Chapter 7 Resource Masters - Math Problem Solvingjaeproblemsolving.weebly.com/uploads/5/1/9/6/51966985/algebra_1... · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide

Chapter 7Resource Masters

CONSUMABLE WORKBOOKS Many of the worksheets contained in the ChapterResource Masters booklets are available as consumable workbooks in bothEnglish and Spanish.

Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7Reading to Learn Mathematics Workbook 0-07-861060-5

ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

StudentWorks™ This CD-ROM includes the entire Student Edition text alongwith the English workbooks listed above.

TeacherWorks™ All of the materials found in this booklet are included for view-ing and printing in the Glencoe Algebra 1 TeacherWorks CD-ROM.

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, teachers, and families without charge; and be used solely in conjunction with Glencoe Algebra 1. 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: 0-07-827731-0 Glencoe Algebra 1Chapter 7 Resource Masters

3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 1

Contents

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

Lesson 7-1Study Guide and Intervention . . . . . . . . 403–404Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 405Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 406Reading to Learn Mathematics . . . . . . . . . . 407Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 408





Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 433–434Chapter 7 Test, Form 2A . . . . . . . . . . . 435–436Chapter 7 Test, Form 2B . . . . . . . . . . . 437–438Chapter 7 Test, Form 2C . . . . . . . . . . . 439–440Chapter 7 Test, Form 2D . . . . . . . . . . . 441–442Chapter 7 Test, Form 3 . . . . . . . . . . . . 443–444Chapter 7 Open-Ended Assessment . . . . . . 445Chapter 7 Vocabulary Test/Review . . . . . . . 446Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 447Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 448Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 449Chapter 7 Cumulative Review . . . . . . . . . . . 450Chapter 7 Standardized Test Practice . . 451–452Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 453–454

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

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A27

© Glencoe/McGraw-Hill iv Glencoe Algebra 1

Teacher’s Guide to Using theChapter 7 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. 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 1 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 7-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 1 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 1

Assessment OptionsThe assessment masters in the Chapter 7Resources 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 1. 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 404–405. 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 _____

77

© Glencoe/McGraw-Hill vii Glencoe Algebra 1

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.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

consistent

kuhn·SIHS·tuhnt

dependent

elimination

ih·LIH·muh·NAY·shuhn

independent

inconsistent

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 1

Vocabulary Term Found on Page Definition/Description/Example

substitution

SUHB·stuh·TOO·shuhn

system of equations

system of inequalities

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

77

Study Guide and InterventionGraphing Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

© Glencoe/McGraw-Hill 403 Glencoe Algebra 1

Less

on

7-1

Number of Solutions Two or more linear equations involving the same variables forma system of equations. A solution of the system of equations is an ordered pair of numbersthat satisfies both equations. The table below summarizes information about systems oflinear equations.

Graph of a System intersecting lines same line parallel lines

Number of Solutions exactly one solution infinitely many solutions no solution

Terminologyconsistent and consistent and

inconsistent independent dependent

Use the graph at the right to determine whether the system has no solution, one solution, orinfinitely many solutions.

a. y � �x � 2y � x � 1Since the graphs of y � �x � 2 and y � x � 1 intersect,there is one solution.

b. y � �x � 23x � 3y � �3Since the graphs of y � �x � 2 and 3x � 3y � �3 are parallel, there are no solutions.

c. 3x � 3y � �3y � �x � 1Since the graphs of 3x � 3y � �3 and y � �x � 1 coincide,there are infinitely many solutions.

Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.

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

3. y � �x � 3 4. 2x � 2y � �62x � 2y � 4 3x � y � 3

x

y

O

y � �x � 3

3x � y � 3

2x � 2y � �6

2x � 2y � 4

y � x � 1

x

y

O

y � x � 1

y � �x � 1

3x � 3y � �3

y � �x � 2

x

y

Ox

y

Ox

y

O

ExampleExample

ExercisesExercises


Solve by Graphing One method of solving a system of equations is to graph theequations on the same coordinate plane.

Graph each system of equations. Then determine whether thesystem has no solution, one solution, or infinitely many solutions. If the system hasone solution, name it.

a. x � y � 2x � y � 4The graphs intersect. Therefore, there is one solution. The point (3, �1) seems to lie on both lines. Check this estimate by replacing x with 3 and y with �1 in each equation.

x � y � 23 � (�1) � 2 ✓

x � y � 43 � (�1) � 3 � 1 or 4 ✓The solution is (3, �1).

b. y � 2x � 12y � 4x � 2The graphs coincide. Therefore there are infinitely many solutions.

Graph each system of equations. Then determine whether the system has nosolution, one solution, or infinitely many solutions. If the system has one solution,name it.

1. y � �2 2. x � 2 3. y � x3x � y � �1 2x � y � 1 x � y � 3

4. 2x � y � 6 5. 3x � 2y � 6 6. 2y � �4x � 42x � y � �2 3x � 2y � �4 y � �2x � 2

y � �2x � 22y � �4x � 4

x

y

O

3x � 2y � �4

3x � 2y � 6

x

y

O

2x � y � 6

2x � y � �2

(1, 4)

x

y

O

x � y � 3

y � 12x(2, 1)

x

y

O2x � y � 1

x � 2

(2, –3)

x

y

O

3x � y � �1

y � �2

(–1, –2)x

y

O

1�2

x

y

O

y � 2x � 1 2y � 4x � 2

x

y

O(3, –1)

x � y � 4

x � y � 2

Study Guide and Intervention (continued)

Graphing Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

ExampleExample

ExercisesExercises

Skills PracticeGraphing Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1


1. y � x � 1 2. x � y � �4y � �x � 1 y � x � 4

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


5. 2x � y � 1 6. x � 1 7. 3x � y � �3y � �3 2x � y � 4 3x � y � 3

8. y � x � 2 9. x � 3y � �3 10. y � x � �1x � y � �2 x � 3y � �3 x � y � 3

11. x � y � 3 12. x � 2y � 4 13. y � 2x � 3

x � 2y � 3 y � � x � 2 3y � 6x � 6

y � 2x � 3

3y � 6x � 6

x

y

O

y � �1–2x � 2

x � 2y � 4

x

y

O

x � 2y � 3

(3, 0)

x � y � 3

x

y

O

1�2

x � y � 3

(2, 1)

y � x � �1 x

y

O

x � 3y � �3

(–3, 0)

x � 3y � �3

x

y

O

y � x � 2

x � y � �2

x

y

O

3x � y � 3

3x � y � �3

x

y

O

(1, 2)

x � 1

2x � y � 4x

y

O

(–1, –3)

2x � y � 1

y � �3x

y

O

x

y

Oy � �x � 1

x � y � �4 2x � 2y � 2

y � 2x � 3

y � x � 1

y � x � 4



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

3. x � 3y � 3 4. x � 3y � 3x � y � �3 2x � y � �3


5. 3x � y � �2 6. y � 2x � 3 7. x � 2y � 33x � y � 0 4x � 2y � 6 3x � y � �5

BUSINESS For Exercises 8 and 9, use the following information.Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 inoperating costs per week plus $0.50 to produce each treat.He plans to sell each treat for $1.50.

8. Graph the system of equations y � 0.5x � 20 and y � 1.5x to represent the situation.

9. How many treats does Nick need to sell per week tobreak even?

SALES For Exercises 10–12, use the following information.A used book store also started selling used CDs and videos.In the first week, the store sold 40 used CDs and videos, at$4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $180.00

10. Write a system of equations to represent the situation.

11. Graph the system of equations.

12. How many CDs and videos did the store sell in the firstweek?

4c � 6v � 180

c � v � 40

(30, 10)

y � 1.5x

y � 0.5x � 20(20, 30)

Sales ($)

Dog Treats

Co

st (

$)

5 1510 20 25 30 35 40 450

40

35

30

25

20

15

10

5

3x � y � �5

x � 2y � 3(–1, 2)

4x � 2y � 6

y � 2x � 3

x

y

O3x � y � 0

3x � y � �2

x

y

O

x

y

O

x � y � �3

x � 3y � 3 2x � y � �3

4x � 2y � �6

x � y � 3

Practice Graphing Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Reading to Learn MathematicsGraphing Systems of Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1

Pre-Activity How can you use graphs to compare the sales of two products?

Read the introduction to Lesson 7-1 at the top of page 369 in your textbook.

• What is meant by the term linear function?

• What does it mean to say that two lines intersect?

Reading the Lesson

1. Each figure shows the graph of a system of two equations. Write the letter of the figuresthat illustrate each statement.

A. B.

C. D.

a. A system of two linear equations can have an infinite number of solutions.

b. A system of equations is consistent if there is at least one ordered pair that satisfiesboth equations.

c. If two graphs are parallel, there are no ordered pairs that satisfy both equations.

d. If a system of equations has exactly one solution, it is independent.

e. If a system of equations has an infinite number of solutions, it is dependent.

Helping You Remember

2. Describe how you can solve a system of equations by graphing.

x

y

Ox

y

O

x

y

Ox

y

O


Graphing a TripThe distance formula, d � rt, is used to solve many types of problems. If you graph an equation such as d � 50t, the graph is a model for a car going at 50 mi/h. The time the car travels is t;the distance in miles the car covers is d. The slope of the line is the speed.

Suppose you drive to a nearby town and return. You average 50 mi/h on the trip out but only 25 mi/h on the trip home. The round trip takes 5 hours. How far away is the town?

The graph at the right represents your trip. Notice that the return trip is shown with a negative slope because you are driving in the opposite direction.

Solve each problem.

1. Estimate the answer to the problem in the above example. About how far away is the town?

2. Graph this trip and solve the problem. An airplane has enough fuel for 3 hours of safe flying. On the trip out the pilotaverages 200 mi/h flying against a headwind. On the trip back,the pilot averages 250 mi/h. How long a trip out can the pilot make?

3. Graph this trip and solve the 4. Graph this trip and solve the problem. Youproblem. You drive to a town drive at an average speed of 50 mi/h to a100 miles away. On the trip out you discount shopping plaza, spend 2 hours average 25 mi/h. On the trip back you shopping, and then return at an average average 50 mi/h. How many hours do speed of 25 mi/h. The entire trip takes you spend driving? 8 hours. How far away is the shopping plaza?

t

d

O

50

2t

d

O 2

50

t

d

O 1

100

t

d

O

slope is 50

slope is –25

2

50

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Study Guide and InterventionSubstitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

Substitution One method of solving systems of equations is substitution.

Use substitution tosolve the system of equations.y � 2x4x � y � �4

Substitute 2x for y in the secondequation.

4x � y � �4 Second equation

4x � 2x � �4 y � 2x

2x � �4 Combine like terms.

� Divide each side by 2.

x � �2 Simplify.

Use y � 2x to find the value of y.y � 2x First equation

y � 2(�2) x � �2

y � �4 Simplify.

The solution is (�2, �4).

�4�2

2x�2

Solve for one variable, thensubstitute.x � 3y � 72x � 4y � �6

Solve the first equation for x since the coefficientof x is 1.

x � 3y � 7 First equation

x � 3y � 3y � 7 � 3y Subtract 3y from each side.

x � 7 � 3y Simplify.

Find the value of y by substituting 7 � 3y for xin the second equation.

2x � 4y � �6 Second equation

2(7 � 3y) � 4y � �6 x � 7 � 3y

14 � 6y � 4y � �6 Distributive Property

14 � 10y � �6 Combine like terms.

14 � 10y � 14 � �6 � 14 Subtract 14 from each side.

�10y � �20 Simplify.

� Divide each side by �10.

y � 2 Simplify.

Use y � 2 to find the value of x.x � 7 � 3yx � 7 � 3(2)x � 1

The solution is (1, 2).

�20��10

�10y��10

Example 1Example 1 Example 2Example 2

ExercisesExercises

Use substitution to solve each system of equations. If the system does not haveexactly one solution, state whether it has no solution or infinitely many solutions.

1. y � 4x 2. x � 2y 3. x � 2y � 33x � y � 1 y � x � 2 x � 2y � 4

4. x � 2y � �1 5. c � 4d � 1 6. x � 2y � 03y � x � 4 2c � 8d � 2 3x � 4y � 4

7. 2b � 6a � 14 8. x � y � 16 9. y � �x � 33a � b � 7 2y � �2x � 2 2y � 2x � 4

10. x � 2y 11. x � 2y � �5 12. �0.2x � y � 0.50.25x � 0.5y � 10 x � 2y � �1 0.4x � y � 1.1


Real-World Problems Substitution can also be used to solve real-world problemsinvolving systems of equations. It may be helpful to use tables, charts, diagrams, or graphsto help you organize data.

CHEMISTRY How much of a 10% saline solution should be mixedwith a 20% saline solution to obtain 1000 milliliters of a 12% saline solution?

Let s � the number of milliliters of 10% saline solution.Let t � the number of milliliters of 20% saline solution.Use a table to organize the information.

10% saline 20% saline 12% saline

Total milliliters s t 1000

Milliliters of saline 0.10s 0.20t 0.12(1000)

Write a system of equations.s � t � 10000.10s � 0.20t � 0.12(1000)Use substitution to solve this system.

s � t � 1000 First equation

s � 1000 � t Solve for s.

0.10s � 0.20t � 0.12(1000) Second equation

0.10(1000 � t) � 0.20t � 0.12(1000) s � 1000 � t

100 � 0.10t � 0.20t � 0.12(1000) Distributive Property

100 � 0.10t � 0.12(1000) Combine like terms.

0.10t � 20 Simplify.

� Divide each side by 0.10.

t � 200 Simplify.

s � t � 1000 First equation

s � 200 � 1000 t � 200

s � 800 Solve for s.

800 milliliters of 10% solution and 200 milliliters of 20% solution should be used.

1. SPORTS At the end of the 2000-2001 football season, 31 Super Bowl games had beenplayed with the current two football leagues, the American Football Conference (AFC) andthe National Football Conference (NFC). The NFC won five more games than the AFC.How many games did each conference win? Source: New York Times Almanac

2. CHEMISTRY A lab needs to make 100 gallons of an 18% acid solution by mixing a 12%acid solution with a 20% solution. How many gallons of each solution are needed?

3. GEOMETRY The perimeter of a triangle is 24 inches. The longest side is 4 inches longerthan the shortest side, and the shortest side is three-fourths the length of the middleside. Find the length of each side of the triangle.

20�0.10

0.10t�0.10


Substitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

ExampleExample

ExercisesExercises

Skills PracticeSubstitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2


1. y � 4x 2. y � 2xx � y � 5 x � 3y � �14

3. y � 3x 4. x � �4y2x � y � 15 3x � 2y � 20

5. y � x � 1 6. x � y � 7x � y � 3 x � 8y � 2

7. y � 4x � 1 8. y � 3x � 8y � 2x � 5 5x � 2y � 5

9. 2x � 3y � 21 10. y � 5x � 8y � 3 � x 4x � 3y � 33

11. x � 2y � 13 12. x � 5y � 43x � 5y � 6 3x � 15y � �1

13. 3x � y � 4 14. x � 4y � 82x � 3y � �9 2x � 5y � 29

15. x � 5y � 10 16. 5x � 2y � 142x � 10y � 20 2x � y � 5

17. 2x � 5y � 38 18. x � 4y � 27x � 3y � �3 3x � y � �23

19. 2x � 2y � 7 20. 2.5x � y � �2x � 2y � �1 3x � 2y � 0



1. y � 6x 2. x � 3y 3. x � 2y � 72x � 3y � �20 3x � 5y � 12 x � y � 4

4. y � 2x � 2 5. y � 2x � 6 6. 3x � y � 12y � x � 2 2x � y � 2 y � �x � 2

7. x � 2y � 13 8. x � 2y � 3 9. x � 5y � 36�2x � 3y � �18 4x � 8y � 12 2x � y � �16

10. 2x � 3y � �24 11. x � 14y � 84 12. 0.3x � 0.2y � 0.5x � 6y � 18 2x � 7y � �7 x � 2y � �5

13. 0.5x � 4y � �1 14. 3x � 2y � 11 15. x � 2y � 121�2

Practice Substitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

x � 2.5y � 3.5 x � y � 4 x � 2y � 6

16. x � y � 3 17. 4x � 5y � �7 18. x � 3y � �4

2x � y � 25 y � 5x 2x � 6y � 5

EMPLOYMENT For Exercises 19–21, use the following information.Kenisha sells athletic shoes part-time at a department store. She can earn either $500 permonth plus a 4% commission on her total sales, or $400 per month plus a 5% commission ontotal sales.


20. What is the total price of the athletic shoes Kenisha needs to sell to earn the sameincome from each pay scale?

21. Which is the better offer?

MOVIE TICKETS For Exercises 22 and 23, use the following information.Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased8 tickets for $52.75.


23. How many adult tickets and student tickets were purchased?

1�3

1�2

Reading to Learn MathematicsSubstitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

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


• What is the system of equations?

• Based on the graph, are there 0, 1, or infinitely many solutions of thesystem?

Reading the Lesson

1. Describe how you would use substitution to solve each system of equations.

a. y � �2xx � 3y � 15

b. 3x � 2y � 12x � 2y

c. x � 2y � 72x � 8y � 8

d. �3x � 5y � 812x � y � 24

2. Jess solved a system of equations and her result was �8 � �8. All of her work wascorrect. Describe the graph of the system. Explain.

3. Miguel solved a system of equations and his result was 5 � �2. All of his work wascorrect. Describe the graph of the system. Explain.


4. What is usually the first step in solving a system of equations by substitution?


Equations of Lines and Planes in Intercept FormOne form that a linear equation may take is interceptform. The constants a and b are the x- and y-intercepts of the graph.

� � 1

In three-dimensional space, the equation of a plane takes a similar form.

� � � 1

Here, the constants a, b, and c are the points where theplane meets the x, y, and z-axes.

Solve each problem.

z�c

y�b

x�a

y�b

x�a

z

y

x

O

x–8 � y–7 � z–6 � 1

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

7-27-2

1. Graph the equation � � � 1.

5. Graph the equation � � � 1.

2. For the plane in Exercise 1, write anequation for the line where the planeintersects the xy-plane. Use interceptforms.

� � 1

3. Write an equation for the line wherethe plane intersects the xz-plane.

� � 1

4. Write an equation for the line wherethe plane intersects the yz-plane.

� � 1

6. Write an equation for the xy-plane.

z � 0

7. Write an equation for the yz-plane.

x � 0

8. Write an equation for a plane parallelto the xy-plane with a z-intercept of 2.

z � 2

9. Write an equation for a plane parallelto the yz-plane with an x-intercept of 23.

x � �3

z�1

y�2

z�1

x�3

y�2

x�3

z

y

x

O

z�2

y�4

x�1

z

y

x

O

z�1

y�2

x�3

Study Guide and InterventionElimination Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

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7-3

Elimination Using Addition In systems of equations in which the coefficients of thex or y terms are additive inverses, solve the system by adding the equations. Because one ofthe variables is eliminated, this method is called elimination.

Use addition to solve thesystem of equations.x � 3y � 73x � 3y � 9

Write the equations in column form and addto eliminate y.

x � 3y � 7(�) 3x � 3y � 9

4x � 16Solve for x.

�

x � 4Substitute 4 for x in either equation andsolve for y.

4 � 3y � 74 � 3y � 4 � 7 � 4

�3y � 3

�

y � �1The solution is (4, �1).

3��3

�3y��3

16�4

4x�4

The sum of two numbersis 70 and their difference is 24. Findthe numbers.

Let x represent one number and y representthe other number.

x � y � 70(�) x � y � 24

2x � 94

�

x � 47Substitute 47 for x in either equation.

47 � y � 7047 � y � 47 � 70 � 47

y � 23The numbers are 47 and 23.

94�2

2x�2


ExercisesExercises

Use elimination to solve each system of equations.

1. x � y � �4 2. 2m � 3n � 14 3. 3a � b � �9x � y � 2 m � 3n � �11 �3a � 2b � 0

4. �3x � 4y � �1 5. 3c � d � 4 6. �2x � 2y � 93x � y � �4 2c � d � 6 2x � y � �6

7. 2x � 2y � �2 8. 4x � 2y � �1 9. x � y � 23x � 2y � 12 �4x � 4y � �2 x � y � �3

10. 2x � 3y � 12 11. �0.2x � y � 0.5 12. 0.1x � 0.3y � 0.94x � 3y � 24 0.2x � 2y � 1.6 0.1x � 0.3y � 0.2

13. Rema is older than Ken. The difference of their ages is 12 and the sum of their ages is50. Find the age of each.

14. The sum of the digits of a two-digit number is 12. The difference of the digits is 2. Findthe number if the units digit is larger than the tens digit.


Elimination Using Subtraction In systems of equations where the coefficients of thex or y terms are the same, solve the system by subtracting the equations.

Use subtraction to solve the system of equations.2x � 3y � 115x � 3y � 14

2x � 3y � 11 Write the equations in column form and subtract.

(�) 5x � 3y � 14�3x � �3 Subtract the two equations. y is eliminated.


x � 1 Simplify.

2(1) � 3y � 11 Substitute 1 for x in either equation.

2 � 3y � 11 Simplify.

2 � 3y � 2 � 11 � 2 Subtract 2 from each side.

�3y � 9 Simplify.



The solution is (1, �3).


1. 6x � 5y � 4 2. 3m � 4n � �14 3. 3a � b � 16x � 7y � �20 3m � 2n � �2 a � b � 3

4. �3x � 4y � �23 5. c � 3d � 11 6. x � 2y � 6�3x � y � 2 2c � 3d � 16 x � y � 3

7. 2a � 3b � �13 8. 4x � 2y � 6 9. 5s � t � 62a � 2b � 7 4x � 4y � 10 5s � 2t � 3

10. 6x � 3y � 12 11. x � 2y � 3.5 12. 0.2x � y � 0.74x � 3y � 24 x � 3y � �9 0.2x � 2y � 1.2

13. The sum of two numbers is 70. One number is ten more than twice the other number.Find the numbers.

14. GEOMETRY Two angles are supplementary. The measure of one angle is 10° more thanthree times the other. Find the measure of each angle.

9��3

�3y��3

�3��3

�3x��3


Elimination Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

Skills PracticeElimination Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3


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

3. x � 4y � 11 4. �x � 3y � 6x � 6y � 11 x � 3y � 18

5. 3x � 4y � 19 6. x � 4y � �83x � 6y � 33 x � 4y � �8

7. 3a � 4b � 2 8. 3c � d � �14a � 4b � 12 �3c � d � 5

9. 2x � 3y � 9 10. x � y � 4�5x � 3y � 30 2x � y � �4

11. 3m � n � 26 12. 5x � y � �6�2m � n � �24 �x � y � 2

13. 6x � 2y � 32 14. 3x � 2y � �194x � 2y � 18 �3x � 5y � 25

15. 7m � 4n � 2 16. 2x � 5y � �287m � 2n � 8 4x � 5y � 4

17. The sum of two numbers is 28 and their difference is 4. What are the numbers?

18. Find the two numbers whose sum is 29 and whose difference is 15.


20. Find the two numbers whose sum is 54 and whose difference is 4.

21. Two times a number added to another number is 25. Three times the first number minusthe other number is 20. Find the numbers.



1. x � y � 1 2. p � q � �2 3. 4x � y � 23x � y � �9 p � q � 8 3x � y � 12

4. 2x � 5y � �3 5. 3x � 2y � �1 6. 5x � 3y � 222x � 2y � 6 4x � 2y � �6 5x � 2y � 2

7. 5x � 2y � 7 8. 3x � 9y � �12 9. �4c � 2d � �2�2x � 2y � �14 3x � 15y � �6 2c � 2d � �14

10. 2x � 6y � 6 11. 7x � 2y � 2 12. 4.25x � 1.28y � �9.22x � 3y � 24 7x � 2y � �30 x � 1.28y � 17.6

13. 2x � 4y � 10 14. 2.5x � y � 10.7 15. 6m � 8n � 3x � 4y � �2.5 2.5x � 2y � 12.9 2m � 8n � �3

16. 4a � b � 2 17. � x � y � �2 18. x � y � 84a � 3b � 10

x � y � 4 x � y � 19


20. Four times one number added to another number is 36. Three times the first numberminus the other number is 20. Find the numbers.

21. One number added to three times another number is 24. Five times the first numberadded to three times the other number is 36. Find the numbers.

22. LANGUAGES English is spoken as the first or primary language in 78 more countriesthan Farsi is spoken as the first language. Together, English and Farsi are spoken as afirst language in 130 countries. In how many countries is English spoken as the firstlanguage? In how many countries is Farsi spoken as the first language?

23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hatsfor $43.00. Tori bought two pairs of gloves and two hats for $30.00. What were the pricesfor the gloves and hats?

1�2

3�2

2�3

1�3

1�2

3�4

4�3

1�3

Practice Elimination Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

Reading to Learn MathematicsElimination Using Addition and Subtraction

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Pre-Activity How can you use a system of equations to solve problems aboutweather?


What fact explains why the variable d gets eliminated from the system ofequations?

Reading the Lesson

1. Write addition or subtraction to tell which operation it would be easiest to use toeliminate a variable of the system. Explain your choice.

System of Equations Operation Explanation

a. 3x � 5y � 12�3x � 2y � 6

b. 3x � 5y � 73x � 2y � 8

c. �x � 4y � 94x � 4y � 6

d. 5x � 7y � 178x � 7y � 9


2. Tell how you can decide whether to use addition or subtraction to eliminate a variable ina system of equations.


Rózsa PéterRózsa Péter (1905–1977) was a Hungarian mathematician dedicated toteaching others about mathematics. As professor of mathematics at ateachers’ college in Budapest, she wrote several mathematics textbooksand championed reforms in the teaching of mathematics. In 1945 shewrote Playing with Infinity: Mathematical Explorations and Excursions,a popular work in which she attempted to convey the spirit ofmathematics to the general public.

By far Péter’s greatest contribution to mathematics was her pioneeringresearch in the field of recursive function theory. When you evaluate afunction recursively, you begin with one initial value of x. Working fromthis single number, you can use the function to generate an entiresequence of numbers. For instance, here is how you use an initialvalue of x � 1 to evaluate the function f(x) � 3x recursively.

f(1) � 3(1) � 3 ↓

f(3) � 3(3) � 9 ↓

f(9) � 3(9) � 27 ↓

f(27) � 3(27) � 81 ↓

f(81) � 3(81) � 243 ↓

The first five numbers of the sequence generated by this function are 3, 9, 27, 81, and 243.

Write the first five numbers of the sequence generated by each function, using the given number as the initial value of x.

1. f(x) � 3x; x � 2 2. g(x) � x � 5; x � 1

3. f(x) � 2x � 1; x � �3 4. f(x) � x2; x � 2

5. h(x) � �x; x � 3 6. k(x) � ; x � 10 1�x

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

Study Guide and InterventionElimination Using Multiplication

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Elimination Using Multiplication Some systems of equations cannot be solvedsimply by adding or subtracting the equations. In such cases, one or both equations mustfirst be multiplied by a number before the system can be solved by elimination.

Use elimination to solvethe system of equations.x � 10y � 34x � 5y � 5

If you multiply the second equation by �2,you can eliminate the y terms.

x � 10y � 3(�) �8x � 10y � �10

�7x � �7

�

x � 1Substitute 1 for x in either equation.

1 � 10y � 31 � 10y � 1 � 3 � 1

10y � 2

�

y �

The solution is �1, �.1�5

1�5

2�10

10y�10

�7��7

�7x��7

Use elimination to solvethe system of equations.3x � 2y � �72x � 5y � 10

If you multiply the first equation by 2 andthe second equation by �3, you caneliminate the x terms.

6x � 4y � �14(�) �6x � 15y � �30

11y � �44

�

y � �4Substitute �4 for y in either equation.3x � 2(�4) � �7

3x � 8 � �73x � 8 �8 � �7 �8

3x � �15

�

x � �5The solution is (�5, �4).

�15�3

3x�3

�44�11

11y�11


ExercisesExercises


1. 2x � 3y � 6 2. 2m � 3n � 4 3. 3a � b � 2x � 2y � 5 �m � 2n � 5 a � 2b � 3

4. 4x � 5y � 6 5. 4c � 3d � 22 6. 3x � 4y � �46x � 7y � �20 2c � d � 10 x � 3y � �10

7. 4s � t � 9 8. 4a � 3b � �8 9. 2x � 2y � 55s � 2t � 8 2a � 2b � 3 4x � 4y � 10

10. 6x � 4y � �8 11. 4x � 2y � �5 12. 2x � y � 3.54x � 2y � �3 �2x � 4y � 1 �x � 2y � 2.5

13. GARDENING The length of Sally’s garden is 4 meters greater than 3 times the width.The perimeter of her garden is 72 meters. What are the dimensions of Sally’s garden?

14. Anita is 4 years older than Basilio. Three times Anita’s age added to six times Basilio’s

age is 36. How old are Anita and Basilio?

1�2


Determine the Best Method The methods to use for solving systems of linearequations are summarized in the table below.

Method The Best Time to Use

Graphing to estimate the solution, since graphing usually does not give an exact solution

Substitution if one of the variables in either equation has a coefficient of 1 or �1

Elimination Using Addition if one of the variables has opposite coefficients in the two equations

Elimination Using Subtraction if one of the variables has the same coefficient in the two equations

Elimination Using Multiplicationif none of the coefficients are 1 or �1 and neither of the variables can beeliminated by simply adding or subtracting the equations

Determine the best method to solve the system of equations. Thensolve the system.6x � 2y � 20�2x � 4y � �16

Since the coefficients of x will be additive inverses of each other if you multiply the secondequation by 3, use elimination.


Elimination Using Multiplication

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

ExampleExample

ExercisesExercises

6x � 2y � 20(�) �6x � 12y � �48 Multiply the second equation by 3.

14y � �28 Add the two equations. x is eliminated.



�28�14

14y�14

6x � 2(�2) � 20 Substitute �2 for y in

either equation.

6x � 4 � 20 Simplify.

6x � 4 � 4 � 20 � 4 Add 4 to each side.

6x � 24 Simplify.


x � 4 Simplify.

24�6

6x�6

The solution is (4, �2).

Determine the best method to solve each system of equations. Then solve the system.

1. x � 2y � 3 2. m � 6n � �8 3. a � b � 6x � y � 1 m � 2n � 8 a � 2b � 7

4. 4x � y � 15 5. 3c � d � 14 6. x � 2y � �9�x � 3y � �12 c � d � 2 y � 4x

7. 4x � 2y � 10 8. x � �2y 9. 2s � 3t � 42x � 2y � 5 4x � 4y � �10 3s � 2t � 24

10. 4a � 4b � �10 11. 4x � 10y � �6 12. 2x � y � 32a � 4b � �2 �2x � 10y � 2 �x � y � 0

Skills PracticeElimination Using Multiplication

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4


1. x � y � �9 2. 3x � 2y � �95x � 2y � 32 x � y � �13

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

5. 4x � 2y � �14 6. 2x � y � 03x � y � �8 5x � 3y � 2

7. 5x � 3y � �10 8. 2x � 3y � 143x � 5y � �6 3x � 4y � 4

9. 2x � 3y � 21 10. 3x � 2y � �265x � 2y � 25 4x � 5y � �4

11. 3x � 6y � �3 12. 5x � 2y � �32x � 4y � 30 3x � 3y � 9

13. Two times a number plus three times another number equals 13. The sum of the twonumbers is 7. What are the numbers?

14. Four times a number minus twice another number is �16. The sum of the two numbersis �1. Find the numbers.


15. 2x � 3y � 10 16. 8x � 7y � 185x � 2y � �8 3x � 7y � 26

17. y � 2x 18. 3x � y � 63x � 2y � 35 3x � y � 3

19. 3x � 4y � 17 20. y � 3x � 14x � 5y � 2 3x � y � �1



1. 2x � y � �1 2. 5x � 2y � �10 3. 7x � 4y � �43x � 2y � 1 3x � 6y � 66 5x � 8y � 28

4. 2x � 4y � �22 5. 3x � 2y � �9 6. 4x � 2y � 323x � 3y � 30 5x � 3y � 4 �3x � 5y � �11

7. 3x � 4y � 27 8. 0.5x � 0.5y � �2 9. 2x � y � �75x � 3y � 16 x � 0.25y � 6

x � y � 0

10. Eight times a number plus five times another number is �13. The sum of the twonumbers is 1. What are the numbers?

11. Two times a number plus three times another number equals 4. Three times the firstnumber plus four times the other number is 7. Find the numbers.

Determine the best method to solve each system of equations. Then solve thesystem.

12. 5x � 7y � 3 13. 7x � 2y � 2 14. �6x � 2y � 142x � 7y � �38 2x � 3y � �28 6x � 8y � �20

15. x � 2y � 6 16. 4x � 3y � �2 17. y � x1�2

1�2

3�4

Practice Elimination Using Multiplication

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

x � y � 3 4x � 3y � 3x � 2y � 9

18. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% inone year, and the other rose 9% in one year. If Gunther’s investment rose a total of $684in one year, how much did he invest in each mutual fund?

19. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. Thereturn trip took three hours. Find the rate at which Laura and Brent paddled the canoein still water.

20. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits arereversed, the new number is 45 more than the original number. Find the number.

5�2

1�2

Reading to Learn MathematicsElimination Using Multiplication

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Pre-Activity How can a manager use a system of equations to plan employee time?


Can the system of equations be solved by elimination with addition orsubtraction? Explain.

Reading the Lesson

1. Could elimination by multiplication be used to solve the system shown below? Explain.3x � 5y � 15�6x � 7y � 11

2. Tell whether it would be easiest to use substitution, elimination by addition, eliminationby subtraction, or elimination by multiplication to solve the system. Explain your choice.

System of Equations Solution Method Explanation

a. �3x � 4y � 23x � 2y � 10

b. x � 2y � 05x � 4y � 8

c. 6x � 5y � �182x � 10y � 27

d. �2x � 3y � 93x � 3y � 12


3. If you are going to solve a system by elimination, how do you decide whether you willneed to multiply one or both equations by a number?


George Washington Carver and Percy Julian In 1990, George Washington Carver and Percy Julian became the first AfricanAmericans elected to the National Inventors Hall of Fame. Carver (1864–1943)was an agricultural scientist known worldwide for developing hundreds of usesfor the peanut and the sweet potato. His work revitalized the economy of thesouthern United States because it was no longer dependent solely upon cotton.Julian (1898–1975) was a research chemist who became famous for inventinga method of making a synthetic cortisone from soybeans. His discovery hashad many medical applications, particularly in the treatment of arthritis.

There are dozens of other African American inventors whose accomplishmentsare not as well known. Their inventions range from common household itemslike the ironing board to complex devices that have revolutionizedmanufacturing. The exercises that follow will help you identify just a few ofthese inventors and their inventions.

Match the inventors with their inventions by matching each systemwith its solution. (Not all the solutions will be used.)

1. Sara Boone x � y � 2 A. (1, 4) automatic traffic signal x � y � 10

2. Sarah Goode x � 2 � y B. (4, �2) eggbeater 2y � x � 9

3. Frederick M. y � 2x � 6 C. (�2, 3) fire extinguisher Jones y � �x � 3

4. J. L. Love 2x � 3y � 8 D. (�5, 7) folding cabinet bed 2x � y � �8

5. T. J. Marshall y � 3x � 9 E. (6, �4) ironing board 2y � x � 4

6. Jan Matzeliger y � 4 � 2x F. (�2, 4) pencil sharpener 6x � 3y � 12

7. Garrett A. 3x � 2y � �5 G. (�3, 0) portable X-ray machine Morgan 3y � 4x � 8

8. Norbert Rillieux 3x � y � 12 H. (2, �3) player piano y � 3x � 15

I. no solution evaporating pan for refining sugar

J. infinitely lasting (shaping) many machine for solutions manufacturing shoes

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

Study Guide and InterventionGraphing Systems of Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Systems of Inequalities The solution of a system of inequalities is the set of allordered pairs that satisfy both inequalities. If you graph the inequalities in the samecoordinate plane, the solution is the region where the graphs overlap.

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

The solution includes the ordered pairs in the intersection of thegraphs. This region is shaded at the right. The graphs of y � x � 2and y � �2x � 1 are boundaries of this region. The graph of y � x � 2 is dashed and is not included in the graph of y � x � 2.

Solve the system of inequalities by graphing.x � y � 4x � y � �1

The graphs of x � y � 4 and x � y � �1 are parallel. Because the two regions have no points in common, the system of inequalities has no solution.

Solve each system of inequalities by graphing.

1. y � �1 2. y � �2x � 2 3. y � x � 1x � 0 y � x � 1 3x � 4y 12

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

5x � 2y � 6

y � �x � 1

x

y

O

y � 2x � 3

y � �1 � 2xx

y

O

2x � y � 1

x � y � �2

x

y

O

y � x � 1

3x � 4y � 12

x

y

O

y � x � 1

y � �2x � 2

x

y

O

x � 0

y � �1

x

y

O

x

y

O

x � y � �1

x � y � 4

x

y

O

y � x � 2

y � �2x � 1

Example 1Example 1

Example 2Example 2

ExercisesExercises


Real-World Problems In real-world problems, sometimes only whole numbers makesense for the solution, and often only positive values of x and y make sense.

BUSINESS AAA Gem Company produces necklaces and bracelets. In a 40-hour week, the company has 400 gems to use. A necklace requires 40 gems and a bracelet requires 10 gems. It takes 2 hours to produce anecklace and a bracelet requires one hour. How many of each type can be produced in a week?

Let n � the number of necklaces that will be produced and b � thenumber of bracelets that will be produced. Neither n or b can be anegative number, so the following system of inequalities represents the conditions of the problems.

n 0b 0b � 2n � 4010b � 40n � 400

The solution is the set ordered pairs in the intersection of the graphs. This region is shadedat the right. Only whole-number solutions, such as (5, 20) make sense in this problem.

For each exercise, graph the solution set. List three possible solutions to theproblem.

1. HEALTH Mr. Flowers is on a restricted 2. RECREATION Maria had $150 in gift diet that allows him to have between certificates to use at a record store. She 1600 and 2000 Calories per day. His bought fewer than 20 recordings. Each daily fat intake is restricted to between tape cost $5.95 and each CD cost $8.95.45 and 55 grams. What daily Calorie How many of each type of recording might and fat intakes are acceptable? she have bought?

t � c � 20

5.95t � 8.95c � 150

30

25

20

15

10

5

Tapes

Co

mp

act

Dis

cs

5 10 15 20 25 300

60

50

40

30

20

10

Calories

Fat

Gra

ms

1000 2000 30000

Necklaces

Bra

cele

ts

10 20 30 40 500

50

40

30

20

10

10b � 40n � 400

b � 2n � 40


Graphing Systems of Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

ExampleExample

ExercisesExercises

Skills PracticeGraphing Systems of Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5


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

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

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

Write a system of inequalities for each graph.

10. 11. 12.

x

y

Ox

y

Ox

y

O

y � x � 1

y � 2x � 4

x

y

O

y � 2x � 2

y � �x � 2

x

y

O

y � �x � 1y � x � 1

x

y

O

x � y � 2

y � x � 4x

y

Ox

y

O

y � x � 2x � 2

x

y

O

y � x � 3

y � �1

x

y

O

y � 2

x � �2x

y

O

x � �1

y � �3x

y

O



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

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

FITNESS For Exercises 7 and 8, use the following information.Diego started an exercise program in which each week he works out at the gym between 4.5 and 6 hours and walksbetween 9 and 12 miles.

7. Make a graph to show the number of hours Diego works out at the gym and the number of miles he walks per week.

8. List three possible combinations of working out and walking that meet Diego’s goals.

SOUVENIRS For Exercises 9 and 10, use the following information.Emily wants to buy turquoise stones on her trip to New Mexicoto give to at least 4 of her friends. The gift shop sells stones foreither $4 or $6 per stone. Emily has no more than $30 to spend.

9. Make a graph showing the numbers of each price of stoneEmily can purchase.

10. List three possible solutions.

Gym (hours)

Diego’s Routine

Wak

ing

(m

iles)

1 32 4 5 6 7 80

16

14

12

10

8

6

4

2

2x � y � 2

x � 2y � 2

y � x � 4

2x � y � 2y � 2x � 1

y � 2 � x

x � y � 1

x � 2y � 1 x

y

O

y � 2x � 3

y � x � 2

x

y

O

y � x � 2y � x

x

y

O

Practice Graphing Systems of Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Reading to Learn MathematicsGraphing Systems of Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Pre-Activity How can you use a system of inequalities to plan a sensible diet?


The green section on the graph represents a range of

Calories a day and grams of fat per day.

Reading the LessonWrite the inequality symbols that you need to get a system whose graph looks likethe one shown. Use �, �, �, or �.

1. 2.

y x � 2 y x � 2

y �2x � 1 y �2x � 1

3. 4.

y x � 2 y x � 2

y �2x � 1 y �2x � 1


5. Describe how you would explain the process of using a graph to solve a system ofinequalities to a friend who missed Lesson 7-5.

x

y

O

y � x � 2

y � �2x � 1

x

y

O

y � x � 2

y � �2x � 1

x

y

O

y � x � 2

y � �2x � 1

x

y

O

y � x � 2y � �2x � 1


Describing RegionsThe shaded region inside the triangle can be described with a system of three inequalities.

y � 2x � 1

y � x � 3

y � 29x � 31

Write systems of inequalities to describe each region. You may first need to divide a region into triangles or quadrilaterals.

1. 2.

3.

x

y

O

x

y

Ox

y

O

1�3 x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Chapter 7 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

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

Use the graph for Questions 1–4.

1. The ordered pair (3, 2) is the solution of which system of equations?

A. y � ��13�x � 3 B. y � ��

13�x � 3

y � �3x � 2 y � 2x

C. y � ��13�x � 3 D. y � �3x � 2 1.

y � 2x �4 y � 2x � 4

For Questions 2–4, find how many solutions exist for each system of equations.

A. no solution B. one solutionC. infinitely many solutions D. cannot be determined

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

5. When solving the system of equations, r � 4 � swhich expression could be substituted 3r � 2s � 15for r in the second equation? A. 4 � s B. 4 � r C. s � 4 D. �

4s� 5.

6. If x � 2 and 3x � y � 5, what is the value of y?A. 0 B. �1 C. 11 D. 10 6.

7. Use substitution to solve the system n � 3m �11of equations. 2m � 3n � 0A. (�2, 3) B. (�3, 2) C. (3, �2) D. (2, �3) 7.

8. Use elimination to solve the system x � y � 5of equations. x � y � 3A. (4, 1) B. (4, �1) C. (�4, 1) D. (�4, �1) 8.

9. Use elimination to solve the system x � 6y � 10of equations. x � 5y � 9 A. (1, 4) B. (4, 1) C. (�1, �4) D. (�4, �1) 9.

10. Use elimination to find the value of x in 2x � 2y � 10the solution for the system of equations. 2x � 3y � 5A. 1 B. 10 C. 4 D. �2 10.

11. To eliminate the variable y in the system of equations, 6x � 4y � 22multiply the second equation by which number? 2x � y � 1A. 3 B. 9 C. 22 D. 4 11.

12. Use elimination to solve the system 2x � 5y � 7of equations. 3x � 6y � 3A. (�9, 5) B. (5, �9) C. (�1, 1) D. (1, �1) 12.

77

2.

3.

y

xO

y � 2x � 0

y � 2x � 4

y ��3x � 2

y � 2x

y � � x � 313


Chapter 7 Test, Form 1 (continued)

For Questions 13 and 14, determine the best method to solve the system of equations.

A. substitution B. elimination using additionC. elimination using subtraction D. elimination using multiplication

13. 5x � 2y � 4 14. y � 3x � 12 13.2x � 2y � 8 2x � y � 16

15. Determine which ordered pair represents 14.

a solution of the system of inequalities graphed at the right.A. (2, 1) B. (�3, �1)C. (1, 3) D. (�2, 3) 15.

16. Which system of inequalities is represented by the graph at the right? A. y � 0 B. x � 0

y � x y � xC. x � 0 D. x � 0 16.

y � x y � x

17. An airport shuttle company owns sedans that have a maximum capacity of 3 passengers and vans that have a maximum capacity of 8 passengers.Their 12 vehicles have a combined maximum capacity of 61 passengers.How many vans does the company own?A. 5 B. 8 C. 12 D. 7 17.

18. Find the two numbers whose sum is 26 and whose difference is 12.A. 26 and 12 B. 19 and 7 C. 14 and 12 D. 31 and 19 18.

19. Yancy wrote two novels that together contain 580 pages. His longer novel has 160 pages more than his shorter novel. How many pages are contained in Yancy’s shorter novel?A. 290 B. 370 C. 210 D. 130 19.

20. The Foxtail Toy Company makes toy cars and toy dump trucks. They are scaled so that the door handles and wheels are interchangeable. The table gives the door handle and wheel requirements of each type of toy. Which system of inequalities represents the given information?

A. 4x � 2y � 28 B. 4x � 4y ≤ 28 C. 28x � 16y 10 D. 4x � 6y 28 20.4x � 6y � 16 6x � 2y 16 28x � 16y � 6 4x � 2y 16

Bonus Determine the best method to solve x � 3y � 0 B:the system of equations. Then solve 2x � 7y � 0the system.

NAME DATE PERIOD

77

Wheels Door Handles

Parts per Toy Car 4 4

Parts per Toy Dump truck 6 2

Total Available Parts per day 28 16

y

xO

Chapter 7 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm


Use the graph for Questions 1–4.For Questions 1 and 2, determine how many solutions exist for each system of equations.

A. no solutionB. one solutionC. infinitely many solutionsD. cannot be determined

1. y � 3x � 3 2. x � 2y � �1 1.3x � y � 2 2x � 3y � 0

3. The solution to which system of equations has an x value larger than 2?2.

A. x � 2y � �1 B. 3x � y � 2 C. y � 3x � 3 D. 2x � 3y � 0 3.y � 3x � 3 x � 2y � �1 2x � 3y � 0 x � 2y � �1

4. The solution to which system of equations has a positive y value?A. x � 2y � �1 B. 3x � y � 2 C. y � 3x � 3 D. 2x � 3y � 0 4.

y � 3x � 3 x � 2y � �1 2x � 3y � 0 x � 2y � �1

5. When solving the system of equations, x � 2y � 15which expression could be substituted 5x � y � 21for x in the second equation?

A. 15 � 2y B. 21 � 5x C. �15

2� x� D. �

215� y� 5.

6. If x � 2y � 3 and 4x � 5y � 9, what is the value of y?A. 2 B. 1 C. �1 D. �2 6.

7. Use elimination to solve the system x � 7y � 16 and 3x � 7y � 4 for x.A. 3 B. 4 C. 5 D. �6 7.

8. Use elimination to solve the system x � 5y � 20 and x � 3y � �4 for x.A. 5 B. �3 C. 10 D. �40 8.

9. Use elimination to solve the system 8x � 7y � 5 and 3x � 5y � 9 for y.A. �2 B. 8 C. �3 D. �1 9.

10. Use elimination to solve the system 4x � 6y � 10 and 2x � 5y � 1 for x.

A. 11 B. 5�12� C. �2 D. ��

12� 10.

11. The substitution method should be used to solve which system of equations?A. 4x � 3y � 6 B. 2x � 5y � 1 C. 6x � 2y � 1 D. y � 3x � 1 11.

5x � 3y � 2 2x � 3y � 4 3x � 4y � 7 2x � 4y � 5

12. The elimination method using multiplication should be used to solve which system of equations?A. x � 7y � 1 B. 3x � 2y � 1 C. x � y � 16 D. 3x � y � �15 12.

2x � y � 8 4x � 3y � 12 2x � y � 9 3x � 5y � 10

77

y

xO

2x � 3y � 0

x � 2y � �1

y � 3x � 3

y � 3x � 2

3x �y � 2


Chapter 7 Test, Form 2A (continued)

13. State which region in the graph is the solution of the system.y � �x � 1

y �23�x � 2

A. A B. BC. C D. D 13.

14. Which system of inequalities is represented by the graph?A. x � y � 1 B. y � x � 1

y � x � 2 x � y � 2C. y � x � 2 D. x � y � 1 14.

y � x � 1 x � y � 2

For Questions 15 and 16, solve the system and find values of y.

15. 3x � 5y � �35 A. 4 B. �45� C. �4 D. ��

45� 15.

2x � 5y � �30

16. 3x � 4y � �30 A. �6 B. 6 C. �12 D. 12 16.2x � 5y � 72

17. Two times one number added to three times a second number is 21.Five times the first number added to three times the second number is 30.What are the numbers?A. 21 and 30 B. 3 and 2 C. 6 and 1 D. 3 and 5 17.

18. Coffee Cafe makes 90 pounds of coffee that costs $6 per pound. The types of coffee used to make this mixture cost $7 per pound and $4 per pound.How many pounds of the $7-per-pound coffee should be used in this mixture?A. 30 lb B. 40 lb C. 50 lb D. 60 lb 18.

19. In 1999, there were 59,549 physicians specializing in Pediatrics in the United States and its possessions. The number of male physicians minus the number of female physicians in this category is 2551. How many female physicians were there that specialized in Pediatrics in the United States?A. 29,774 B. 28,499 C. 31,050 D. 27,223 19.

20. Laurie and Maya make get well cards and friendship cards. They have enough materials to make 50 cards. Laurie and Maya will make at least 25 friendship cards. They also want to make more than 10 get well cards.Which system of inequalities represents this information?A. 10 � g 25 B. 10 � f 50 C. 10 � g 50 D. g � f 50 20.

25 � f � 40 g � 25 50 25 f 50 f � 25g � 10

Bonus Where on the graph of 2x � 6y � 7 is the x-coordinate B:twice the y-coordinate?

NAME DATE PERIOD

77

y

xO

A C

B

D

y

xO

Chapter 7 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm


Use the graph for Questions 1–4.

For Questions 1 and 2, determine how many solutions exist for each system of equations.


1. 3x � 3y � �6 2. x � y � 0 1.y � x � 2 3x � y � 2

3. The solution to which system of equations has a positive x value?2.

A. x � y � 0 B. 3x � y � 2 C. x � y � 0 D. y � �x � 2 3.3x � y � 2 y � �x � 2 3x � 3y � �6 x � y � 0

4. The solution to which system of equations has a y value less than �1?A. x � y � 0 B. 3x � y � 2 C. x � y � 0 D. y � �x � 2 4.

3x � y � 2 y � �x � 2 3x � 3y � �6 x � y � 0

5. When solving the system of equations, 3x � y � 14which expression could be substituted x � 4y � 3for y in the second equation?

A. 3 � 4y B. �3 �

4x

� C. �14

3� y� D. 14 � 3x 5.

6. If x � 5y � 1 and 2x � 5y � �32, what is the value of y?A. �2 B. 2 C. 1 D. �1 6.

7. Use elimination to solve the system 3x � 5y � 16 and 8x � 5y � 28 for x.

A. �6 B. 5 C. 4 D. �45� 7.

8. Use elimination to solve the system x � 4y � 1 and x � 2y � 19 for x.A. �11 B. 3 C. 25 D. 13 8.

9. Use elimination to solve the system 4x � 7y � �14 and 8x � 5y � 8 for x.

A. 3 �12� B. �1�

12� C. 8 D. �4 9.

10. Use elimination to solve the system 5x � 4y � �10 and 3x � 6y � �6 for y.A. �2 B. �5 C. 0 D. 2 10.

11. The substitution method should be used to solve which system of equations?A. 5x � 7y � 16 B. 4x � 3y � �5 C. x � 3y � 1 D. 2x � 6y � 3 11.

2x � 7y � 12 6x � 3y � 2 2x � y � 7 3x � 2y � �1

12. The elimination method using addition should be used to solve which system of equations?A. y � �4x � 1 B. x � 2y � �4 C. 5x � y � �6 D. x � 4y � �9 12.

x � 2y � 7 3x � 2y � �1 5x � 2y � 3 �8x � y � 1

77

y

xO

y � x � 2

y � �x � 2

x � y � 0

3x � y � 2

3x � 3y � � 6


Chapter 7 Test, Form 2B (continued)

13. State which region in the graph is the solution of the system.x � 1 y�x � 1 yA. A B. BC. C D. D 13.

14. Which system of inequalities is represented by the graph? A. y � �3x � 3 B. y � �3x � 3

y � �3x � 1 y �3x � 1C. y �3x � 3 D. y � �3x � 3 14.

y � �3x � 1 y � �3x � 1

For Questions 15 and 16, solve the system and find the value of y.

15. 2x � 3y � 1 A. �8�37� B. 8�

37� C. �3 D. 3 15.

5x � 4y � �32

16. 6x � 3y � 12 A. �20 B. �1�191�

C. 20 D. 1�191�

16.5x � 3y � 0

17. Five times one number minus two times a second number is 11.Three times the first number minus two times the second number is 1.What are the numbers?A. 5 and 7 B. 2 and 5 C. 11 and 1 D. 4 and �6 17.

18. Colortime Bakers wants to make 30 pounds of a berry mix that costs $3 per pound to use in their pancake mix. They are using blueberries that cost $2 per pound and blackberries that cost $3.50 per pound. How many pounds of blackberries should be used in this mixture?A. 15 lb B. 20 lb C. 10 lb D. 30 lb 18.

19. In 1999, there were 69,063 physicians specializing in Family Practice in the United States and its possessions. The number of male physicians minus the number of female physicians in this category is 31,289. How many female physicians were there that specialized in Family Practice in the United States?A. 50,176 B. 34,531 C. 3243 D. 18,887 19.

20. Beng and Shim make boat-shaped candles and dog-shaped candles. Theyhave enough materials to make 40 candles. Beng and Shim will make at most 20 boat-shaped candles. They will also make less than 35 dog-shaped candles. Which system of inequalities represents this information?A. 5 � b 20 B. 20 � d 40 C. b � d 40 D. 0 b � 20 20.

20 d � 35 b � 35 40 0 b 20 0 d 350 d � 35

Bonus Manuel is 8 years older than his sister. Three years ago B:he was 3 times older than his sister. How old is each now?

NAME DATE PERIOD

77

y

xO

AB

C

D

y

xO

Chapter 7 Test, Form 2C


Use the graph at the right to determine whether each system has no solution, onesolution, or infinitely many solutions.

1. y � �xx � y � 3

2. x � 2y � �34x � y � 6

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitelymany solutions. If the system has one solution, name it.

3. y � �x � 4y � x � 4

4. 2x � y � �36x � 3y � �9

5. x � y � �2x � y � 3

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions.

6. y � 3x 7. 5x � y � 10x � y � 4 7x � 2y � 11

8. x � 6y � 4 9. x � 5y � 103x � 18y � 4 2x � 10y � 20


10. x � 4y � �8 11. 2x � 5y � 3x � 4y � �8 �x � 3y � �7

12. 2x � 5y � �16 13. 2x � 5y � 9�2x � 3y � 12 2x � y � 13

14. 2x � 3y � 1 15. x � 3y � 105x � 4y � 14 x � 2y � 15


16. x � 2y � 1 17. 5x � y � 173x � y � 11 3x � y � 13

NAME DATE PERIOD

SCORE 77

Ass

essm

ent

y

xO

y ��x

x � y � 0

x � y � 3

x � 2y � �3

4x � y � 6

1.

2.3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

y

xO

y

xO

y

xO


Chapter 7 Test, Form 2C (continued)

For Questions 18 and 19, solve each system of inequalitiesby graphing.

18. y � x � 1y 2x � 1

19. x � y � 22x � y � 1

20. The sum of two numbers is 17 and their difference is 29.What are the two numbers?

21. Adult tickets for the school musical sold for $3.50 and student tickets sold for $2.50. Three hundred twenty-one tickets were sold altogether for $937.50. How many of each kind of ticket were sold?

22. Ayana has $2.35 in nickels and dimes. If she has 33 coins in all, find the number of nickels and dimes.

23. The largest county in the state of New York is 1769 square miles larger than the smallest county in the same state. The size of the largest county is 64 times the size of the smallest county plus five square miles. How large is the smallest county in the state of New York?

For Questions 24 and 25, use the following information. 24.

The Martinez Company manufactures two types of industrial fans, standard and economy. These items are built using machines and manual labor. The table gives the time requirements at each type of workstation for each type of fan.

24. Make a graph showing the number of standard fans and the number of economy fans that can be made in a week.

25. List three possible solutions. 25.

Bonus Mavis is 5 years older than her brother. Five years ago B:she was 2 times older than her brother. How old is each now?

NAME DATE PERIOD

77

18.

19.

20.

21.

22.

23.

y

xO

y

xO

Hours per Standard Fan

Hours per Economy Fan

Total AvailableHours Each Week

Machines 3 3 1500

Manual Labor 2 1 800

e

sO

200

400

600

800

200 400 600 800

Chapter 7 Test, Form 2D


Use the graph at the right to determine whether each system has no solution, onesolution, or infinitely manysolutions.

1. x � y � 3y � �x � 3

2. 3x � y � �32x � 3y � �2

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely manysolutions. If the system has one solution, name it.

3. y � �x � 3y � x � 3

4. 2x � y � 54x � 2y � 10

5. x � y � 0x � y � 2


6. y � 2x 7. 2x � y � 32x � y � 8 5x � 7y � 17

8. x � 5y � 2 9. x � 3y � �24x � 20y � 8 4x � 12y � 7


10. 2x � 3y � 19 11. 6x � 4y � 202x � 3y � 1 4x � 2y � 4

12. 2x � 2y � 6 13. 7x � 3y � 13x � 2y � �11 9x � 3y � �3

14. 2x � 3y � 23 15. x � 3y � 03x � 5y � 6 x � 5y � 16


16. y � 3x � 1 17. 5x � 15y � �20x � 2y � 8 5x � 4y � �9

NAME DATE PERIOD

SCORE 77

Ass

essm

ent

y

xO

2x � 3y � �2x � y � 3

y � �x � 3

x � y � �1

3x � y � �3

1.

2.3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

y

xO

y

xO

y

xO


Chapter 7 Test, Form 2D (continued)

For Questions 18 and 19, solve each system of inequalities by graphing.

18. y � �2x � 1y x � 1

19. 2x � y � �32x � y � 1

20. The sum of two numbers is 16 and their difference is 20.What are the two numbers?

21. Kyle just started a new job that pays $7 per hour. He had been making $5 per hour at his old job. Kyle worked a total of 54 hours last month and made $338 before deductions.How many hours did he work at his new job?

22. So far this basketball season, all of Nikki’s points have come from two-point and three-point field goals. She has scored a total of 43 points. The number of three-point field goals she has made is one more than twice as many two-point field goals. How many of each type of field goal has Nikki made?

23. The largest county in the state of Texas is 6064 square miles larger than the smallest county in the same state. The size of the largest county is 48 times the size of the smallest county plus one square mile. How large is the smallest county in the state of Texas?

For Questions 24 and 25, use the following information. 24.

The Tadashi Corporation manufactures a large speaker and a small speaker for phone headsets. The speakers are used on the Tadashi 200 and the Tadashi 500 phone systems. The table gives the speaker requirement for each phone system.

24. Make a graph showing the number of Tadashi 200 and Tadashi 500 phone systems that can be made in a week.

25. List three possible solutions. 25.

Bonus Find the point on the graph of 3x � 4y� 9 where the B:y-coordinate is 3 times the x-coordinate.

NAME DATE PERIOD

77

18.

19.

20.

21.

22.

23.

y

xO

y

xO

y

xO

100

200

300

400

100 200 300 400

Speakers on Tadashi 200

Speakers on Tadashi 500

Total AvailableSpeakers Each Week

Large Speaker 2 3 900

Small Speaker 2 4 1000

Chapter 7 Test, Form 3


Determine whether each system has no solution, onesolution, or infinitely many solutions by graphing each system.

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

Graph each system of equations. Determine whether the system has no solution, one solution, or infinitely manysolutions. If the system has one solution, name it.

3. �13�y � x

y � x � 4 � 0

4. x � 3y � 33y � �x � 9

5. �15� � �

75�y � x

10x � 14y � 2


6. y � 2x � 7 7. 4y � 3x � 53x � 4y � 8 �

34�x � y � 4

8. �12�x � 5y � 19 9. 0.5x � 3.5y � 1

x � 2y � �10 x � 2 � 7y


10. 6x � 7y � 21 11. 0.2x � 0.5y � 0.73x � 7y � 6 �0.2x � 0.6y � �1.4

12. 2x � �23�y � �8 13. �

12�x � �

25�y � � 10

�12�x � �

13�y � 1 3x � 6y � �6

14. 0.4x � 0.1y � 1 15. �34�x � �

12�y � 1�

12�

0.5x � 0.1y � 1.6�34�x � y � 4�

12�


16. x � y � 147 17. 7y � 2�12� � 2x

25x � 10y � 2415 5x � 3y � 4

NAME DATE PERIOD

SCORE 77

Ass

essm

ent1.

2.3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

y

xO

y

xO

y

xO


Chapter 7 Test, Form 3 (continued)

For Questions 18 and 19, solve each system of inequalities by graphing.

18. y � 2xy � x � 1x � 3y � �12

19. x 1x � y � 2x � 2y � 3y 3x � 4

20. Three times one number added to five times a second number is 68. Three times the second number minus four times the first number is 6. What are the two numbers?

21. The difference of two numbers is 5. Five times the lesser number minus the greater number is 9. What are the two numbers?

22. The sum of the digits of a 2-digit number is 13. If the digits are reversed, the new number is 9 more than the original number. Find the original number.

23. A trail mix that costs $2.45 per pound is mixed with a trail mix that costs $2.30 per pound. How much of each type of trail mix must be used to have 30 pounds of a trail mix that costs $2.35 per pound?

24. A boat travels 60 miles downstream in the same time it takes to go 36 miles upstream. The speed of the boat in still water is 15 mph greater than the speed of the current.Find the speed of the current.

25. Mrs. Lewis needs to buy two types of grain, oats and barley, to mix as a feed supplement for her cattle. She has $15,000 to spend on grain, and wants the mixture to be at most 3 parts oats per 2 parts barley. She can buy oats for $1.10 per bushel and barley for $2.05 per bushel. If Mrs. Lewis needs at least 7500 bushels of grain, make a graph showing the number of bushels of oats and the number of bushels of barley that she should buy, and list three possible solutions.

Bonus Graph the solution set of �3 4x � y � 1.

NAME DATE PERIOD

77

18.

19.

20.

21.

22.

23.

24.

25.

B: y

xO

y

xO

y

xO

y

xO

Chapter 7 Open-Ended Assessment


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

1. Ruth invests $10,000 in two accounts. One account has an annualinterest rate of 7%, and the other account has an annual interestrate of 5%. Let I represent the total interest earned in one year.Then Ruth’s investments can be modeled by the system ofequations x � y � 10,000 and 0.07x � 0.05y � I.a. Determine a solution for this system of equations that

represents a possible investment, and find the value of Ithat corresponds to your solution.

b. Find the solution for the system if I � 800. Explain why thissolution does not represent a possible investment.

2. A car rental company wants to charge a rate of $A per day plus$B per mile to rent a compact car. Their leading competitorcharges $15 per day plus $0.25 per mile to rent a compact car.a. Explain how the system of equations y � A � Bx and

y � 15 � 0.25x compares the total cost of renting a compactcar for one day from this company and from their leadingcompetitor.

b. Describe how the value of A and the value of B affects thecomparison of the total cost of any one-day rental from thesetwo companies.

3. A bookstore makes a profit of $2.50 on each book they sell, and$0.75 on each magazine they sell. Each week the store sells x books and y magazines. Let $P be the weekly profit, and $S bethe weekly sales for the bookstore.a. Write a system of equations that models the possible weekly

sales and weekly profit for the book store. Then describepossible values for P and S.

b. Replace the equal signs in the system of equations from part awith inequality symbols to create a system of inequalities.Explain how these two systems differ in modeling the weeklyprofit and weekly sales of the bookstore. Explain how thesetwo systems are the same.

c. Choose a value for P and a value for S, and substitute thevalues into the system of inequalities from part b. Make agraph of this system of inequalities, and describe what thegraph represents.

NAME DATE PERIOD

SCORE 77

Ass

essm

ent


Chapter 7 Vocabulary Test/Review

Underline or circle the correct term to complete each sentence.

1. Two equations, such as y � 3x � 6 and y � 12 � 4x, together arecalled a .system of inequalities system of equations solution of a system

2. If the graphs of the equations of a system intersect or coincide,the system of equations is .consistent inconsistent independent

3. If the graphs of the equations of a system are parallel, the systemis .consistent inconsistent dependent

4. If a system of equations has exactly one solution, the system is.

inconsistent dependent independent

5. If a system of equations has an infinite number of solutions, thesystem is .inconsistent dependent independent

6. Sometimes adding two equations of a system will give an equationwith only one variable. This is helpful when you are solving thesystem by .graphing substitution elimination

7. For a system where one variable is solved in terms of the othervariable, you can solve the system by using .addition substitution elimination

In your own words —Define the term.

8. system of inequalities

?

?

?

?

?

?

?

consistentdependent

eliminationinconsistent

independentsubstitution

system of equationssystem of inequalities

NAME DATE PERIOD

SCORE 77

Chapter 7 Quiz (Lessons 7–1 and 7–2)

77


Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution,name it.

1. y � �32�x

y � �x � 5

2. x � 2y � �2x � 2y � 3

For Questions 3 and 4, use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solutions or infinitely many solutions. 3.

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

5. In order for José and Marty to compete against each other 5.during the wrestling season next year they need to be in the same weight category. José weighs 180 pounds and plans to gain 2 pounds per week. Marty weighs 250 pounds and plans to lose 1 pound per week. In which week will they weigh the same?

NAME DATE PERIOD

SCORE

Chapter 7 Quiz (Lesson 7–3)

For Questions 1–4, use elimination to solve each system of equations.

1. x � y � 4 2. �2x � y � 5x � y � 7 2x � 3y � 3

3. 5r � 3s � 17 4. 3x � 2 � 7y2r � 3s � 9 �4x � 30 � 7y

5. Standardized Test Practice If x � 2y � 7 and 3x � 2y � 1,what is the value of y?

A. �3 B. �5 C. 2 D. �2�12� 5.

NAME DATE PERIOD

SCORE 77

Ass

essm

ent

1.

2. y

xO

y

xO

1.

2.

3.

4.



1. x � 4y � 11 1.5x � 7y � �10

2. 2r � 3s � 9 2.3r � 2s � 12

3. 4c � 6d � �10 3.8c � 3d � �5

Determine the best method to solve each system of 4.equations. Then solve the system.

4. x � 2y � 1 5. 7x � 5y � 29 5.3x � y � 17 21x � 25y � �33


For Questions 1–3, solve each system of inequalities by 1.graphing.

1. y � 2x � y � 3

2. y � 2x � 1 2.y � �x � 4

3. Adult tickets to the school musical are $5 and 3.student tickets are $2. There are 300 seats in the auditorium where the musical is being performed.The goal for ticket sales for one performance is at least $900. Make a graph showing the number of each kind of ticket needed to be sold to reach the goal. List three possible solutions.

y

xO

y

xO

NAME DATE PERIOD

SCORE


77

NAME DATE PERIOD

SCORE

77

s

aO

100

200

300

400

100 200 300 400

Chapter 7 Mid-Chapter Test (Lessons 7–1 through 7–3)


Write the letter for the correct answer in the blank at the right of each question.Use the graph for Questions 1–3.For Questions 1 and 2, determine how many solutions exist for each system of equations.


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

13�x � �

23� 2.

3. The solution to which system of equations has a positive y value?A. x � y � 3 B. x � y � 3 C. x � 3y � �2 D. x � y � 3 3.

x � 3y � �2 x � y � �2 x � y � �2 x � y � 3

4. If y � 5x � 3 and 3x � y � �1, what is the value of y?A. 2 B. �1 C. 7 D. �8 4.

5. Use elimination to solve the system x � 5y � �6of equations for y. x � 2y � 8

A. ��23� B. 2 C. 4 D. 4�

23� 5.

6. How many solutions does the system 2y � 10x � 14 and 5x � y � 7 have?A. one B. two C. none D. infinitely many 6.

7. Graph the system of equations. Then determine whether it has no solution, one solution, or infinitely many solutions. If the system has one solution name it.

x � 3y � �3x � 3y � 9

8. Use substitution to solve the 4x � y � 0system of equations. 2y � x � �7

For Questions 9–11, use elimination to solve each system of equations.

9. �12�x � y � 7 10. 2x � 7y � 20

��12�x � 3y � �11

3x � 7y � �5

11. 9x � 2y � �17�11x � 2y � 3

12. At the end of the 2000 WNBA regular season, the HoustonComets had 22 more victories than losses. The number of victories they had was three less than six times the number of losses. How many regular season games did the Houston Comets play during the 2000 WNBA season?

Part II

Part I

NAME DATE PERIOD

SCORE 77

Ass

essm

ent

y

xO

x � y � 3

x � 3y � �2

x � y � 3x � y � �2

y � x � 13

23

7.

8.

9.

10.

11.

12.

y

xO


Chapter 7 Cumulative Review (Chapters 1–7)

1. Solve �a6� � 5 � 12. (Lesson 3-4) 1.

2. Find the coordinates of the vertices of triangle DEF with 2.D(1, 1), E(5, 1), and F(2, 4) rotated 90° clockwise about the origin. (Lesson 4-2)

3. Solve y � �14�x � 1 if the domain is {�4, �2, 0, 2, 4}. (Lesson 4-4) 3.

4. Write the slope-intercept form of an equation of the line 4.that passes through (0, �4) and is parallel to the graph of 4x � y � 7. (Lesson 5-6)

5. Solve �45�a � �12. (Lesson 6-2) 5.

6. Write an open sentence 6.involving absolute value for the graph. (Lesson 6-5)

7. Graph the system of equations. Then determine whether 7.the system has no solution, one solution, or infinitely manysolutions. (Lesson 7-1)

3x � y � 1y � 3x � 1

8. Jim’s Brakes charges $25 for parts and $55 per hour to fix 8.the brakes on a car. Myron’s Auto charges $40 for parts and $30 per hour to do the same job. What length of job in hours would have the same cost at both shops? (Lesson 7-2)

9. The sum of two numbers is 21, and their difference is 7. 9.What are the numbers? (Lesson 7-3)

10. Use elimination to solve the system of equations. (Lesson 7-4) 10.2x � 4y � 263x � 2y � 15

11. Solve the system of inequalities by graphing. (Lesson 7-5) 11.y � xy � x � 4

y

xO

y

xO

NAME DATE PERIOD

77

�1�2�3�5�4 0 1 2 63 4 5

Standardized Test Practice (Chapters 1–7)


1. Which equation is not equivalent to x � 7 � 12? (Lesson 3-2)

A. x � 9 � 14 B. x � 10 � 9 C. x � 19 D. x � 3 � 16 1.

2. Find the value of y so that the line through (2, 3) and (5, y) has a slope of �2. (Lesson 5-1)

E. �3 F. �32� G. 9 H. �

92� 2.

3. Solve ��53�r � �

190�. (Lesson 6-2)

A. �r � r � �195�� B. �r � r � �

195�� C. �r � r � ��

23�� D. �r � r � �

23�� 3.

4. Write a compound inequality for the graph. (Lesson 6-4)

E. x � �2 or x 3 F. x � �2 and x � 3G. �2 � x � 3 H. �2 � x � 3 4.

5. Which inequality is represented by a dashed line through (0, 1) and (2, 0), and is satisfied by (2, 1)? (Lesson 6-6)

A. x � 2y � 2 B. x � 2y � 2 C. x � 2y � 2 D. x � 2y 2 5.

6. How many solutions exist for the system of equations? (Lesson 7-1)

2x � 3y � 144x � 6y � 21E. no solutions F. one solutionG. two solutions H. infinitely many solutions 6.

7. When solving the following system, which expression could be substituted for y? (Lesson 7-2)

5x � 12y � 67x � y � 3A. 7x � 3 B. �7x � 3 C. 5x � 6 D. �5x � 6 7.

8. If 4x � 5y � 6 and 7x � 5y � 3, what is the value of y? (Lesson 7-3)

E. �1 F. 2 G. 1 H. �3 8.

9. Which ordered pair satisfies the system of inequalities? (Lesson 7-5)

x � 2y 83x � 2y � 10A. (0, 4) B. (3, 3) C. (4, �2) D. (5, 1) 9. DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

77

Ass

essm

ent

NAME DATE PERIOD

Part 1: Multiple Choice

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

�1�2�3�5�4 0 1 2 63 4 5


Standardized Test Practice (continued)

10. Find ��89� (�64). (Lesson 2-4) 10. 11.

11. If f(x) � x2 � 4x, find f(�3). (Lesson 4-6)

12. What is the slope of a line parallel to the 12. 13.line that passes through (�2, 1) and (3, 7)?(Lesson 5-6)

13. If 4x � 7y � �3 and 3x � 2y � 14, what is the value of x? (Lesson 7-4)

Column A Column B

14. 14.

(Lesson 2-7)

15. Given: parallelogram ABCD with A(�1, 3), B(3, 2), C(2, �2), and 15.D(�2, �1) rotated 90° counterclockwise about the origin. (Lesson 4-2)

16. Given: �2x � �4 and y � 6 �3. (Lessons 6-1 and 6-2) 16.

yx

DCBA

the y-coordinate of D�the x-coordinate of B�

DCBA

DCBA��191��

57

��

Part 3: Quantitative Comparison

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

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.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

77

NAME DATE PERIOD NAME DATE PERIOD

Part 2: Grid In

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

A

D

C

B

NAME DATE PERIOD

SCORE Unit 2 Test(Chapters 4–7)


1. Trapezoid WXYZ with W(�2, 1), X(�4, 1), Y(�4, 4), and 1.Z(�2, 5) is translated 3 units right. Find the coordinates of the vertices of the figure after the transformation is performed. Then graph the preimage and its image.

2. Express the relation shown in the 2.mapping as a set of ordered pairs. Then write the inverse of the relation.

3. Solve 2x � 3 � y if the domain is {�2, �1, 0, 3, 5}. 3.

4. Graph 3x � y � 1. 4.

5. If g(x) � 2x2 � 3, find g(�4). 5.

6. Find the 25th term of the arithmetic sequence with first 6.term 7 and common difference �2.

7. Find the next three terms in the sequence 5, 6, 8, 11, … . 7.

8. A giraffe can travel 800 feet in 20 seconds. Write a direct 8.variation equation for the distance traveled in any time.

9. Write an equation of the line whose slope is 2 and whose 9.y-intercept is 9.

10. Write an equation of the line that passes through (�1, �7) 10.and (1, 3).

11. Write y � 4 � ��32

�(x � 6) in standard form. 11.

12. Write the slope-intercept form of an equation of the line 12.that passes through (�2, 0) and is parallel to the graph of y � �3x � 2.

y

xO

y

xO

X

1234

23455

Y

Ass

essm

ent


Unit 2 Test (continued)

13. The table below shows the distance driven during four 13.different trips and the duration of each trip. Draw a scatter plot and determine what relationship exists, if any, in the data. Write an equation for a line of fit for the data.

Solve each inequality.

14. 4x � 5 � 7x � 10 14.

15. 2(5a � 4) � 3(6 � 2a) � 6 15.

Solve each compound inequality.

16. 5 � 2t � 7 � 11 16.

17. 13 � 4 � 3v or 2v � 14 8 17.

For Questions 18 and 19, solve each open sentence. Then graph the solution set.

18. � 3b � 5 � � 7 18.

19. � w � 5 � 1 19.

20. Use a graph to determine whether the system x � y � 4 20.and y � x has no solution, one solution, or infinitely manysolutions.

For Questions 21–24, determine the best method to solve each system of equations. Then solve the system.

21. x � y � 2 22. �x � 5y � 7y � 2x �1 x � y � 1

23. 3x � y � �6 24. 3x � 3y � �63x � 3y � 18 7x � 4y � 1

25. Solve the system of inequalities by graphing. 25.2x � y 3x � 2y � 4

y

xO

�7�6�5�4�8 �3�2�1 0

�3�2�1 0 1 2 4 53

40

0

80

120

160

1 2 3 4

Dis

tan

ce (

mile

s)

Time (hours)

NAME DATE PERIOD

21.

22.

23.

24.

Time (hours) 1 2 2.5 4

Distance (miles) 50 85 120 180

Standardized Test PracticeStudent Record Sheet (Use with pages 404–405 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 1

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

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 11–13, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 (grid in) 11 12 13

12 (grid in)

13 (grid in)

14

15

16

17

Record your answers for Questions 18–19 on the back of this paper.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

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0 0 0

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.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

NAME DATE PERIOD

77

An

swer

s

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

Part 3 Extended ResponsePart 3 Extended Response

Part 1 Multiple ChoicePart 1 Multiple Choice


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mb

er o

f S

olu

tio

ns

exac

tly o

ne s

olut

ion

infin

itely

man

y so

lutio

nsno

sol

utio

n

Term

ino

log

yco

nsis

tent

and

cons

iste

nt a

ndin

cons

iste

nt

inde

pend

ent

depe

nden

t

Use

th

e gr

aph

at

the

righ

t to

det

erm

ine

wh

eth

er t

he

syst

em h

as n

oso

luti

on,o

ne

solu

tion

,or

infi

nit

ely

ma

ny

solu

tion

s.

a.y

��

x�

2y

�x

�1

Sin

ce t

he

grap

hs

of y

��

x�

2 an

d y

�x

�1

inte

rsec

t,th

ere

is o

ne

solu

tion

.

b.

y�

�x

�2

3x�

3y�

�3

Sin

ce t

he

grap

hs

of y

��

x�

2 an

d 3x

�3y

��

3 ar

e pa

rall

el,t

her

e ar

e n

o so

luti

ons.

c.3x

�3y

��

3y

��

x�

1S

ince

th

e gr

aph

s of

3x

�3y

��

3 an

d y

��

x�

1 co

inci

de,

ther

e ar

e in

fin

itel

y m

any

solu

tion

s.

Use

th

e gr

aph

at

the

righ

t to

det

erm

ine

wh

eth

er e

ach

sy

stem

has

no

solu

tion

,on

eso

luti

on,o

r in

fin

itel

y m

an

yso

luti

ons.

1.y

��

x�

32.

2x�

2y�

�6

y�

x�

1y

��

x�

3o

ne

infi

nit

ely

man

y

3.y

��

x�

34.

2x�

2y�

�6

2x�

2y�

43x

�y

�3

no

ne

on

e

x

y

O

y �

�x

� 33x

� y

� 3

2x �

2y

� �

6

2x �

2y

� 4

y �

x �

1

x

y O

y �

x �

1

y �

�x

� 1

3x �

3y

� �

3

y �

�x

� 2

x

y

Ox

y

Ox

y

O

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

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w-H

ill40

4G

lenc

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lgeb

ra 1

Solv

e b

y G

rap

hin

gO

ne

met

hod

of

solv

ing

a sy

stem

of

equ

atio

ns

is t

o gr

aph

th

eeq

uat

ion

s on

th

e sa

me

coor

din

ate

plan

e.

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s.T

hen

det

erm

ine

wh

eth

er t

he

syst

em h

as n

oso

luti

on,o

ne

solu

tion

,or

infi

nit

ely

ma

ny

solu

tion

s.If

th

e sy

stem

has

one

solu

tion

,nam

e it

.

a.x

�y

�2

x�

y�

4T

he

grap

hs

inte

rsec

t.T

her

efor

e,th

ere

is o

ne

solu

tion

.Th

e po

int

(3,�

1) s

eem

s to

lie

on

bot

h l

ines

.Ch

eck

this

est

imat

e by

rep

laci

ng

xw

ith

3 a

nd

yw

ith

�1

in e

ach

equ

atio

n.

x�

y�

23

�(�

1) �

2 ✓

x�

y�

43

�(�

1) �

3 �

1 or

4 ✓

Th

e so

luti

on i

s (3

,�1)

.

b.

y�

2x�

12y

�4x

�2

Th

e gr

aph

s co

inci

de.T

her

efor

e th

ere

are

infi

nit

ely

man

y so

luti

ons.

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s.T

hen

det

erm

ine

wh

eth

er t

he

syst

em h

as n

oso

luti

on,o

ne

solu

tion

,or

infi

nit

ely

ma

ny

solu

tion

s.If

th

e sy

stem

has

on

e so

luti

on,

nam

e it

.

1.y

��

2o

ne;

(�1,

�2)

2.x

�2

on

e;(2

,�3)

3.y

�x

on

e;(2

,1)

3x�

y�

�1

2x�

y�

1x

�y

�3

4.2x

�y

�6

on

e;(1

,4)

5.3x

�2y

�6

no

so

luti

on

6.2y

��

4x�

4in

fin

itel

y 2x

�y

��

23x

�2y

��

4y

��

2x�

2m

any

y �

�2x

� 2

2y �

�4x

� 4 x

y

O

3x �

2y

� �

4

3x �

2y

� 6

x

y

O

2x �

y �

6

2x �

y �

�2

( 1, 4

)

x

y

O

x �

y �

3

y �

1 2x( 2

, 1)

x

y

O2x

� y

� 1

x �

2

( 2, –

3)

x

y

O

3x �

y �

�1

y �

�2

( –1,

–2)

x

y

O

1 � 2

x

y O

y �

2x

� 1

2y �

4x

� 2x

y

O( 3

, –1) x �

y �

4

x �

y �

2

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Gra

ph

ing

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-1

7-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 7-1)


An

swer

s

Skil

ls P

ract

ice

Gra

ph

ing

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-1

7-1

©G

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ill40

5G

lenc

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lgeb

ra 1

Lesson 7-1

Use

th

e gr

aph

at

the

righ

t to

det

erm

ine

wh

eth

er

each

sys

tem

has

no

solu

tion

,on

eso

luti

on,o

r in

fin

itel

y m

an

yso

luti

ons.

1.y

�x

�1

2.x

�y

��

4in

fin

itel

yy

��

x�

1o

ne

y�

x�

4m

any

3.y

�x

�4

4.y

�2x

�3

2x�

2y�

22x

�2y

�2

no

so

luti

on

on

e

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s.T

hen

det

erm

ine

wh

eth

er t

he

syst

em h

as n

oso

luti

on,o

ne

solu

tion

,or

infi

nit

ely

ma

ny

solu

tion

s.If

th

e sy

stem

has

on

e so

luti

on,

nam

e it

.

5.2x

�y

�1

6.x

�1

7.3x

�y

��

3n

o

y�

�3

on

e;(�

1,�

3)2x

�y

�4

on

e;(1

,2)

3x�

y�

3so

luti

on

8.y

�x

�2

infi

nit

ely

9.x

�3y

��

3o

ne;

10.y

�x

��

1x

�y

��

2m

any

x�

3y�

�3

(�3,

0)x

�y

�3

on

e;(2

,1)

11.x

�y

�3

12.x

�2y

�4

infi

nit

ely

13.y

�2x

�3

no

so

luti

on

x�

2y�

3o

ne;

(3,0

)y

��

x�

2m

any

3y�

6x�

6

y �

2x

� 3

3y �

6x

� 6x

y

O

y �

�1 – 2x

� 2

x �

2y

� 4 x

y

O

x �

2y

� 3

( 3, 0

)

x �

y �

3

x

y

O

1 � 2

x �

y �

3

( 2, 1

)

y �

x �

�1

x

y

O

x �

3y

� �

3

( –3,

0)

x �

3y

� �

3x

y

O

y �

x �

2

x �

y �

�2

x

y

O

3x �

y �

3

3x �

y �

�3

x

y

O

( 1, 2

)

x �

1 2x �

y �

4x

y

O

( –1,

–3)

2 x �

y �

1

y �

�3

x

y

O

x

y

Oy

� �

x �

1

x �

y �

�4

2x �

2y

� 2

y �

2x

� 3

y �

x �

1

y �

x �

4

©G

lenc

oe/M

cGra

w-H

ill40

6G

lenc

oe A

lgeb

ra 1

Use

th

e gr

aph

at

the

righ

t to

det

erm

ine

wh

eth

er

each

sys

tem

has

no

solu

tion

,on

eso

luti

on,o

r in

fin

itel

y m

an

yso

luti

ons.

1.x

�y

�3

2.2x

�y

��

3x

�y

��

34x

�2y

��

6n

o s

olu

tio

nin

fin

itel

y m

any

3.x

�3y

�3

4.x

�3y

�3

x�

y�

�3

on

e2x

�y

��

3o

ne

Gra

ph

eac

h s

yste

m o

f eq

uat

ion

s.T

hen

det

erm

ine

wh

eth

er t

he

syst

em h

as n

oso

luti

on,o

ne

solu

tion

,or

infi

nit

ely

ma

ny

solu

tion

s.If

th

e sy

stem

has

on

e so

luti

on,

nam

e it

.

5.3x

�y

��

2n

o

6.y

�2x

�3

infi

nit

ely

7.x

�2y

�3

on

e;3x

�y

�0

solu

tio

n4x

�2y

�6

man

y3x

�y

��

5(�

1,2)

BU

SIN

ESS

For

Exe

rcis

es 8

an

d 9

,use

th

e fo

llow

ing

info

rmat

ion

.N

ick

plan

s to

sta

rt a

hom

e-ba

sed

busi

nes

s pr

odu

cin

g an

d se

llin

g go

urm

et d

og t

reat

s.H

e fi

gure

s it

wil

l co

st $

20 i

nop

erat

ing

cost

s pe

r w

eek

plu

s $0

.50

to p

rodu

ce e

ach

tre

at.

He

plan

s to

sel

l ea

ch t

reat

for

$1.

50.

8.G

raph

th

e sy

stem

of

equ

atio

ns

y�

0.5x

�20

an

d y

�1.

5xto

rep

rese

nt

the

situ

atio

n.

9.H

ow m

any

trea

ts d

oes

Nic

k n

eed

to s

ell

per

wee

k to

brea

k ev

en?

20

SALE

SF

or E

xerc

ises

10–

12,u

se t

he

foll

owin

g in

form

atio

n.

A u

sed

book

sto

re a

lso

star

ted

sell

ing

use

d C

Ds

and

vide

os.

In t

he

firs

t w

eek,

the

stor

e so

ld 4

0 u

sed

CD

s an

d vi

deos

,at

$4.0

0 pe

r C

D a

nd

$6.0

0 pe

r vi

deo.

Th

e sa

les

for

both

CD

s an

d vi

deos

tot

aled

$18

0.00

10.W

rite

a s

yste

m o

f eq

uat

ion

s to

re

pres

ent

the

situ

atio

n.

11.G

raph

th

e sy

stem

of

equ

atio

ns.

12.H

ow m

any

CD

s an

d vi

deos

did

the

sto

re s

ell

in t

he f

irst

wee

k?30

CD

s an

d 1

0 vi

deo

sc�

v�

404c

�6v

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� 6

v �

180

c �

v �

40

( 30,

10)

CD

Sal

es (

$)

CD

an

d V

ideo

Sale

s

Video Sales ($)

515

1020

2530

3540

450

40 35 30 25 20 15 10 5

y �

1.5

x

y �

0.5

x �

20

( 20,

30) Sa

les

($)

Do

g T

reats

Cost ($)

515

1020

2530

3540

450

40 35 30 25 20 15 10 5

3x �

y �

�5

x �

2y

� 3 x

y O

( –1,

2)

4x �

2y

� 6

y �

2x

� 3

x

y

O3x

� y

� 0

3x �

y �

�2

x

y

O

x

y

O

x �

y �

�3

x �

3y

� 3

2x �

y �

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4x �

2y

� �

6

x �

y �

3

Pra

ctic

e (

Ave

rag

e)

Gra

ph

ing

Sys

tem

s o

f E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-1

7-1



Readin

g t

o L

earn

Math

em

ati

csG

rap

hin

g S

yste

ms

of

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-1

7-1

©G

lenc

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cGra

w-H

ill40

7G

lenc

oe A

lgeb

ra 1

Lesson 7-1

Pre-

Act

ivit

yH

ow c

an y

ou u

se g

rap

hs

to c

omp

are

the

sale

s of

tw

o p

rod

uct

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-1

at

the

top

of p

age

369

in y

our

text

book

.

•W

hat

is

mea

nt

by t

he

term

lin

ear

fun

ctio

n?

Sam

ple

an

swer

:a

fun

ctio

n w

ho

se g

rap

h is

a li

ne

•W

hat

doe

s it

mea

n t

o sa

y th

at t

wo

lin

es i

nte

rsec

t?S

amp

le a

nsw

er:

Th

e lin

es c

ross

.

Rea

din

g t

he

Less

on

1.E

ach

fig

ure

sh

ows

the

grap

h o

f a

syst

em o

f tw

o eq

uat

ion

s.W

rite

th

e le

tter

of

the

figu

res

that

ill

ust

rate

eac

h s

tate

men

t.

A.

B.

C.

D.

a.A

sys

tem

of

two

lin

ear

equ

atio

ns

can

hav

e an

in

fin

ite

nu

mbe

r of

sol

uti

ons.

D

b.

A s

yste

m o

f eq

uat

ion

s is

con

sist

ent

if t

her

e is

at

leas

t on

e or

dere

d pa

ir t

hat

sat

isfi

esbo

th e

quat

ion

s.B

,C,D

c.If

tw

o gr

aph

s ar

e pa

rall

el,t

her

e ar

e n

o or

dere

d pa

irs

that

sat

isfy

bot

h e

quat

ion

s.A

d.

If a

sys

tem

of

equ

atio

ns

has

exa

ctly

on

e so

luti

on,i

t is

in

depe

nde

nt.

B,C

e.If

a s

yste

m o

f eq

uat

ion

s h

as a

n i

nfi

nit

e n

um

ber

of s

olu

tion

s,it

is

depe

nde

nt.

D

Hel

pin

g Y

ou

Rem

emb

er

2.D

escr

ibe

how

you

can

sol

ve a

sys

tem

of

equ

atio

ns

by g

raph

ing.

Sam

ple

an

swer

:G

rap

h t

he

equ

atio

ns

on

th

e sa

me

coo

rdin

ate

pla

ne.

Lo

cate

any

po

ints

of

inte

rsec

tio

n.

x

y

Ox

y

O

x

y

Ox

y O

©G

lenc

oe/M

cGra

w-H

ill40

8G

lenc

oe A

lgeb

ra 1

Gra

ph

ing

a T

rip

Th

e di

stan

ce f

orm

ula

,d �

rt,i

s u

sed

to s

olve

man

y ty

pes

of

prob

lem

s.If

you

gra

ph a

n e

quat

ion

su

ch a

s d

�50

t,th

e gr

aph

is

a m

odel

for

a c

ar g

oin

g at

50

mi/h

.Th

e ti

me

the

car

trav

els

is t

;th

e di

stan

ce i

n m

iles

th

e ca

r co

vers

is

d.T

he

slop

e of

th

e li

ne

is

the

spee

d.

Su

ppos

e yo

u d

rive

to

a n

earb

y to

wn

an

d re

turn

.You

ave

rage

50

mi/h

on

th

e tr

ip o

ut

but

only

25

mi/h

on

th

e tr

ip h

ome.

Th

e ro

un

d tr

ip t

akes

5 h

ours

.How

far

aw

ay i

s th

e to

wn

?

Th

e gr

aph

at

the

righ

t re

pres

ents

you

r tr

ip.N

otic

e th

at t

he

retu

rn

trip

is

show

n w

ith

a n

egat

ive

slop

e be

cau

se y

ou a

re d

rivi

ng

in t

he

oppo

site

dir

ecti

on.

Sol

ve e

ach

pro

ble

m.

1.E

stim

ate

the

answ

er t

o th

e pr

oble

m i

n t

he

abov

e ex

ampl

e.A

bou

t h

ow f

ar a

way

is

the

tow

n?

abo

ut

80 m

iles

2.G

raph

th

is t

rip

and

solv

e th

e pr

oble

m.A

n a

irpl

ane

has

en

ough

fu

el f

or 3

hou

rs o

f sa

fe f

lyin

g.O

n t

he

trip

ou

t th

e pi

lot

aver

ages

200

mi/h

fly

ing

agai

nst

a h

eadw

ind.

On

th

e tr

ip b

ack,

the

pilo

t av

erag

es 2

50 m

i/h.H

ow l

ong

a tr

ip o

ut

can

th

e pi

lot

mak

e?

abo

ut

1h

ou

rs a

nd

330

mile

s

3.G

raph

th

is t

rip

and

solv

e th

e

4.G

raph

th

is t

rip

and

solv

e th

e pr

oble

m.Y

oupr

oble

m.Y

ou d

rive

to

a to

wn

dr

ive

at a

n a

vera

ge s

peed

of

50 m

i/h t

o a

100

mil

es a

way

.On

th

e tr

ip o

ut

you

di

scou

nt

shop

pin

g pl

aza,

spen

d 2

hou

rs

aver

age

25 m

i/h.O

n t

he

trip

bac

k yo

u

shop

pin

g,an

d th

en r

etu

rn a

t an

ave

rage

av

erag

e 50

mi/h

.How

man

y h

ours

do

sp

eed

of 2

5 m

i/h.T

he

enti

re t

rip

take

s

you

spe

nd

driv

ing?

8 h

ours

.How

far

aw

ay i

s th

e sh

oppi

ng

plaz

a?6

ho

urs

100

mile

s

t

d

O50

2t

d

O2

50

2 � 3t

d

O1

100

t

d O

slop

e is

50

slop

e is

–25

2

50

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-1

7-1



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Su

bst

itu

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

©G

lenc

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cGra

w-H

ill40

9G

lenc

oe A

lgeb

ra 1

Lesson 7-2

Sub

stit

uti

on

On

e m

eth

od o

f so

lvin

g sy

stem

s of

equ

atio

ns

is s

ub

stit

uti

on.

Use

su

bst

itu

tion

to

solv

e th

e sy

stem

of

equ

atio

ns.

y�

2x4x

�y

��

4

Su

bsti

tute

2x

for

yin

th

e se

con

deq

uat

ion

.4x

�y

��

4S

econ

d eq

uatio

n

4x�

2x�

�4

y�

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2x�

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bine

like

ter

ms.

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ivid

e ea

ch s

ide

by 2

.

x�

�2

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plify

.

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y�

2xto

fin

d th

e va

lue

of y

.y

�2x

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t eq

uatio

n

y�

2(�

2)x

��

2

y�

�4

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plify

.

Th

e so

luti

on i

s (�

2,�

4).

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�2

2x � 2

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ve f

or o

ne

vari

able

,th

ensu

bst

itu

te.

x�

3y�

72x

�4y

��

6

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ve t

he

firs

t eq

uat

ion

for

xsi

nce

th

e co

effi

cien

tof

xis

1.

x�

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irst

equa

tion

x�

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3y�

7 �

3yS

ubtr

act

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om e

ach

side

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x�

7 �

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impl

ify.

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d th

e va

lue

of y

by s

ubs

titu

tin

g 7

�3y

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th

e se

con

d eq

uat

ion

.2x

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econ

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uatio

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2(7

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4y�

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x�

7 �

3y

14 �

6y�

4y�

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trib

utiv

e P

rope

rty

14 �

10y

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6C

ombi

ne li

ke t

erm

s.

14 �

10y

�14

��

6 �

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ubtr

act

14 f

rom

eac

h si

de.

�10

y�

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plify

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ivid

e ea

ch s

ide

by �

10.

y�

2S

impl

ify.

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y�

2 to

fin

d th

e va

lue

of x

.x

�7

�3y

x�

7 �

3(2)

x�

1

Th

e so

luti

on i

s (1

,2).

�20

� �10

�10

y� �

10

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ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Use

su

bst

itu

tion

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.If

th

e sy

stem

doe

s n

oth

ave

exac

tly

one

solu

tion

,sta

te w

het

her

it

has

no

solu

tion

or

infi

nit

ely

ma

ny

solu

tion

s.

1.y

�4x

2.x

�2y

3.x

�2y

�3

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no

so

luti

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�2y

��

15.

c�

4d�

1in

fin

itel

y 6.

x�

2y�

03y

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)2c

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man

y3x

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(4,�

2)

7.2b

�6a

�14

infi

nit

ely

8.x

�y

�16

9.y

��

x�

3n

o3a

�b

�7

man

y2y

��

2x�

2n

o s

olu

tio

n2y

�2x

�4

solu

tio

n

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(20,

10)

11.x

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.�0.

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lenc

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Rea

l-W

orl

d P

rob

lem

sS

ubs

titu

tion

can

als

o be

use

d to

sol

ve r

eal-

wor

ld p

robl

ems

invo

lvin

g sy

stem

s of

equ

atio

ns.

It m

ay b

e h

elpf

ul

to u

se t

able

s,ch

arts

,dia

gram

s,or

gra

phs

to h

elp

you

org

aniz

e da

ta.

CH

EMIS

TRY

How

mu

ch o

f a

10%

sal

ine

solu

tion

sh

ould

be

mix

edw

ith

a 2

0% s

alin

e so

luti

on t

o ob

tain

100

0 m

illi

lite

rs o

f a

12%

sal

ine

solu

tion

?

Let

s�

the

nu

mbe

r of

mil

lili

ters

of

10%

sal

ine

solu

tion

.L

et t

�th

e n

um

ber

of m

illi

lite

rs o

f 20

% s

alin

e so

luti

on.

Use

a t

able

to

orga

niz

e th

e in

form

atio

n.

10%

sal

ine

20%

sal

ine

12%

sal

ine

Tota

l mill

ilite

rss

t10

00

Mill

ilite

rs o

f sa

line

0.10

s0.

20t

0.12

(100

0)

Wri

te a

sys

tem

of

equ

atio

ns.

s�

t�

1000

0.10

s�

0.20

t�

0.12

(100

0)U

se s

ubs

titu

tion

to

solv

e th

is s

yste

m.

s�

t�

1000

Firs

t eq

uatio

n

s�

1000

�t

Sol

ve f

or s

.

0.10

s�

0.20

t�

0.12

(100

0)S

econ

d eq

uatio

n

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(100

0 �

t) �

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t�

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(100

0)s

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00 �

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000)

Dis

trib

utiv

e P

rope

rty

100

�0.

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000)

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bine

like

ter

ms.

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t�

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impl

ify.

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ivid

e ea

ch s

ide

by 0

.10.

t�

200

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plify

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s�

t�

1000

Firs

t eq

uatio

n

s�

200

�10

00t�

200

s�

800

Sol

ve f

or s

.

800

mil

lili

ters

of

10%

sol

uti

on a

nd

200

mil

lili

ters

of

20%

sol

uti

on s

hou

ld b

e u

sed.

1.SP

OR

TSA

t th

e en

d of

the

200

0-20

01 f

ootb

all s

easo

n,31

Sup

er B

owl g

ames

had

bee

npl

ayed

wit

h th

e cu

rren

t tw

o fo

otba

ll le

ague

s,th

e A

mer

ican

Foo

tbal

l Con

fere

nce

(AF

C)

and

the

Nat

ion

al F

ootb

all

Con

fere

nce

(N

FC

).T

he

NF

C w

on f

ive

mor

e ga

mes

th

an t

he

AF

C.

How

man

y ga

mes

did

eac

h c

onfe

ren

ce w

in?

Sour

ce:N

ew Y

ork

Times

Alm

anac

AF

C 1

3;N

FC

18

2.C

HEM

ISTR

YA

lab

nee

ds t

o m

ake

100

gall

ons

of a

n 1

8% a

cid

solu

tion

by

mix

ing

a 12

%ac

id s

olu

tion

wit

h a

20%

sol

uti

on.H

ow m

any

gall

ons

of e

ach

sol

uti

on a

re n

eede

d?25

gal

of

12%

so

luti

on

an

d 7

5 g

al o

f 20

% s

olu

tio

n

3.G

EOM

ETRY

Th

e pe

rim

eter

of

a tr

ian

gle

is 2

4 in

ches

.Th

e lo

nge

st s

ide

is 4

in

ches

lon

ger

than

th

e sh

orte

st s

ide,

and

the

shor

test

sid

e is

th

ree-

fou

rth

s th

e le

ngt

h o

f th

e m

iddl

esi

de.F

ind

the

len

gth

of

each

sid

e of

th

e tr

ian

gle.

6 in

.,8

in.,

10 in

.

20� 0.

100.

10t

� 0.10

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Su

bst

itu

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Su

bst

itu

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

©G

lenc

oe/M

cGra

w-H

ill41

1G

lenc

oe A

lgeb

ra 1

Lesson 7-2

Use

su

bst

itu

tion

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.If

th

e sy

stem

doe

s n

oth

ave

exac

tly

one

solu

tion

,sta

te w

het

her

it

has

no

solu

tion

or

infi

nit

ely

ma

ny

solu

tion

s.

1.y

�4x

2.y

�2x

x�

y�

5(1

,4)

x�

3y�

�14

(�2,

�4)

3.y

�3x

4.x

��

4y2x

�y

�15

(3,9

)3x

�2y

�20

(8,�

2)

5.y

�x

�1

6.x

�y

�7

x�

y�

3(2

,1)

x�

8y�

2(�

6,1)

7.y

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8.y

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y�

2x�

5(�

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9)5x

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(�1,

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10.y

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y�

3 �

x(6

,�3)

4x�

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33(3

,7)

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3x�

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6(7

,3)

3x�

15y

��

1n

o s

olu

tio

n

13.3

x�

y�

414

.x�

4y�

82x

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9(3

,5)

2x�

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29(1

2,�

1)

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16.5

x�

2y�

142x

�10

y�

20in

fin

itel

y m

any

2x�

y�

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,3)

17.2

x�

5y�

3818

.x�

4y�

27x

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3(9

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3x�

y�

�23

(�5,

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19.2

x�

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720

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x�

y�

�2

x�

2y�

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3x�

2y�

0(�

2,3)

3 � 2

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lenc

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w-H

ill41

2G

lenc

oe A

lgeb

ra 1

Use

su

bst

itu

tion

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.If

th

e sy

stem

doe

s n

oth

ave

exac

tly

one

solu

tion

,sta

te w

het

her

it

has

no

solu

tion

or

infi

nit

ely

ma

ny

solu

tion

s.

1.y

�6x

2.x

�3y

3.x

�2y

�7

2x�

3y�

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(�1,

�6)

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5y�

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,3)

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y�

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5.y

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y�

x�

2(4

,6)

2x�

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2n

o s

olu

tio

ny

��

x�

2(7

,�9)

7.x

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8)8.

x�

2y�

3in

fin

itel

y9.

x�

5y�

36(�

4,�

8)�

2x�

3y�

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8y�

12m

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2x�

y�

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x�

3y�

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y�

8412

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4,5)

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)

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15.

x�

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121 � 2

Pra

ctic

e (

Ave

rag

e)

Su

bst

itu

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

x�

2.5y

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5(6

,�1)

x�

y�

4(5

,2)

x�

2y�

6(1

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317

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(12,

1)�

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MP

LO

YM

EN

TF

or E

xerc

ises

19–

21,u

se t

he

foll

owin

g in

form

atio

n.

Ken

ish

a se

lls

ath

leti

c sh

oes

part

-tim

e at

a d

epar

tmen

t st

ore.

Sh

e ca

n e

arn

eit

her

$50

0 pe

rm

onth

plu

s a

4% c

omm

issi

on o

n h

er t

otal

sal

es,o

r $4

00 p

er m

onth

plu

s a

5% c

omm

issi

on o

nto

tal

sale

s.

19.W

rite

a s

yste

m o

f eq

uat

ion

s to

rep

rese

nt

the

situ

atio

n.

y�

0.04

x�

500

and

y�

0.05

x�

400

20.W

hat

is

the

tota

l pr

ice

of t

he

ath

leti

c sh

oes

Ken

ish

a n

eeds

to

sell

to

earn

th

e sa

me

inco

me

from

eac

h p

ay s

cale

?$1

0,00

0

21.W

hic

h i

s th

e be

tter

off

er?

the

firs

t o

ffer

if s

he

exp

ects

to

sel

l les

s th

an$1

0,00

0 in

sh

oes

,an

d t

he

seco

nd

off

er if

sh

e ex

pec

ts t

o s

ell m

ore

th

an$1

0,00

0 in

sh

oes

MO

VIE

TIC

KET

SF

or E

xerc

ises

22

and

23,

use

th

e fo

llow

ing

info

rmat

ion

.T

icke

ts t

o a

mov

ie c

ost

$7.2

5 fo

r ad

ult

s an

d $5

.50

for

stu

den

ts.A

gro

up

of f

rien

ds p

urc

has

ed8

tick

ets

for

$52.

75.

22.W

rite

a s

yste

m o

f eq

uat

ion

s to

rep

rese

nt

the

situ

atio

n.

x�

y�

8 an

d 7

.25x

�5.

5y�

52.7

5

23.H

ow m

any

adu

lt t

icke

ts a

nd

stu

den

t ti

cket

s w

ere

purc

has

ed?

5 ad

ult

an

d 3

stu

den

t

1 � 123 � 4

2 � 31 � 3

1 � 3

1 � 2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csS

ub

stit

uti

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

©G

lenc

oe/M

cGra

w-H

ill41

3G

lenc

oe A

lgeb

ra 1

Lesson 7-2

Pre-

Act

ivit

yH

ow c

an a

sys

tem

of

equ

atio

ns

be

use

d t

o p

red

ict

med

ia u

se?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-2

at

the

top

of p

age

376

in y

our

text

book

.

•W

hat

is

the

syst

em o

f eq

uat

ion

s?y

��

2.8x

�17

0,y

�14

.4x

�2

•B

ased

on

th

e gr

aph

,are

th

ere

0,1,

or i

nfi

nit

ely

man

y so

luti

ons

of t

he

syst

em?

1 so

luti

on

Rea

din

g t

he

Less

on

1.D

escr

ibe

how

you

wou

ld u

se s

ubs

titu

tion

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.

a.y

��

2xx

�3y

�15

b.

3x�

2y�

12x

�2y

c.x

�2y

�7

2x�

8y�

8

d.

�3x

�5y

�81

2x�

y�

24

2.Je

ss s

olve

d a

syst

em o

f eq

uat

ion

s an

d h

er r

esu

lt w

as �

8 �

�8.

All

of

her

wor

k w

asco

rrec

t.D

escr

ibe

the

grap

h o

f th

e sy

stem

.Exp

lain

.It

is a

lin

e.T

her

e ar

e in

fin

itel

ym

any

solu

tio

ns,

sin

ce �

8 �

�8

is a

lway

s tr

ue.

3.M

igu

el s

olve

d a

syst

em o

f eq

uat

ion

s an

d h

is r

esu

lt w

as 5

��

2.A

ll o

f h

is w

ork

was

corr

ect.

Des

crib

e th

e gr

aph

of

the

syst

em.E

xpla

in.

Th

e g

rap

h h

as t

wo

par

alle

llin

es,s

ince

5 �

�2

is a

lway

s fa

lse.

Hel

pin

g Y

ou

Rem

emb

er

4.W

hat

is

usu

ally

th

e fi

rst

step

in

sol

vin

g a

syst

em o

f eq

uat

ion

s by

su

bsti

tuti

on?

Sam

ple

answ

er:

So

lve

on

e o

f th

e eq

uat

ion

s fo

r o

ne

vari

able

in t

erm

s o

f th

e o

ther

.

So

lve

the

seco

nd

eq

uat

ion

fo

r y.

Su

bst

itu

te t

he

exp

ress

ion

yo

u f

ind

fo

r y

in t

he

firs

t eq

uat

ion

.Sim

plif

yan

d s

olv

e fo

r x.

Th

en u

se t

he

valu

e o

f x

to f

ind

th

eva

lue

of

y.

So

lve

the

firs

t eq

uat

ion

fo

r x.

Su

bst

itu

te t

he

exp

ress

ion

you

fin

d f

or

xin

th

e se

con

d e

qu

atio

n.S

imp

lify

and

so

lve

for

y.T

hen

use

th

e va

lue

of

yto

fin

d t

he

valu

e o

f x.

Su

bst

itu

te 2

yfo

r x

in t

he

firs

t eq

uat

ion

.Sim

plif

y an

dso

lve

for

y.T

hen

use

th

e va

lue

of

yto

fin

d t

he

valu

e o

f x.

Su

bst

itu

te �

2xfo

r y

in t

he

seco

nd

eq

uat

ion

.Sim

plif

yan

d s

olv

e fo

r x.

Th

en u

se t

he

valu

e o

f x

to f

ind

th

eva

lue

of

y.

©G

lenc

oe/M

cGra

w-H

ill41

4G

lenc

oe A

lgeb

ra 1

Eq

uat

ion

s o

f L

ines

an

d P

lan

es in

Inte

rcep

t F

orm

On

e fo

rm t

hat

a l

inea

r eq

uat

ion

may

tak

e is

in

terc

ept

form

.Th

e co

nst

ants

aan

d b

are

the

x- a

nd

y-in

terc

epts

of

th

e gr

aph

.

��

1

In t

hre

e-di

men

sion

al s

pace

,th

e eq

uat

ion

of

a pl

ane

take

s a

sim

ilar

for

m.

��

�1

Her

e,th

e co

nst

ants

a,b

,an

d c

are

the

poin

ts w

her

e th

epl

ane

mee

ts t

he

x,y,

and

z-ax

es.

Sol

ve e

ach

pro

ble

m.

z � cy � b

x � a

y � bx � a

z

y

x

O

x – 8 � y – 7 �

z – 6 � 1

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2

1.G

raph

th

e eq

uat

ion

�

��

1.

5.G

raph

th

e eq

uat

ion

�

��

1.

2.F

or t

he

plan

e in

Exe

rcis

e 1,

wri

te a

neq

uat

ion

for

th

e li

ne

wh

ere

the

plan

ein

ters

ects

th

e xy

-pla

ne.

Use

in

terc

ept

form

s.

��

1

3.W

rite

an

equ

atio

n f

or t

he

lin

e w

her

eth

e pl

ane

inte

rsec

ts t

he

xz-p

lan

e.

��

1

4.W

rite

an

equ

atio

n f

or t

he

lin

e w

her

eth

e pl

ane

inte

rsec

ts t

he

yz-p

lan

e.

��

1

6.W

rite

an

equ

atio

n f

or t

he

xy-p

lan

e.

z�

0

7.W

rite

an

equ

atio

n f

or t

he

yz-p

lan

e.

x�

0

8.W

rite

an

equ

atio

n f

or a

pla

ne

para

llel

to t

he

xy-p

lan

e w

ith

a z

-in

terc

ept

of 2

.

z�

2

9.W

rite

an

equ

atio

n f

or a

pla

ne

para

llel

to t

he y

z-pl

ane

wit

h an

x-i

nter

cept

of

23.

x�

�3z �

y �

z �x �

y �x �

z

y

x

O

z � 2y � 4

x � 1

z

y

x

O

z � 1y � 2

x � 3



Stu

dy G

uid

e a

nd I

nte

rven

tion

Elim

inat

ion

Usi

ng

Add

itio

n a

nd

Su

btr

acti

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3

©G

lenc

oe/M

cGra

w-H

ill41

5G

lenc

oe A

lgeb

ra 1

Lesson 7-3

Elim

inat

ion

Usi

ng

Ad

dit

ion

In s

yste

ms

of e

quat

ion

s in

wh

ich

th

e co

effi

cien

ts o

f th

ex

or y

term

s ar

e ad

diti

ve i

nve

rses

,sol

ve t

he

syst

em b

y ad

din

g th

e eq

uat

ion

s.B

ecau

se o

ne

ofth

e va

riab

les

is e

lim

inat

ed,t

his

met

hod

is

call

ed e

lim

inat

ion

.

Use

ad

dit

ion

to

solv

e th

esy

stem

of

equ

atio

ns.

x�

3y�

73x

�3y

�9

Wri

te t

he

equ

atio

ns

in c

olu

mn

for

m a

nd

add

to e

lim

inat

e y.

x�

3y�

7(�

) 3x

�3y

�9

4x�

16S

olve

for

x.

�

x�

4S

ubs

titu

te 4

for

xin

eit

her

equ

atio

n a

nd

solv

e fo

r y.

4 �

3y�

74

�3y

�4

�7

�4

�3y

�3

�

y�

�1

Th

e so

luti

on i

s (4

,�1)

.

3� �

3�

3y� �

3

16 � 44x � 4

Th

e su

m o

f tw

o n

um

ber

sis

70

and

th

eir

dif

fere

nce

is

24.F

ind

the

nu

mb

ers.

Let

xre

pres

ent

one

nu

mbe

r an

d y

repr

esen

tth

e ot

her

nu

mbe

r.x

�y

�70

(�)

x�

y�

242x

�94

�

x�

47S

ubs

titu

te 4

7 fo

r x

in e

ith

er e

quat

ion

.47

�y

�70

47 �

y�

47 �

70 �

47y

�23

Th

e n

um

bers

are

47

and

23.

94 � 22x � 2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.x

�y

��

42.

2m �

3n�

143.

3a�

b�

�9

x�

y�

2(�

1,�

3)m

�3n

��

11(1

,�4)

�3a

�2b

�0

(�2,

3)

4.�

3x�

4y�

�1

5.3c

�d

�4

6.�

2x�

2y�

93x

�y

��

4(�

1,1)

2c�

d�

6(2

,�2)

2x�

y�

�6

��,3

�7.

2x�

2y�

�2

8.4x

�2y

��

19.

x�

y�

23x

�2y

�12

(2,�

3)�

4x�

4y�

�2

��1,

��

x�

y�

�3

��,�

�10

.2x

�3y

�12

11.�

0.2x

�y

�0.

512

.0.1

x�

0.3y

�0.

94x

�3y

�24

(6,0

)0.

2x�

2y�

1.6

(1,0

.7)

0.1x

�0.

3y�

0.2

�,

�13

.Rem

a is

old

er t

han

Ken

.Th

e di

ffer

ence

of

thei

r ag

es i

s 12

an

d th

e su

m o

f th

eir

ages

is

50.F

ind

the

age

of e

ach

.R

ema

is 3

1 an

d K

en is

19.

14.T

he

sum

of

the

digi

ts o

f a

two-

digi

t n

um

ber

is 1

2.T

he

diff

eren

ce o

f th

e di

gits

is

2.F

ind

the

nu

mbe

r if

th

e u

nit

s di

git

is l

arge

r th

an t

he

ten

s di

git.

57

7 � 611 � 25 � 2

1 � 23 � 2

3 � 2

©G

lenc

oe/M

cGra

w-H

ill41

6G

lenc

oe A

lgeb

ra 1

Elim

inat

ion

Usi

ng

Su

btr

acti

on

In s

yste

ms

of e

quat

ion

s w

her

e th

e co

effi

cien

ts o

f th

ex

or y

term

s ar

e th

e sa

me,

solv

e th

e sy

stem

by

subt

ract

ing

the

equ

atio

ns.

Use

su

btr

acti

on t

o so

lve

the

syst

em o

f eq

uat

ion

s.2x

�3y

�11

5x�

3y�

14

2x�

3y�

11W

rite

the

equa

tions

in c

olum

n fo

rm a

nd s

ubtr

act.

(�)

5x�

3y�

14�

3x�

�3

Sub

trac

t th

e tw

o eq

uatio

ns.

yis

elim

inat

ed.

�D

ivid

e ea

ch s

ide

by �

3.

x�

1S

impl

ify.

2(1)

�3y

�11

Sub

stitu

te 1

for

xin

eith

er e

quat

ion.

2 �

3y�

11S

impl

ify.

2 �

3y�

2 �

11 �

2S

ubtr

act

2 fr

om e

ach

side

.

�3y

�9

Sim

plify

.

�D

ivid

e ea

ch s

ide

by �

3.

y�

�3

Sim

plify

.

Th

e so

luti

on i

s (1

,�3)

.

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.6x

�5y

�4

2.3m

�4n

��

143.

3a�

b�

16x

�7y

��

203m

�2n

��

2a

�b

�3

(�1,

2)(�

2,2)

(�1,

4)

4.�

3x�

4y�

�23

5.c

�3d

�11

6.x

�2y

�6

�3x

�y

�2

2c�

3d�

16x

�y

�3

(1,5

)(5

,�2)

(4,�

1)

7.2a

�3b

��

138.

4x�

2y�

69.

5s�

t�

62a

�2b

�7

4x�

4y�

105s

�2t

�3

��,4

��

,2�

(1,�

1)

10.6

x�

3y�

1211

.x�

2y�

3.5

12.0

.2x

�y

�0.

74x

�3y

�24

x�

3y�

�9

0.2x

�2y

�1.

2(�

6,�

16)

(�1.

5,2.

5)(1

,0.5

)

13.T

he

sum

of

two

nu

mbe

rs i

s 70

.On

e n

um

ber

is t

en m

ore

than

tw

ice

the

oth

er n

um

ber.

Fin

d th

e n

um

bers

.50

;20

14.G

EOM

ETRY

Tw

o an

gles

are

su

pple

men

tary

.Th

e m

easu

re o

f on

e an

gle

is 1

0°m

ore

than

thre

e ti

mes

th

e ot

her

.Fin

d th

e m

easu

re o

f ea

ch a

ngl

e.42

.5�;

137.

5�

1 � 21 � 2

9� �

3�

3y� �

3

�3

� �3

�3x

� �3

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Elim

inat

ion

Usi

ng

Add

itio

n a

nd

Su

btr

acti

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Elim

inat

ion

Usi

ng

Add

itio

n a

nd

Su

btr

acti

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3

©G

lenc

oe/M

cGra

w-H

ill41

7G

lenc

oe A

lgeb

ra 1

Lesson 7-3

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.x

�y

�1

2.�

x�

y�

1x

�y

�3

(2,1

)x

�y

�11

(5,6

)

3.x

�4y

�11

4.�

x�

3y�

6x

�6y

�11

(11,

0)x

�3y

�18

(6,4

)

5.3x

�4y

�19

6.x

�4y

��

83x

�6y

�33

(�3,

7)x

�4y

��

8(�

8,0)

7.3a

�4b

�2

8.3c

�d

��

14a

�4b

�12

(2,�

1)�

3c�

d�

5(�

1,�

2)

9.2x

�3y

�9

10.x

�y

�4

�5x

�3y

�30

(�3,

�5)

2x�

y�

�4

(0,�

4)

11.3

m�

n�

2612

.5x

�y

��

6�

2m�

n�

�24

(10,

4)�

x�

y�

2(�

1,1)

13.6

x�

2y�

3214

.3x

�2y

��

194x

�2y

�18

(7,5

)�

3x�

5y�

25(�

5,�

2)

15.7

m�

4n�

216

.2x

�5y

��

287m

�2n

�8

(2,�

3)4x

�5y

�4

(�4,

4)

17.T

he

sum

of

two

nu

mbe

rs i

s 28

an

d th

eir

diff

eren

ce i

s 4.

Wh

at a

re t

he

nu

mbe

rs?

12,1

6

18.F

ind

the

two

nu

mbe

rs w

hos

e su

m i

s 29

an

d w

hos

e di

ffer

ence

is

15.

7,22

19.T

he

sum

of

two

nu

mbe

rs i

s 24

an

d th

eir

diff

eren

ce i

s 2.

Wh

at a

re t

he

nu

mbe

rs?

13,1

1

20.F

ind

the

two

nu

mbe

rs w

hos

e su

m i

s 54

an

d w

hos

e di

ffer

ence

is

4.25

,29

21.T

wo

tim

es a

nu

mbe

r ad

ded

to a

not

her

nu

mbe

r is

25.

Th

ree

tim

es t

he

firs

t n

um

ber

min

us

the

oth

er n

um

ber

is 2

0.F

ind

the

nu

mbe

rs.

9,7

©G

lenc

oe/M

cGra

w-H

ill41

8G

lenc

oe A

lgeb

ra 1

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.x

�y

�1

2.p

�q

��

23.

4x�

y�

23x

�y

��

9p

�q

�8

3x�

y�

12

(�4,

�5)

(3,�

5)(5

,3)

4.2x

�5y

��

35.

3x�

2y�

�1

6.5x

�3y

�22

2x�

2y�

64x

�2y

��

65x

�2y

�2

(6,�

3)(�

5,7)

(2,4

)

7.5x

�2y

�7

8.3x

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��

129.

�4c

�2d

��

2�

2x�

2y�

�14

3x�

15y

��

62c

�2d

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14

(3,�

4)(�

7,�

1)(�

2,5)

10.2

x�

6y�

611

.7x

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�2

12.4

.25x

�1.

28y

��

9.2

2x�

3y�

247x

�2y

��

30x

�1.

28y

�17

.6

(9,2

)(�

2,8)

(1.6

,12.

5)

13.2

x�

4y�

1014

.2.5

x�

y�

10.7

15.6

m�

8n�

3x

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��

2.5

2.5x

�2y

�12

.92m

�8n

��

3

(2.5

,1.2

5)(3

.4,2

.2)

�1,

�16

.4a

�b

�2

17.�

x�

y�

�2

18.

x�

y�

84a

�3b

�10

x�

y�

4x

�y

�19

(10,

�1)

(12,

2)

19.T

he

sum

of

two

nu

mbe

rs i

s 41

an

d th

eir

diff

eren

ce i

s 5.

Wh

at a

re t

he

nu

mbe

rs?

18,2

3

20.F

our

tim

es o

ne

nu

mbe

r ad

ded

to a

not

her

nu

mbe

r is

36.

Th

ree

tim

es t

he

firs

t n

um

ber

min

us

the

oth

er n

um

ber

is 2

0.F

ind

the

nu

mbe

rs.

8,4

21.O

ne

nu

mbe

r ad

ded

to t

hre

e ti

mes

an

oth

er n

um

ber

is 2

4.F

ive

tim

es t

he

firs

t n

um

ber

adde

d to

th

ree

tim

es t

he

oth

er n

um

ber

is 3

6.F

ind

the

nu

mbe

rs.

3,7

22.L

AN

GU

AG

ESE

ngl

ish

is

spok

en a

s th

e fi

rst

or p

rim

ary

lan

guag

e in

78

mor

e co

un

trie

sth

an F

arsi

is

spok

en a

s th

e fi

rst

lan

guag

e.T

oget

her

,En

glis

h a

nd

Fars

i ar

e sp

oken

as

afi

rst

lan

guag

e in

130

cou

ntr

ies.

In h

ow m

any

cou

ntr

ies

is E

ngl

ish

spo

ken

as

the

firs

tla

ngu

age?

In

how

man

y co

un

trie

s is

Far

si s

poke

n a

s th

e fi

rst

lan

guag

e?E

ng

lish

:10

4 co

un

trie

s,Fa

rsi:

26 c

ou

ntr

ies

23.D

ISC

OU

NTS

At

a sa

le o

n w

inte

r cl

oth

ing,

Cod

y bo

ugh

t tw

o pa

irs

of g

love

s an

d fo

ur

hat

sfo

r $4

3.00

.Tor

i bo

ugh

t tw

o pa

irs

of g

love

s an

d tw

o h

ats

for

$30.

00.W

hat

wer

e th

e pr

ices

for

the

glov

es a

nd

hat

s?g

love

s:$8

.50,

hat

s:$6

.50.

1 � 23 � 2

2 � 31 � 3

1 � 23 � 4

4 � 31 � 3

3 � 41 � 2

Pra

ctic

e (

Ave

rag

e)

Elim

inat

ion

Usi

ng

Add

itio

n a

nd

Su

btr

acti

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3 ��

,4�

1 � 2



Readin

g t

o L

earn

Math

em

ati

csE

limin

atio

n U

sin

g A

ddit

ion

an

d S

ub

trac

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3

©G

lenc

oe/M

cGra

w-H

ill41

9G

lenc

oe A

lgeb

ra 1

Lesson 7-3

Pre-

Act

ivit

yH

ow c

an y

ou u

se a

sys

tem

of

equ

atio

ns

to s

olve

pro

ble

ms

abou

tw

eath

er?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-3

at

the

top

of p

age

382

in y

our

text

book

.

Wh

at f

act

expl

ain

s w

hy

the

vari

able

dge

ts e

lim

inat

ed f

rom

th

e sy

stem

of

equ

atio

ns?

d�

d�

0

Rea

din

g t

he

Less

on

1.W

rite

ad

dit

ion

or s

ubt

ract

ion

to t

ell

wh

ich

ope

rati

on i

t w

ould

be

easi

est

to u

se t

oel

imin

ate

a va

riab

le o

f th

e sy

stem

.Exp

lain

you

r ch

oice

.

Sys

tem

of

Eq

uat

ion

sO

per

atio

nE

xpla

nat

ion

a.3x

�5y

�12

add

itio

nT

he

coef

fici

ents

of

the

xte

rms

are

�3x

�2y

�6

add

itiv

e in

vers

es.

b.

3x�

5y�

7su

btr

acti

on

Th

e co

effi

cien

ts o

f th

e x

term

s ar

e 3x

�2y

�8

the

sam

e.

c.�

x�

4y�

9su

btr

acti

on

Th

e co

effi

cien

ts o

f th

e y

term

s ar

e 4x

�4y

�6

the

sam

e.

d.

5x�

7y�

17ad

dit

ion

Th

e co

effi

cien

ts o

f th

e y

term

s ar

e 8x

�7y

�9

add

itiv

e in

vers

es.

Hel

pin

g Y

ou

Rem

emb

er

2.T

ell

how

you

can

dec

ide

wh

eth

er t

o u

se a

ddit

ion

or

subt

ract

ion

to

elim

inat

e a

vari

able

in

a sy

stem

of

equ

atio

ns.

Sam

ple

an

swer

:L

oo

k at

th

e co

effi

cien

ts o

f ea

chva

riab

le.I

f th

e co

effi

cien

ts o

f o

ne

of

the

vari

able

s ar

e ad

dit

ive

inve

rses

,yo

u c

an u

se a

dd

itio

n t

o e

limin

ate

the

vari

able

.If

the

coef

fici

ents

of

on

eva

riab

le a

re t

he

sam

e,yo

u c

an u

se s

ub

trac

tio

n t

o e

limin

ate

the

vari

able

.

©G

lenc

oe/M

cGra

w-H

ill42

0G

lenc

oe A

lgeb

ra 1

Ró

zsa

Pét

erR

ózsa

Pét

er (

1905

–197

7) w

as a

Hun

gari

an m

athe

mat

icia

n de

dica

ted

tote

ach

ing

oth

ers

abou

t m

ath

emat

ics.

As

prof

esso

r of

mat

hem

atic

s at

ate

ache

rs’ c

olle

ge in

Bud

apes

t,sh

e w

rote

sev

eral

mat

hem

atic

s te

xtbo

oks

and

cham

pion

ed r

efor

ms

in t

he

teac

hin

g of

mat

hem

atic

s.In

194

5 sh

ew

rote

Pla

ying

wit

h In

fini

ty:M

athe

mat

ical

Exp

lora

tion

s an

d E

xcur

sion

s,a

popu

lar

wor

k in

wh

ich

sh

e at

tem

pted

to

con

vey

the

spir

it o

fm

ath

emat

ics

to t

he

gen

eral

pu

blic

.

By

far

Pét

er’s

gre

ates

t co

ntri

buti

on t

o m

athe

mat

ics

was

her

pio

neer

ing

rese

arch

in

th

e fi

eld

of r

ecu

rsiv

e fu

nct

ion

th

eory

.Wh

en y

ou e

valu

ate

afu

ncti

on r

ecur

sive

ly,y

ou b

egin

wit

h on

e in

itia

l val

ue o

f x.

Wor

king

fro

mth

is s

ingl

e n

um

ber,

you

can

use

th

e fu

nct

ion

to

gen

erat

e an

en

tire

sequ

ence

of

nu

mbe

rs.F

or i

nst

ance

,her

e is

how

you

use

an

in

itia

lva

lue

of x

�1

to e

valu

ate

the

fun

ctio

n f

(x)

�3x

recu

rsiv

ely.

f(1)

�3(

1) �

3 ↓f(

3) �

3(3)

�9 ↓

f(9)

�3(

9) �

27

↓f(

27)

�3(

27)

�81

↓

f(81

) �

3(81

) �

243 ↓

Th

e fi

rst

five

nu

mbe

rs o

f th

e se

quen

ce g

ener

ated

by

this

fu

nct

ion

are

3,

9,27

,81,

and

243.

Wri

te t

he

firs

t fi

ve n

um

ber

s of

th

e se

qu

ence

gen

erat

ed b

y ea

ch

fun

ctio

n,u

sin

g th

e gi

ven

nu

mb

er a

s th

e in

itia

l va

lue

of x

.

1.f(

x) �

3x;x

�2

2.g(

x) �

x�

5;x

�1

6,18

,54,

162,

486

�4,

�9,

�14

,�19

,�24

3.f(

x) �

2x�

1;x

��

34.

f(x)

�x2

;x�

2

�5,

�9,

�17

,�33

,�65

4;16

;25

6;65

,536

;4,

294,

967,

296

5.h

(x)

��

x;x

�3

6.k(

x) �

;x�

10

�3,

3,�

3,3,

�3

,10,

,10,

1 � 101 � 10

1 � 10

1 � x

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-H

ill42

1G

lenc

oe A

lgeb

ra 1

Lesson 7-4

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

Som

e sy

stem

s of

equ

atio

ns

can

not

be

solv

edsi

mpl

y by

add

ing

or s

ubt

ract

ing

the

equ

atio

ns.

In s

uch

cas

es,o

ne

or b

oth

equ

atio

ns

mu

stfi

rst

be m

ult

ipli

ed b

y a

nu

mbe

r be

fore

th

e sy

stem

can

be

solv

ed b

y el

imin

atio

n.

Use

eli

min

atio

n t

o so

lve

the

syst

em o

f eq

uat

ion

s.x

�10

y�

34x

�5y

�5

If y

ou m

ult

iply

th

e se

con

d eq

uat

ion

by

�2,

you

can

eli

min

ate

the

yte

rms.

x�

10y

�3

(�)

�8x

�10

y�

�10

�7x

��

7

�

x�

1S

ubs

titu

te 1

for

xin

eit

her

equ

atio

n.

1 �

10y

�3

1 �

10y

�1

�3

�1

10y

�2

�

y�

Th

e so

luti

on i

s �1,

�.1 � 5

1 � 52 � 1010

y� 10

�7

� �7

�7x

� �7

Use

eli

min

atio

n t

o so

lve

the

syst

em o

f eq

uat

ion

s.3x

�2y

��

72x

�5y

�10

If y

ou m

ult

iply

th

e fi

rst

equ

atio

n b

y 2

and

the

seco

nd

equ

atio

n b

y �

3,yo

u c

anel

imin

ate

the

xte

rms.

6x�

4y�

�14

(�)

�6x

�15

y�

�30

11y

��

44

�

y�

�4

Su

bsti

tute

�4

for

yin

eit

her

equ

atio

n.

3x�

2(�

4) �

�7

3x�

8 �

�7

3x�

8 �

8 �

�7

�8

3x�

�15

�

x�

�5

Th

e so

luti

on i

s (�

5,�

4).

�15

�3

3x � 3

�44

�11

11y

� 11

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.2x

�3y

�6

2.2m

�3n

�4

3.3a

�b

�2

x�

2y�

5(�

3,4)

�m

�2n

�5

(�1,

2)a

�2b

�3

(1,1

)

4.4x

�5y

�6

5.4c

�3d

�22

6.3x

�4y

��

46x

�7y

��

20(�

1,2)

2c�

d�

10(4

,�2)

x�

3y�

�10

(�4,

�2)

7.4s

�t

�9

8.4a

�3b

��

89.

2x�

2y�

55s

�2t

�8

(2,�

1)2a

�2b

�3

��,2

�4x

�4y

�10

�,0

�10

.6x

�4y

��

811

.4x

�2y

��

512

.2x

�y

�3.

5(0

.9,1

.7)

4x�

2y�

�3

��1,

��

2x�

4y�

1��

,�

�x

�2y

�2.

5

13.G

AR

DEN

ING

Th

e le

ngt

h o

f S

ally

’s g

arde

n i

s 4

met

ers

grea

ter

than

3 t

imes

th

e w

idth

.T

he

peri

met

er o

f h

er g

arde

n i

s 72

met

ers.

Wh

at a

re t

he

dim

ensi

ons

of S

ally

’s g

arde

n?

28 m

by

8 m

14.A

nit

a is

4ye

ars

olde

r th

an B

asil

io.T

hre

e ti

mes

An

ita’

s ag

e ad

ded

to s

ix t

imes

Bas

ilio

’s

age

is 3

6.H

ow o

ld a

re A

nit

a an

d B

asil

io?

1 � 2

1 � 23 � 2

1 � 2

5 � 21 � 2

An

ita:

7 yr

;B

asili

o:

2yr

1 � 2

©G

lenc

oe/M

cGra

w-H

ill42

2G

lenc

oe A

lgeb

ra 1

Det

erm

ine

the

Bes

t M

eth

od

Th

e m

eth

ods

to u

se f

or s

olvi

ng

syst

ems

of l

inea

req

uat

ion

s ar

e su

mm

ariz

ed i

n t

he

tabl

e be

low

.

Met

ho

dT

he

Bes

t Tim

e to

Use

Gra

ph

ing

to e

stim

ate

the

solu

tion,

sin

ce g

raph

ing

usua

lly d

oes

not g

ive

an e

xact

sol

utio

n

Su

bst

itu

tio

nif

one

of t

he v

aria

bles

in e

ither

equ

atio

n ha

s a

coef

ficie

nt o

f 1

or �

1

Elim

inat

ion

Usi

ng

Ad

dit

ion

if on

e of

the

var

iabl

es h

as o

ppos

ite c

oeffi

cien

ts in

the

tw

o eq

uatio

ns

Elim

inat

ion

Usi

ng

Su

btr

acti

on

if on

e of

the

var

iabl

es h

as t

he s

ame

coef

ficie

nt in

the

tw

o eq

uatio

ns

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

if no

ne o

f th

e co

effic

ient

s ar

e 1

or �

1 an

d ne

ither

of

the

varia

bles

can

be

elim

inat

ed b

y si

mpl

y ad

ding

or

subt

ract

ing

the

equa

tions

Det

erm

ine

the

bes

t m

eth

od t

o so

lve

the

syst

em o

f eq

uat

ion

s.T

hen

solv

e th

e sy

stem

.6x

�2y

�20

�2x

�4y

��

16

Sin

ce t

he

coef

fici

ents

of

xw

ill

be a

ddit

ive

inve

rses

of

each

oth

er i

f yo

u m

ult

iply

th

e se

con

deq

uat

ion

by

3,u

se e

lim

inat

ion

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

6x�

2y�

20(�

) �

6x�

12y

��

48M

ultip

ly t

he s

econ

d eq

uatio

n by

3.

14y

��

28A

dd th

e tw

o eq

uatio

ns. x

is e

limin

ated

.

�D

ivid

e ea

ch s

ide

by 1

4.

y�

�2

Sim

plify

.

�28

�14

14y

� 14

6x�

2(�

2) �

20S

ubst

itute

�2

for

yin

eith

er e

quat

ion.

6x�

4 �

20S

impl

ify.

6x�

4 �

4 �

20 �

4A

dd 4

to

each

sid

e.

6x�

24S

impl

ify.

�D

ivid

e ea

ch s

ide

by 6

.

x�

4S

impl

ify.

24 � 66x � 6

Th

e so

luti

on i

s (4

,�2)

.

Det

erm

ine

the

bes

t m

eth

od t

o so

lve

each

sys

tem

of

equ

atio

ns.

Th

en s

olve

th

e sy

stem

.

1.x

�2y

�3

2.m

�6n

��

83.

a�

b�

6x

�y

�1

m�

2n�

8a

�2b

�7

elim

inat

ion

(�

);(�

1,2)

sub

stitu

tion

;(4,

�2)

sub

stitu

tion

;(5,

�1)

4.4x

�y

�15

5.3c

�d

�14

6.x

�2y

��

9�

x�

3y�

�12

c�

d�

2y

�4x

sub

stitu

tion

;(3,

3)el

imin

atio

n (

�);

(6,4

)su

bst

itutio

n;(

�1,

�4)

7.4x

�2y

�10

8.x

��

2y9.

2s�

3t�

42x

�2y

�5

4x�

4y�

�10

3s�

2t�

24su

bst

itutio

n;(

�1,

3)su

bst

itutio

n; ��

5,�

elim

inat

ion

(�

);(1

2,�

6)

10.4

a�

4b�

�10

11.4

x�

10y

��

612

.2x

�y

�3

2a�

4b�

�2

�2x

�10

y�

2�

x�

y�

0el

imin

atio

n (

�);

��2,

�el

imin

atio

n (

�);

��2,

�su

bst

itutio

n;(

�3,

�3)

1 � 51 � 2

5 � 2



Skil

ls P

ract

ice

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-H

ill42

3G

lenc

oe A

lgeb

ra 1

Lesson 7-4

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.x

�y

��

92.

3x�

2y�

�9

5x�

2y�

32(2

,�11

)x

�y

��

13(�

7,6)

3.2x

�5y

�3

4.2x

�y

�3

�x

�3y

��

7(4

,�1)

�4x

�4y

��

8(1

,1)

5.4x

�2y

��

146.

2x�

y�

03x

�y

��

8(�

1,5)

5x�

3y�

2(�

2,4)

7.5x

�3y

��

108.

2x�

3y�

143x

�5y

��

6(�

2,0)

3x�

4y�

4(4

,2)

9.2x

�3y

�21

10.3

x�

2y�

�26

5x�

2y�

25(3

,�5)

4x�

5y�

�4

(�6,

�4)

11.3

x�

6y�

�3

12.5

x�

2y�

�3

2x�

4y�

30(7

,4)

3x�

3y�

9(�

3,6)

13.T

wo

tim

es a

nu

mbe

r pl

us

thre

e ti

mes

an

oth

er n

um

ber

equ

als

13.T

he

sum

of

the

two

nu

mbe

rs i

s 7.

Wh

at a

re t

he

nu

mbe

rs?

8,�

1

14.F

our

tim

es a

nu

mbe

r m

inu

s tw

ice

anot

her

nu

mbe

r is

�16

.Th

e su

m o

f th

e tw

o n

um

bers

is �

1.F

ind

the

nu

mbe

rs.

�3,

2

Det

erm

ine

the

bes

t m

eth

od t

o so

lve

each

sys

tem

of

equ

atio

ns.

Th

en s

olve

th

e sy

stem

.

15.2

x�

3y�

10el

imin

atio

n (

�);

16.8

x�

7y�

18el

imin

atio

n (

�);

5x�

2y�

�8

(�4,

6)3x

�7y

�26

(4,2

)

17.y

�2x

sub

stit

uti

on

;18

.3x

�y

�6

elim

inat

ion

(�

);3x

�2y

�35

(5,1

0)3x

�y

�3

no

so

luti

on

19.3

x�

4y�

17el

imin

atio

n (

�);

20.y

�3x

�1

sub

stit

uti

on

;4x

�5y

�2

(3,�

2)3x

�y

��

1in

finite

ly m

any

solu

tion

s

©G

lenc

oe/M

cGra

w-H

ill42

4G

lenc

oe A

lgeb

ra 1

Use

eli

min

atio

n t

o so

lve

each

sys

tem

of

equ

atio

ns.

1.2x

�y

��

12.

5x�

2y�

�10

3.7x

�4y

��

43x

�2y

�1

3x�

6y�

665x

�8y

�28

(�3,

�5)

(2,1

0)(�

4,6)

4.2x

�4y

��

225.

3x�

2y�

�9

6.4x

�2y

�32

3x�

3y�

305x

�3y

�4

�3x

�5y

��

11(3

,7)

(�1,

�3)

(7,�

2)

7.3x

�4y

�27

8.0.

5x�

0.5y

��

29.

2x�

y�

�7

5x�

3y�

16x

�0.

25y

�6

x�

y�

0(5

,3)

(4,�

8)(�

2,4)

10.E

igh

t ti

mes

a n

um

ber

plu

s fi

ve t

imes

an

oth

er n

um

ber

is �

13.T

he

sum

of

the

two

nu

mbe

rs i

s 1.

Wh

at a

re t

he

nu

mbe

rs?

�6,

7

11.T

wo

tim

es a

nu

mbe

r pl

us

thre

e ti

mes

an

oth

er n

um

ber

equ

als

4.T

hre

e ti

mes

th

e fi

rst

nu

mbe

r pl

us

fou

r ti

mes

th

e ot

her

nu

mbe

r is

7.F

ind

the

nu

mbe

rs.

5,�

2

Det

erm

ine

the

bes

t m

eth

od t

o so

lve

each

sys

tem

of

equ

atio

ns.

Th

en s

olve

th

esy

stem

.

12.5

x�

7y�

313

.7x

�2y

�2

14.�

6x�

2y�

142x

�7y

��

382x

�3y

��

286x

�8y

��

20el

imin

atio

n (

�);

elim

inat

ion

(�

);el

imin

atio

n (

�);

(�5,

4)(�

2,8)

(�2,

�1)

15.x

�2y

�6

16.4

x�

3y�

�2

17.y

�x

1 � 21 � 2

3 � 4

Pra

ctic

e (

Ave

rag

e)

Elim

inat

ion

Usi

ng

Mu

ltip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4 x

�y

�3

4x�

3y�

3x

�2y

�9

sub

stit

uti

on

;in

fin

itel

y el

imin

atio

n (

�);

sub

stit

uti

on

;(6

,3)

man

y so

luti

on

sn

o s

olu

tio

n

18.F

INA

NC

EG

un

ther

in

vest

ed $

10,0

00 i

n t

wo

mu

tual

fu

nds

.On

e of

th

e fu

nds

ros

e 6%

in

one

year

,an

d th

e ot

her

ros

e 9%

in

on

e ye

ar.I

f G

un

ther

’s i

nve

stm

ent

rose

a t

otal

of

$684

in o

ne

year

,how

mu

ch d

id h

e in

vest

in

eac

h m

utu

al f

un

d?$7

200

in t

he

6% f

un

d a

nd

$28

00 in

th

e 9%

fu

nd

19.C

AN

OEI

NG

Lau

ra a

nd

Bre

nt

padd

led

a ca

noe

6 m

iles

ups

trea

m i

n f

our

hou

rs.T

he

retu

rn t

rip

took

th

ree

hou

rs.F

ind

the

rate

at

wh

ich

Lau

ra a

nd

Bre

nt

padd

led

the

can

oein

sti

ll w

ater

.1.

75 m

i/h

20.N

UM

BER

TH

EORY

Th

e su

m o

f th

e di

gits

of

a tw

o-di

git

nu

mbe

r is

11.

If t

he

digi

ts a

rere

vers

ed,t

he

new

nu

mbe

r is

45

mor

e th

an t

he

orig

inal

nu

mbe

r.F

ind

the

nu

mbe

r.38

5 � 2

1 � 2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csE

limin

atio

n U

sin

g M

ult

iplic

atio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-H

ill42

5G

lenc

oe A

lgeb

ra 1

Lesson 7-4

Pre-

Act

ivit

yH

ow c

an a

man

ager

use

a s

yste

m o

f eq

uat

ion

s to

pla

n e

mp

loye

e ti

me?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-4

at

the

top

of p

age

387

in y

our

text

book

.

Can

th

e sy

stem

of

equ

atio

ns

be s

olve

d by

eli

min

atio

n w

ith

add

itio

n o

rsu

btra

ctio

n?

Exp

lain

.N

o;

nei

ther

var

iab

le h

as c

oef

fici

ents

th

at a

read

dit

ive

inve

rses

or

equ

al.

Rea

din

g t

he

Less

on

1.C

ould

eli

min

atio

n b

y m

ult

ipli

cati

on b

e u

sed

to s

olve

th

e sy

stem

sh

own

bel

ow?

Exp

lain

.3x

�5y

�15

�6x

�7y

�11

Yes;

you

can

mu

ltip

ly t

he

firs

t eq

uat

ion

by

2 to

mak

e th

e co

effi

cien

ts o

fth

e x

term

s ad

dit

ive

inve

rses

.

2.T

ell

wh

eth

er i

t w

ould

be

easi

est

to u

se s

ubs

titu

tion

,eli

min

atio

n b

y ad

diti

on,e

lim

inat

ion

by s

ubt

ract

ion

,or

elim

inat

ion

by

mu

ltip

lica

tion

to

solv

e th

e sy

stem

.Exp

lain

you

r ch

oice

.

Sys

tem

of

Eq

uat

ion

sS

olu

tio

n M

eth

od

Exp

lan

atio

n

a.�

3x�

4y�

2el

imin

atio

n b

y T

he

coef

fici

ents

of

the

xte

rms

are

3x�

2y�

10ad

dit

ion

add

itiv

e in

vers

es.

b.

x�

2y�

0su

bst

itu

tio

nIt

is e

asy

to s

olv

e th

e fi

rst

equ

atio

n

5x�

4y�

8fo

r x.

c.6x

�5y

��

18el

imin

atio

n b

y S

amp

le a

nsw

er:Y

ou

can

mu

ltip

ly t

he

2x�

10y

�27

mu

ltip

licat

ion

firs

t eq

uat

ion

by

2 to

elim

inat

e y

byad

dit

ion

.

d.

�2x

�3y

�9

elim

inat

ion

by

Th

e co

effi

cien

ts o

f th

e y

term

s ar

e 3x

�3y

�12

sub

trac

tio

nth

e sa

me.

Hel

pin

g Y

ou

Rem

emb

er

3.If

you

are

goi

ng

to s

olve

a s

yste

m b

y el

imin

atio

n,h

ow d

o yo

u d

ecid

e w

het

her

you

wil

ln

eed

to m

ult

iply

on

e or

bot

h e

quat

ion

s by

a n

um

ber?

Sam

ple

an

swer

:If

bo

thad

din

g t

he

equ

atio

ns

and

su

btr

acti

ng

th

e eq

uat

ion

s re

sult

in e

qu

atio

ns

that

sti

ll h

ave

two

var

iab

les,

you

will

nee

d t

o u

se m

ult

iplic

atio

n.

©G

lenc

oe/M

cGra

w-H

ill42

6G

lenc

oe A

lgeb

ra 1

Geo

rge

Was

hin

gto

n C

arve

r an

d P

ercy

Ju

lian

In

199

0,G

eorg

e W

ash

ingt

on C

arve

r an

d P

ercy

Ju

lian

bec

ame

the

firs

t A

fric

anA

mer

ican

s el

ecte

d to

the

Nat

iona

l In

vent

ors

Hal

l of

Fam

e.C

arve

r (1

864–

1943

)w

as a

n ag

ricu

ltur

al s

cien

tist

kno

wn

wor

ldw

ide

for

deve

lopi

ng h

undr

eds

of u

ses

for

the

pean

ut

and

the

swee

t po

tato

.His

wor

k re

vita

lize

d th

e ec

onom

y of

th

eso

uthe

rn U

nite

d S

tate

s be

caus

e it

was

no

long

er d

epen

dent

sol

ely

upon

cot

ton.

Juli

an (

1898

–197

5) w

as a

res

earc

h c

hem

ist

wh

o be

cam

e fa

mou

s fo

r in

ven

tin

ga

met

hod

of

mak

ing

a sy

nth

etic

cor

tiso

ne

from

soy

bean

s.H

is d

isco

very

has

had

man

y m

edic

al a

ppli

cati

ons,

part

icu

larl

y in

th

e tr

eatm

ent

of a

rth

riti

s.

Th

ere

are

doze

ns

of o

ther

Afr

ican

Am

eric

an i

nve

nto

rs w

hos

e ac

com

plis

hm

ents

are

not

as

wel

l kn

own

.Th

eir

inve

nti

ons

ran

ge f

rom

com

mon

hou

seh

old

item

sli

ke t

he

iron

ing

boar

d to

com

plex

dev

ices

th

at h

ave

revo

luti

oniz

edm

anu

fact

uri

ng.

Th

e ex

erci

ses

that

fol

low

wil

l h

elp

you

ide

nti

fy ju

st a

few

of

thes

e in

ven

tors

an

d th

eir

inve

nti

ons.

Mat

ch t

he

inve

nto

rs w

ith

th

eir

inve

nti

ons

by

mat

chin

g ea

ch s

yste

mw

ith

its

sol

uti

on.(

Not

all

th

e so

luti

ons

wil

l b

e u

sed

.)

1.S

ara

Boo

ne

x�

y�

2E

A.(

1,4)

auto

mat

ic t

raff

ic s

ign

al

x�

y�

10

2.S

arah

Goo

dex

�2

�y

DB

.(4,

�2)

eggb

eate

r 2y

�x

�9

3.F

rede

rick

M.

y�

2x�

6G

C.(

�2,

3)fi

re e

xtin

guis

her

Jo

nes

y�

�x

�3

4.J.

L.L

ove

2x�

3y �

8F

D.(

�5,

7)fo

ldin

g ca

bin

et b

ed

2x�

y�

�8

5.T.

J.M

arsh

all

y�

3x�

9C

E.(

6,�

4)ir

onin

g bo

ard

2y�

x�

4

6.Ja

n M

atze

lige

ry

�4

�2x

JF.

(�2,

4)pe

nci

l sh

arpe

ner

6x

�3y

�12

7.G

arre

tt A

.3x

�2y

��

5A

G.(

�3,

0)po

rtab

le X

-ray

mac

hin

e M

orga

n3y

�4x

�8

8.N

orbe

rt R

illi

eux

3x�

y�

12I

H.(

2,�

3)pl

ayer

pia

no

y�

3x�

15

I.n

o so

luti

onev

apor

atin

g pa

n f

or r

efin

ing

suga

r

J.in

fin

itel

yla

stin

g (s

hap

ing)

m

any

mac

hin

e fo

r so

luti

ons

man

ufa

ctu

rin

g sh

oes

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

Sys

tem

s o

f In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

©G

lenc

oe/M

cGra

w-H

ill42

7G

lenc

oe A

lgeb

ra 1

Lesson 7-5

Syst

ems

of

Ineq

ual

itie

sT

he

solu

tion

of

a sy

stem

of

ineq

ual

itie

sis

th

e se

t of

all

orde

red

pair

s th

at s

atis

fy b

oth

in

equ

alit

ies.

If y

ou g

raph

th

e in

equ

alit

ies

in t

he

sam

eco

ordi

nat

e pl

ane,

the

solu

tion

is

the

regi

on w

her

e th

e gr

aph

s ov

erla

p.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by

grap

hin

g.y

�x

�2

y

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stri

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ple

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00 C

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ries

,S

amp

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nsw

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apes

,9 C

Ds;

45 f

at g

ram

s;18

00 C

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2000

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i

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IRS

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form

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7-5

7-5



Readin

g t

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earn

Math

em

ati

csG

rap

hin

g S

yste

ms

of

Ineq

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NA

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7-5

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Pre-

Act

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an y

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ineq

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pla

n a

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e in

trod

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to

Les

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7-5

at

the

top

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age

394

in y

our

text

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.

Th

e gr

een

sec

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on

th

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aph

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a r

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s a

day

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gram

s of

fat

per

day

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sym

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wh

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7-5

7-5



1.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. B

D

B

C

A

C

C

A

C

D

B

A

substitution; (0, 0)

D

C

B

A

D

C

A

B

A

D

C

B

B

C

B

A

B

C

Chapter 7 Assessment Answer KeyForm 1 Form 2APage 433 Page 434 Page 435

An

swer

s

(continued on the next page)

2.

3. C

A


13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

C

D

B

A

A

D

C

B

B

C

C

A

D

C

A

D

B

A

BC

��7, �3�12

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D

B

D

D

C

A

A

C

Chapter 7 Assessment Answer KeyForm 2A (continued) Form 2BPage 436 Page 437 Page 438

Manuel is 15 years old;his sister is 7 years old.


1.

2.3.

one solution; (4, 0)4.

infinitely many solutions5.

no solution

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

28 square miles

14 dimes;19 nickels

186 student tickets;135 adult tickets

23 and �6

y

xO

x � y � 2

2x � y � 1


(13, 1)(2, 1)

(7, �1)(�3, 2)(4, �1)(�8, 0)

infinitely many solutions

no solution(3, 5)(1, 3)

y

xO

x � y � 3

x � y ��2

y

xO

y ��x � 4

y �x � 4

one solution

no solution

Chapter 7 Assessment Answer KeyForm 2CPage 439 Page 440

An

swer

s

elimination withsubtraction; (2, �7)

y

xO

2x � y � �3

6x � 3y � �9

y

xO

y � 2x � 1

y � x � 1

Mavis is 15 years old; her brother is10 years old

Sample answers: 100 standardfans, 400 economy fans; 200standard fans, 300 economy fans;400 standard fans, 0 economy fans

e

sO

200

400

600

800

200 400 600 800Standard Fans

Eco

no

my

Fan

s

3s � 3e � 1500

2s � e � 800


1.

2.3.

one solution; (3, 0)

4.

infinitely many solutions5.

no solution

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: (�1, �3)

129 square miles

5 two-point goals;11 three-point goals

34 hours

18 and �2

y

xO

2x � y � �3

2x � y � 1

y

xO

y ��2x � 1

y � x � 1

substitution; (�2, �5)

(6, �2)(7, �3)(�2, 5)(�1, 4)(2, 2)(5, 3)

no solution


(2, 1)(2, 4)

y

xO

4x � 2y � 10

2x � y � 5

y

xO

y � x � 3

y ��x � 3

one solution


Chapter 7 Assessment Answer KeyForm 2DPage 441 Page 442

y

xO

x � y � 2

x � y � 0

elimination with subtraction; (�1, 1)

y

xO

100

200

300

400

100 200 300 400Tadashi 200

Tad

ash

i 500 2x � 3y � 900

2x � 4y � 1000

Sample answers:100 Tadashi 200, 200 Tadashi 500;250 Tadashi 200, 125 Tadashi 500;350 Tadashi 200, 75 Tadashi 500


1.

2.3.

one solution; (�1, �3)4.

no solution5.


6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: y

xO

4x � y � 1

4x � y � �3

5 mph

67

3�12

�, 8�12

�

6, 10

y

xO

x � 3y � �12

y � 2x y � x � 1


(�2, 6)

(6, 14)

(�32, 15)

(�2, �6)

(�14, 7)


(�22, �6)

no solution

(4, 1)

y

xO

3y � �x � 9

x � 3y � 3

y

xO

y �x 13

y � x � 4 � 0

no solution

one solution

An

swer

s

Chapter 7 Assessment Answer KeyForm 3Page 443 Page 444

y

xO

y � x15

75�

10x � 14y � 2

elimination using

multiplication; ��12

�, �12

��

�3, ��37

��

y

xO

x � 2y � 3x � 1

y � 3x � 4x � y � 2

10 lb of $2.45 mix;20 lb of $2.30 mix

y

xO 3000 6000 9000 12000Bushels of Oats

Bu

nch

es o

f B

arle

y(T

ho

usa

nd

s)

3y � 2x

x � y � 7500

1.1x � 2.05y � 15000

2

4

6

8

Sample answers:2000 bushels of oat, 6000 bushels of barley;4000 bushels of oats, 4000 bushels of barley;6000 bushels of oats, 4000 bushels of barley


Chapter 7 Assessment Answer KeyPage 445, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of solvingsystems of equations and solving systems of inequalities.

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

• Shows an understanding of the concepts of solvingsystems of equations and solving systems of inequalities.

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

• Shows an understanding of most of the concepts ofsolving systems of equations and solving systems ofinequalities.

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

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

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

• Shows little or no understanding of most of the conceptsof solving systems of equations and solving systems ofinequalities.

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

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

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

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

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

Chapter 7 Assessment Answer KeyPage 445, Open-Ended Assessment

Sample Answers


1a. Sample answer: (5000, 5000); $6001b. (15,000, �5000); To represent a

possible investment both x and ymust be positive. Thus, there is nointerpretation for a negative value for y.

2a. The student should recognize that the total cost of a one-day rental of a compact car from the car rentalcompany is modeled by the equationy � A � Bx, where y is the total costand x is the number of miles drivenduring the day. Likewise, the total costof a one-day rental of a compact carfrom the leading competitor is modeledby the equation y � 15 � 0.25x.The solution to the system is thenumber of miles driven during the day that makes the cost of renting acompact car from the two companiesfor one day equal, and thecorresponding total cost.

2b. The value of A determines if the one-day fee for the rental of the compactcar from the car rental company willbe less than, greater than, or equal tothe one-day fee from their leadingcompetitor. The value of B determineswhether the number of miles drivenwill keep the total cost less than,greater than, or equal to the total costof renting from the leading competitor,or change which company has thehigher total cost for the rental.

3a. x � y � S2.5x � 0.75y � PBoth S and P must be positive.The student may recognize that themaximum profit is 2.5 times theweekly sales, and the minimum profitis 0.75 times the weekly sales. i.e.,0.75S � P � 2.5S.

3b. Sample answer: x � y � S2.5x � 0.75y � P

The solution of the system of equationsis the ordered pair that corresponds toweekly sales of S and weekly profit of$P. The solution of the system ofinequalities is the set of ordered pairsthat correspond to weekly sales lessthan or equal to S and weekly profitgreater than or equal to $P. The graphof the system of equations correspondsto the boundaries of the graph of thesystem of inequalities. The twosystems consist of the same terms.

3c. Sample answer: x � y � 4002.5x � 0.75y � 350

The graph of this system ofinequalities is the set of ordered pairsthat correspond to weekly sales lessthan or equal to 400, and weekly profitgreater than or equal to $350.

y

xO

100

200

300

400

100 200 300 400Books

Mag

azin

es

x � y � 400

2.5x � 0.75y � 350

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

An

swer

s


Chapter 7 Assessment Answer KeyVocabulary Test/Review Quiz (Lessons 7-1 and 7-2) Quiz (Lesson 7-4)

Page 446 Page 447 Page 448

1. system of equations

2. consistent

3. inconsistent

4. independent

5. dependent

6. elimination

7. substitution

8. Sample answer:two or moreinequalities that are used together

1.

one solution; (2, 3)2.

no solution3.

4.

5.

Quiz (Lesson 7-3)

Page 447

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

Quiz (Lesson 7-5)

Page 448

1.

2.

3.

Answers:200 adult, 50 student;150 adult, 100 student;250 adult, 250 student

y

xO

y � �x � 4

y � 2x � 1

y

xO

y � 2

x � y � 3

elimination using multiplication; (2, 3)


(�1, �1)

�3�35

�, �35

��

(�9, �5)

B

(�4, 2)

�2�23

�, �1�29

��1�

12

�, 2��5�

12

�, �1�12

��

in the 23rd week


(1, 5)

y

xO

x � 2y � 3

x � 2y � �2

y

xO

y � �x � 5

y � x32

s

aO

100

200

300

400

100 200 300 400Adult Tickets

Stu

den

t Ti

cket

s 5a � 2s � 900

a � s � 300


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. y

xO

y � x � 4

y � x

(7, �3)

7, 14

�35

� hr

y

xO

3x � y � 1y � 3x � 1

no solution

� x � 1 � � 4

a � �15

y � 4x � 4

{(�4, �2), (�2, �1.5),(0, �1), (2, �0.5), (4, 0)};

D�(1, �1),E�(1, �5), F�(4, �2)

102

32 games

(�1, �4)

(3, �2)

(16, �1)

(1, �4)

one solution; (3, 2)

y

xO

x � 3y � 9

x � 3y � �3

Part II

D

B

C

A

C

B

Part I

Chapter 7 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 449 Page 450

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10. 11.

12. 13.

14.

15.

16. DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

8

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

6 / 5

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 1

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 7 2/

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 7 Assessment Answer KeyStandardized Test Practice

Page 451 Page 452


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25. y

xO

elimination (�); (3, �5)

elimination (�); (�6, 12)

elimination (�); (3, �2)


no solution�7�6�5�4�8 �3�2�1 0

{w � w �6 or w �4}

�3�2�1 0 1 2 4 53

��23

�, 4�

{v � v �3 or v 11}

{t � �1 t 2}

{a � a � 8}

{x � x �5}

positive; Sample

answer: y � �13

30

�x � �230�

40

0

80

120

160

1 2 3 4

Dis

tan

ce (

mile

s)

Time (hours)

y � �3x � 6

3x � 2y � �10

y � 5x � 2

y � 2x � 9

d � 40t

15, 20, 26

�41

29

y

xO

{(�2, �7), (�1, �5),(0, �3), (3, 3), (5, 7)}

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

(4, 2), (5, 4)}; {(4, 1), (3,2), (5, 3), (2, 4),(4, 5)}

y

xO

Z�

X�

Y

X W

Z

W�

Y�

W’(1, 1), X’(�1, 1),Y’(�1, 4), Z’(1, 5);

Chapter 7 Assessment Answer KeyUnit 2 TestPage 453 Page 454

An

swer

s

Documents

Chapter 7 Resource Masters - Math Problem Solvingjaeproblemsolving.weebly.com/uploads/5/1/9/6/51966985/algebra_1... · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide