

1) #

z"

2R

epla

ce x

with

!1.

!5

#z

"2

Mul

tiply

.z

"7

Add

5 to

bot

h si

des.

The

res

ult

so f

ar is

x"

!1

and

z"

7.

Step

3Su

bsti

tute

!1

for

xan

d 7

for

zin

one

of

the

orig

inal

equ

atio

ns w

ith

thre

e va

riab

les.

3x#

y!

z"

!6

Orig

inal

equ

atio

n w

ith th

ree

varia

bles

3(!

1) #

y!

7 "

!6

Rep

lace

xw

ith !

1 an

d z

with

7.

!3

#y

!7

"!

6M

ultip

ly.

y"

4S

impl

ify.

The

sol

utio

n is

(!1,

4,7)

.

Sol

ve e

ach

sys

tem

of

equ

atio

ns.

1.2x

#3y

!z

"0

2.2x

!y

#4z

"11

3.x

!2y

#z

"8

x!

2y!

4z"

14x

#2y

!6z

"!

112x

#y

!z

"0

3x#

y!

8z"

173x

!2y

!10

z"

113x

!6y

#3z

"24

(4,!

3,!

1)!2,

!5,

"in

finite

ly m

any

solu

tions

4.3x

!y

!z

"5

5.2x

!4y

!z

"10

6.x

!6y

#4z

"2

3x#

2y!

z"

114x

!8y

!2z

"16

2x#

4y!

8z"

166x

!3y

#2z

"!

123x

#y

#z

"12

x!

2y"

5

!,2

,!5 "

no s

olut

ion

!6,,!

"1 $ 4

1 $ 22 $ 3

1 $ 2

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill14

4G

lenc

oe A

lgeb

ra 2

Real

-Wor

ld P

robl

ems

Th

e L

ared

o S

por

ts S

hop

sol

d 1

0 ba

lls,

3 ba

ts,a

nd

2 b

ases

for

$99

on

Mon

day

.On

Tu

esd

ay t

hey

sol

d 4

bal

ls,8

bat

s,an

d 2

bas

es f

or $

78.O

n W

edn

esd

ayth

ey s

old

2 b

alls

,3 b

ats,

and

1 b

ase

for

$33.

60.W

hat

are

th

e p

rice

s of

1 b

all,

1 ba

t,an

d 1

bas

e?

Fir

st d

efin

e th

e va

riab

les.

x"

pric

e of

1 b

all

y"

pric

e of

1 b

atz

"pr

ice

of 1

bas

e

Tra

nsla

te t

he in

form

atio

n in

the

pro

blem

into

thr

ee e

quat

ions

.

10x

#3y

#2z

"99

4x#

8y#

2z"

782x

#3y

#z

"33

.60

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sol

ving

Sys

tem

s of

Equ

atio

ns in

Thr

ee V

aria

bles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

Exam

ple

Exam

ple

Subt

ract

the

sec

ond

equa

tion

fro

m t

he f

irst

equa

tion

to

elim

inat

e z.

10x

#3y

#2z

"99

(!)

4x#

8y#

2z"

786x

!5y

"21

Mul

tipl

y th

e th

ird

equa

tion

by

2 an

dsu

btra

ct f

rom

the

sec

ond

equa

tion

.4x

#8y

#2z

"78

(!) 4

x#

6y#

2z"

67.2

02y

"10

.80

y"

5.40

Subs

titu

te 5

.40

for

yin

the

equ

atio

n 6x

!5y

"21

.

6x!

5(5.

40)

"21

6x"

48x

"8

Subs

titu

te 8

for

xan

d 5.

40 f

or y

in o

ne o

fth

e or

igin

al e

quat

ions

to

solv

e fo

r z.

10x

#3y

#2z

"99

10(8

) #

3(5.

40)

#2z

"99

80 #

16.2

0 #

2z"

992z

"2.

80z

"1.

40So

a b

all c

osts

$8,

a ba

t $5

.40,

and

a ba

se $

1.40

.

1.FI

TNES

S TR

AIN

ING

Car

ly is

tra

inin

g fo

r a

tria

thlo

n.In

her

tra

inin

g ro

utin

e ea

ch w

eek,

she

runs

7 t

imes

as

far

as s

he s

wim

s,an

d sh

e bi

kes

3 ti

mes

as

far

as s

he r

uns.

One

wee

ksh

e tr

aine

d a

tota

l of

232

mile

s.H

ow f

ar d

id s

he r

un t

hat

wee

k?56

mile

s

2.EN

TERT

AIN

MEN

TA

t th

e ar

cade

,Rya

n,Sa

ra,a

nd T

im p

laye

d vi

deo

raci

ng g

ames

,pin

ball,

and

air

hock

ey.R

yan

spen

t $6

for

6 r

acin

g ga

mes

,2 p

inba

ll ga

mes

,and

1 g

ame

of a

irho

ckey

.Sar

a sp

ent

$12

for

3 ra

cing

gam

es,4

pin

ball

gam

es,a

nd 5

gam

es o

f air

hoc

key.

Tim

spen

t $1

2.25

for

2 ra

cing

gam

es,7

pin

ball

gam

es,a

nd 4

gam

es o

f air

hoc

key.

How

muc

h di

dea

ch o

f th

e ga

mes

cos

t?R

acin

g ga

me:

$0.5

0;pi

nbal

l:$0

.75;

air

hock

ey:$

1.50

3.FO

OD

A n

atur

al f

ood

stor

e m

akes

its

own

bran

d of

tra

il m

ix o

ut o

f dr

ied

appl

es,r

aisi

ns,

and

pean

uts.

One

pou

nd o

f th

e m

ixtu

re c

osts

$3.

18.I

t co

ntai

ns t

wic

e as

muc

h pe

anut

sby

wei

ght

as a

pple

s.O

ne p

ound

of

drie

d ap

ples

cos

ts $

4.48

,a p

ound

of

rais

ins

$2.4

0,an

da

poun

d of

pea

nuts

$3.

44.H

ow m

any

ounc

es o

f ea

ch in

gred

ient

are

con

tain

ed in

1 p

ound

of t

he t

rail

mix

?3

oz o

f ap

ples

,7 o

z of

rai

sins

,6 o

z of

pea

nuts

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Sol

ving

Sys

tem

s of

Equ

atio

ns in

Thr

ee V

aria

bles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

3-5

3-5

©G

lenc

oe/M

cGra

w-H

ill14

5G

lenc

oe A

lgeb

ra 2

Lesson 3-5

Sol

ve e

ach

sys

tem

of

equ

atio

ns.

1.2a

#c

"!

10(5

,!5,

!20

)2.

x#

y#

z"

3(0

,2,1

)b

!c

"15

13x

#2z

"2

a!

2b#

c"

!5

!x

!5z

"!

5

3.2x

#5y

#2z

"6

(!3,

2,1)

4.x

#4y

!z

"1

no s

olut

ion

5x!

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

"!

2x

#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"

!6

13x

#2z

"2

r#

s#

4t"

3

9.x

!y

#3z

"3

no s

olut

ion

10.5

m#

3n#

p"

4(!

2,3,

5)!

2x#

2y!

6z"

63m

#2n

"0

y!

5z"

!3

2m!

n#

3p"

8

11.2

x#

2y#

2z"

!2

infin

itely

man

y12

.x#

2y!

z"

4(1

,2,1

)2x

#3y

#2z

"4

3x!

y#

2z"

3x

#y

#z

"!

1!

x#

3y#

z"

6

13.3

x!

2y#

z"

1(5

,7,0

)14

.3x

!5y

#2z

"!

12in

finite

ly m

any

!x

#y

!z

"2

x#

4y!

2z"

85x

#2y

#10

z"

39!

3x#

5y!

2z"

12

15.2

x#

y#

3z"

!2

(!1,

3,!

1)16

.2x

!4y

#3z

"0

(3,0

,!2)

x!

y!

z"

!3

x!

2y!

5z"

133x

!2y

#3z

"!

125x

#3y

!2z

"19

17.!

2x#

y#

2z"

2(1

,!2,

3)18

.x!

2y#

2z"

!1

infin

itely

man

y3x

#3y

#z

"0

x#

2y!

z"

6x

#y

#z

"2

!3x

#6y

!6z

"3

19.T

he s

um o

f th

ree

num

bers

is 1

8.T

he s

um o

f th

e fi

rst

and

seco

nd n

umbe

rs is

15,

and

the

firs

t nu

mbe

r is

3 t

imes

the

thi

rd n

umbe

r.F

ind

the

num

bers

.9,

6,3

©G

lenc

oe/M

cGra

w-H

ill14

6G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

sys

tem

of

equ

atio

ns.

1.2x

!y

#2z

"15

2.x

!4y

#3z

"!

273.

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5 at

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atch

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infi

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at t

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h,an

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are

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less

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3-5

3-5

Documents

Chapter 3 Resource Masters - Math Classmrsstuckeysmathclass.weebly.com/uploads/6/1/1/2/61126813/...©Glencoe/McGraw-Hill iv Glencoe Algebra 2 Teacher’s Guide to Using the Chapter