Chapter 7 Resource Masters - Math Classmrsstuckeysmathclass.weebly.com/uploads/6/1/1/2/61126813/alg1_… · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the

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 # 3x " 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

Cost

($)

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 " 3x

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 equation4x " 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 equationy ! 2("2) x ! "2y ! "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 equationx # 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 equation2(7 " 3y) " 4y ! "6 x ! 7 " 3y

14 " 6y " 4y ! "6 Distributive Property14 " 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 ! 1The solution is (1, 2).

"20$"10

"10y$"10

ExampleExample ExampleExample

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 equations ! 1000 " t Solve for s.

0.10s # 0.20t ! 0.12(1000) Second equation0.10(1000 " t) # 0.20t ! 0.12(1000) s ! 1000 " t

100 " 0.10t # 0.20t ! 0.12(1000) Distributive Property100 # 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 ! 200s ! 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

on 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.


y ! "3 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.

" Divide each side by 14.

y " !2 Simplify.

!28$14

14y$14

6x # 2(!2) " 20 Substitute !2 for y ineither equation.

6x ! 4 " 20 Simplify.6x ! 4 # 4 " 20 # 4 Add 4 to each side.

6x " 24 Simplify.

" Divide each side by 6.

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.

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

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

ExampleExample

ExampleExample

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

Com

pact

Dis

cs

5 10 15 20 25 300

60

50

40

30

20

10

Calories

Fat

Gra

ms

1000 2000 30000

Necklaces

Brac

elet

s

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

(mile

s)

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

© Glencoe/McGraw-Hill A2 Glencoe Algebra 1

Stu

dy

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ide

and I

nte

rven

tion

Gra

phin

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yste

ms

of E

quat

ions

NA

ME

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7-1

7-1

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Lesson 7-1

Num

ber

of S

olut

ions

Tw

o or

mor

e lin

ear

equa

tion

s in

volv

ing

the

sam

e va

riab

les

form

a sy

stem

of

equ

atio

ns.

A s

olut

ion

of t

he s

yste

m o

f equ

atio

ns is

an

orde

red

pair

of n

umbe

rsth

at s

atis

fies

bot

h eq

uati

ons.

The

tab

le b

elow

sum

mar

izes

info

rmat

ion

abou

t sy

stem

s of

linea

r eq

uati

ons.

Gra

ph o

f a

Sys

tem

inte

rsec

ting

lines

sam

e lin

epa

ralle

l lin

es

Num

ber

of S

olut

ions

exac

tly o

ne s

olut

ion

infin

itely

man

y so

lutio

nsno

sol

utio

n

Term

inol

ogy

cons

iste

nt a

ndco

nsis

tent

and

inco

nsis

tent

in

depe

nden

tde

pend

ent

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

Sinc

e th

e gr

aphs

of y

!"

x#

2 an

d y

!x

#1

inte

rsec

t,th

ere

is o

ne s

olut

ion.

b.y

!"

x#

23x

#3y

!"

3Si

nce

the

grap

hs o

f y!

"x

#2

and

3x#

3y!

"3

are

para

llel,

ther

e ar

e no

sol

utio

ns.

c.3x

#3y

!"

3y

!"

x"

1Si

nce

the

grap

hs o

f 3x

#3y

!"

3 an

d y

!"

x"

1 co

inci

de,

ther

e ar

e in

fini

tely

man

y so

luti

ons.

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"

3on

ein

finite

ly m

any

3.y

!"

x"

34.

2x#

2y!

"6

2x#

2y!

43x

#y

!3

none

one

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

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Solv

e by

Gra

phin

gO

ne m

etho

d of

sol

ving

a s

yste

m o

f eq

uati

ons

is t

o gr

aph

the

equa

tion

s on

the

sam

e co

ordi

nate

pla

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

one

solu

tion

,nam

e it

.

a.x

#y

!2

x"

y!

4T

he g

raph

s in

ters

ect.

The

refo

re,t

here

is o

ne s

olut

ion.

The

po

int

(3,"

1) s

eem

s to

lie

on b

oth

lines

.Che

ck t

his

esti

mat

e by

rep

laci

ng x

wit

h 3

and

yw

ith

"1

in e

ach

equa

tion

.x

#y

!2

3 #

("1)

!2

✓x

"y

!4

3 "

("1)

!3

#1

or 4

✓T

he s

olut

ion

is (

3,"

1).

b.y

!2x

#1

2y!

4x#

2T

he g

raph

s co

inci

de.T

here

fore

the

re a

re in

fini

tely

m

any

solu

tion

s.

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

!"

2on

e;("

1,"

2)2.

x!

2on

e;(2

,"3)

3.y

!x

one;

(2,1

)3x

"y

!"

12x

#y

!1

x#

y!

3

4.2x

#y

!6

one;

(1,4

)5.

3x#

2y!

6no

sol

utio

n6.

2y!

"4x

#4

infin

itely

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

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Gra

phin

g S

yste

ms

of E

quat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

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7-1

7-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 7-1)


Ans

wer

s

Skil

ls P

ract

ice

Gra

phin

g S

yste

ms

of E

quat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

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___

7-1

7-1

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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

finite

lyy

!"

x#

1on

ey

!x

#4

man

y

3.y

!x

#4

4.y

!2x

"3

2x"

2y!

22x

"2y

!2

no s

olut

ion

one

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

!"

3no

y

!"

3on

e;("

1,"

3)2x

#y

!4

one;

(1,2

)3x

#y

!3

solu

tion

8.y

!x

#2

infin

itely

9.

x#

3y!

"3

one;

10.y

"x

!"

1x

"y

!"

2m

any

x"

3y!

"3

("3,

0)x

#y

!3

one;

(2,1

)

11.x

"y

!3

12.x

#2y

!4

infin

itely

13

.y!

2x#

3no

sol

utio

nx

"2y

!3

one;

(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

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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

!"

6no

sol

utio

nin

finite

ly m

any

3.x

#3y

!3

4.x

#3y

!3

x#

y!

"3

one

2x"

y!

"3

one

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

!"

2no

6.

y!

2x"

3in

finite

ly

7.x

#2y

!3

one;

3x"

y!

0so

lutio

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

ness

pro

duci

ng a

nd

selli

ng g

ourm

et d

og t

reat

s.H

e fi

gure

s it

will

cos

t $2

0 in

oper

atin

g co

sts

per

wee

k pl

us $

0.50

to

prod

uce

each

tre

at.

He

plan

s to

sel

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eat

for

$1.5

0.

8.G

raph

the

sys

tem

of

equa

tion

s y

!0.

5x#

20 a

nd

y!

1.5x

to r

epre

sent

the

sit

uati

on.

9.H

ow m

any

trea

ts d

oes

Nic

k ne

ed t

o se

ll pe

r w

eek

tobr

eak

even

?20

SALE

SF

or E

xerc

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10–

12,u

se t

he

foll

owin

g in

form

atio

n.

A u

sed

book

sto

re a

lso

star

ted

selli

ng u

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d vi

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.In

the

fir

st w

eek,

the

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e so

ld 4

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ed C

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and

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t$4

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per

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and

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oth

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tot

aled

$18

0.00

10.W

rite

a s

yste

m o

f eq

uati

ons

to

repr

esen

t th

e si

tuat

ion.

11.G

raph

the

sys

tem

of

equa

tion

s.

12.H

ow m

any

CD

s an

d vi

deos

did

the

sto

re s

ell i

n th

e fir

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30 C

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10 v

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sc#

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#6v

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180

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40

( 30,

10)

CD S

ales

($)

CD

an

d V

ideo

Sal

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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

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reat

s

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 !

"3

4x "

2y

! "

6

x #

y !

3

Pra

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e (A

vera

ge)

Gra

phin

g S

yste

ms

of E

quat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

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____

____

____

PE

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7-1

7-1



Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

Sys

tem

s of

Equ

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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ATE

____

____

____

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RIO

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___

7-1

7-1

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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

ucti

on t

o L

esso

n 7-

1 at

the

top

of

page

369

in y

our

text

book

.

•W

hat

is m

eant

by

the

term

line

ar f

unct

ion?

Sam

ple

answ

er:

a fu

nctio

n w

hose

gra

ph is

a li

ne

•W

hat

does

it m

ean

to s

ay t

hat

two

lines

inte

rsec

t?S

ampl

e an

swer

:Th

e lin

es c

ross

.

Read

ing

the

Less

on

1.E

ach

figu

re s

how

s th

e gr

aph

of a

sys

tem

of

two

equa

tion

s.W

rite

the

lett

er o

f the

figu

res

that

illu

stra

te e

ach

stat

emen

t.

A.

B.

C.

D.

a.A

sys

tem

of

two

linea

r eq

uati

ons

can

have

an

infi

nite

num

ber

of s

olut

ions

.D

b.A

sys

tem

of

equa

tion

s is

con

sist

ent

if t

here

is a

t le

ast

one

orde

red

pair

tha

t sa

tisf

ies

both

equ

atio

ns.

B,C

,D

c.If

tw

o gr

aphs

are

par

alle

l,th

ere

are

no o

rder

ed p

airs

tha

t sa

tisf

y bo

th e

quat

ions

.A

d.If

a s

yste

m o

f eq

uati

ons

has

exac

tly

one

solu

tion

,it

is in

depe

nden

t.B

,C

e.If

a s

yste

m o

f eq

uati

ons

has

an in

fini

te n

umbe

r of

sol

utio

ns,i

t is

dep

ende

nt.

D

Hel

ping

You

Rem

embe

r

2.D

escr

ibe

how

you

can

sol

ve a

sys

tem

of

equa

tion

s by

gra

phin

g.S

ampl

e an

swer

:G

raph

the

equ

atio

ns o

n th

e sa

me

coor

dina

te p

lane

.Loc

ate

any

poin

ts o

fin

ters

ectio

n.

x

y

Ox

y

O

x

y

Ox

y O

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Trip

The

dis

tanc

e fo

rmul

a,d

!rt

,is

used

to

solv

e m

any

type

s of

pr

oble

ms.

If y

ou g

raph

an

equa

tion

suc

h as

d!

50t,

the

grap

h is

a

mod

el f

or a

car

goi

ng a

t 50

mi/h

.The

tim

e th

e ca

r tr

avel

s is

t;

the

dist

ance

in m

iles

the

car

cove

rs is

d.T

he s

lope

of t

he li

ne is

th

e sp

eed.

Supp

ose

you

driv

e to

a n

earb

y to

wn

and

retu

rn.Y

ou a

vera

ge

50 m

i/h o

n th

e tr

ip o

ut b

ut o

nly

25 m

i/h o

n th

e tr

ip h

ome.

The

ro

und

trip

tak

es 5

hou

rs.H

ow f

ar a

way

is t

he t

own?

The

gra

ph a

t th

e ri

ght

repr

esen

ts y

our

trip

.Not

ice

that

the

ret

urn

trip

is s

how

n w

ith

a ne

gati

ve s

lope

bec

ause

you

are

dri

ving

in t

he

oppo

site

dir

ecti

on.

Sol

ve e

ach

pro

blem

.

1.E

stim

ate

the

answ

er t

o th

e pr

oble

m in

the

abo

ve e

xam

ple.

Abo

ut

how

far

aw

ay is

the

tow

n?

abou

t 80

mile

s

2.G

raph

thi

s tr

ip a

nd s

olve

the

pro

blem

.An

airp

lane

has

en

ough

fue

l for

3 h

ours

of

safe

fly

ing.

On

the

trip

out

the

pilo

tav

erag

es 2

00 m

i/h f

lyin

g ag

ains

t a

head

win

d.O

n th

e tr

ip b

ack,

the

pilo

t av

erag

es 2

50 m

i/h.H

ow lo

ng a

tri

p ou

t ca

n th

e pi

lot

mak

e?

abou

t 1

hour

s an

d 33

0 m

iles

3.G

raph

thi

s tr

ip a

nd s

olve

the

4.

Gra

ph t

his

trip

and

sol

ve t

he p

robl

em.Y

oupr

oble

m.Y

ou d

rive

to

a to

wn

driv

e at

an

aver

age

spee

d of

50

mi/h

to

a10

0 m

iles

away

.On

the

trip

out

you

di

scou

nt s

hopp

ing

plaz

a,sp

end

2 ho

urs

av

erag

e 25

mi/h

.On

the

trip

bac

k yo

u

shop

ping

,and

the

n re

turn

at

an a

vera

ge

aver

age

50 m

i/h.H

ow m

any

hour

s do

sp

eed

of 2

5 m

i/h.T

he e

ntir

e tr

ip t

akes

yo

u sp

end

driv

ing?

8 ho

urs.

How

far

aw

ay is

the

sho

ppin

g pl

aza?

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

ATE

____

____

____

PE

RIO

D__

___

7-1

7-1



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Sub

stitu

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-2

7-2

©G

lenc

oe/M

cGra

w-H

ill40

9G

lenc

oe A

lgeb

ra 1

Lesson 7-2

Subs

titu

tion

One

met

hod

of s

olvi

ng s

yste

ms

of e

quat

ions

is s

ubs

titu

tion

.

Use

su

bsti

tuti

on t

oso

lve

the

syst

em o

f eq

uat

ion

s.y

!2x

4x"

y!

"4

Subs

titu

te 2

xfo

r y

in t

he s

econ

deq

uati

on.

4x"

y!

"4

Sec

ond

equa

tion

4x"

2x!

"4

y!

2x2x

!"

4C

ombi

ne li

ke te

rms.

!D

ivid

e ea

ch s

ide

by 2

.

x!

"2

Sim

plify

.

Use

y!

2xto

fin

d th

e va

lue

of y

.y

!2x

Firs

t equ

atio

ny

!2(

"2)

x!

"2

y!

"4

Sim

plify

.

The

sol

utio

n is

("2,

"4)

.

"4

$2

2x $ 2

Sol

ve f

or o

ne

vari

able

,th

ensu

bsti

tute

.x

#3y

!7

2x"

4y!

"6

Solv

e th

e fi

rst

equa

tion

for

xsi

nce

the

coef

fici

ent

of x

is 1

.x

#3y

!7

Firs

t equ

atio

nx

#3y

"3y

!7

"3y

Sub

trac

t 3y

from

eac

h si

de.

x!

7 "

3yS

impl

ify.

Fin

d th

e va

lue

of y

by s

ubst

itut

ing

7 "

3yfo

r x

in t

he s

econ

d eq

uati

on.

2x"

4y!

"6

Sec

ond

equa

tion

2(7

"3y

) "4y

!"

6x

!7

"3y

14 "

6y"

4y!

"6

Dis

trib

utiv

e P

rope

rty

14 "

10y

!"

6C

ombi

ne li

ke te

rms.

14 "

10y

"14

!"

6 "

14S

ubtr

act 1

4 fr

om e

ach

side

."

10y

!"

20S

impl

ify.

!D

ivid

e ea

ch s

ide

by "

10.

y!

2S

impl

ify.

Use

y!

2 to

fin

d th

e va

lue

of x

.x

!7

"3y

x!

7 "

3(2)

x!

1T

he s

olut

ion

is (

1,2)

.

"20

$ "10

"10

y$ "

10

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Use

su

bsti

tuti

on t

o so

lve

each

sys

tem

of

equ

atio

ns.

If t

he

syst

em d

oes

not

hav

eex

actl

y on

e so

luti

on,s

tate

wh

eth

er i

t h

as n

oso

luti

on o

r in

fin

itel

y m

an

yso

luti

ons.

1.y

!4x

2.x

!2y

3.x

!2y

"3

3x"

y!

1("

1,"

4)y

!x

"2

(4,2

)x

!2y

#4

no s

olut

ion

4.x

"2y

!"

15.

c"

4d!

1in

finite

ly

6.x

#2y

!0

3y!

x#

4(5

,3)

2c"

8d!

2m

any

3x#

4y!

4(4

,"2)

7.2b

!6a

"14

infin

itely

8.

x#

y!

169.

y!

"x

#3

no3a

"b

!7

man

y2y

!"

2x#

2no

sol

utio

n2y

#2x

!4

solu

tion

10.x

!2y

(20,

10)

11.x

"2y

!"

512

."0.

2x#

y!

0.5

0.25

x#

0.5y

!10

x#

2y!

"1

("3,

1)0.

4x#

y!

1.1

(1,0

.7)

©G

lenc

oe/M

cGra

w-H

ill41

0G

lenc

oe A

lgeb

ra 1

Real

-Wor

ld P

robl

ems

Subs

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.I

t m

ay b

e he

lpfu

l to

use

tabl

es,c

hart

s,di

agra

ms,

or g

raph

sto

hel

p yo

u or

gani

ze d

ata.

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

num

ber

of m

illili

ters

of

10%

sal

ine

solu

tion

.L

et t

!th

e nu

mbe

r of

mill

ilite

rs o

f 20

% s

alin

e so

luti

on.

Use

a t

able

to

orga

nize

the

info

rmat

ion.

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

equa

tion

s.s

#t

!10

000.

10s

#0.

20t

!0.

12(1

000)

Use

sub

stit

utio

n to

sol

ve t

his

syst

em.

s#

t!

1000

Firs

t equ

atio

ns

!10

00 "

tS

olve

for

s.0.

10s

#0.

20t

!0.

12(1

000)

Sec

ond

equa

tion

0.10

(100

0 "

t) #

0.20

t!

0.12

(100

0)s

!10

00 "

t10

0 "

0.10

t#

0.20

t!

0.12

(100

0)D

istr

ibut

ive

Pro

pert

y10

0 #

0.10

t!

0.12

(100

0)C

ombi

ne li

ke te

rms.

0.10

t!

20S

impl

ify.

!D

ivid

e ea

ch s

ide

by 0

.10.

t!

200

Sim

plify

.s

#t

!10

00F

irst e

quat

ion

s#

200

!10

00t!

200

s!

800

Sol

ve fo

r s.

800

mill

ilite

rs o

f 10

% s

olut

ion

and

200

mill

ilite

rs o

f 20

% s

olut

ion

shou

ld b

e us

ed.

1.SP

ORT

SA

t th

e en

d of

the

200

0-20

01 fo

otba

ll se

ason

,31

Supe

r 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) a

ndth

e N

atio

nal F

ootb

all C

onfe

renc

e (N

FC

).T

he N

FC

won

fiv

e m

ore

gam

es t

han

the

AF

C.

How

man

y ga

mes

did

eac

h co

nfer

ence

win

?So

urce

:New

York

Tim

es A

lman

acA

FC 1

3;N

FC 1

8

2.C

HEM

ISTR

YA

lab

need

s to

mak

e 10

0 ga

llons

of

an 1

8% a

cid

solu

tion

by

mix

ing

a 12

%ac

id s

olut

ion

wit

h a

20%

sol

utio

n.H

ow m

any

gallo

ns o

f ea

ch s

olut

ion

are

need

ed?

25 g

al o

f 12

% s

olut

ion

and

75 g

al o

f 20

% s

olut

ion

3.G

EOM

ETRY

The

per

imet

er o

f a

tria

ngle

is 2

4 in

ches

.The

long

est

side

is 4

inch

es lo

nger

than

the

sho

rtes

t si

de,a

nd t

he s

hort

est

side

is t

hree

-fou

rths

the

leng

th o

f th

e m

iddl

esi

de.F

ind

the

leng

th o

f ea

ch s

ide

of t

he t

rian

gle.

6 in

.,8

in.,

10 in

.

20$ 0.

100.

10t

$ 0.10

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Sub

stitu

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-2

7-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Sub

stitu

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-2

7-2

©G

lenc

oe/M

cGra

w-H

ill41

1G

lenc

oe A

lgeb

ra 1

Lesson 7-2

Use

su

bsti

tuti

on t

o so

lve

each

sys

tem

of

equ

atio

ns.

If t

he

syst

em d

oes

not

hav

eex

actl

y on

e so

luti

on,s

tate

wh

eth

er i

t h

as n

oso

luti

on o

r in

fin

itel

y m

an

yso

luti

ons.

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

!4x

"1

8.y

!3x

#8

y!

2x"

5("

2,"

9)5x

#2y

!5

("1,

5)

9.2x

"3y

!21

10.y

!5x

"8

y!

3 "

x(6

,"3)

4x#

3y!

33(3

,7)

11.x

#2y

!13

12.x

#5y

!4

3x"

5y!

6(7

,3)

3x#

15y

!"

1no

sol

utio

n

13.3

x"

y!

414

.x#

4y!

82x

"3y

!"

9(3

,5)

2x"

5y!

29(1

2,"

1)

15.x

"5y

!10

16.5

x"

2y!

142x

"10

y!

20in

finite

ly m

any

2x"

y!

5(4

,3)

17.2

x#

5y!

3818

.x"

4y!

27x

"3y

!"

3(9

,4)

3x#

y!

"23

("5,

"8)

19.2

x#

2y!

720

.2.5

x#

y!

"2

x"

2y!

"1

!2,"

3x#

2y!

0("

2,3)

3 $ 2

©G

lenc

oe/M

cGra

w-H

ill41

2G

lenc

oe A

lgeb

ra 1

Use

su

bsti

tuti

on t

o so

lve

each

sys

tem

of

equ

atio

ns.

If t

he

syst

em d

oes

not

hav

eex

actl

y on

e so

luti

on,s

tate

wh

eth

er i

t h

as n

oso

luti

on o

r in

fin

itel

y m

an

yso

luti

ons.

1.y

!6x

2.x

!3y

3.x

!2y

#7

2x#

3y!

"20

("1,

"6)

3x"

5y!

12(9

,3)

x!

y#

4(1

,"3)

4.y

!2x

"2

5.y

!2x

#6

6.3x

#y

!12

y!

x#

2(4

,6)

2x"

y!

2no

sol

utio

ny

!"

x"

2(7

,"9)

7.x

#2y

!13

("3,

8)8.

x"

2y!

3in

finite

ly9.

x"

5y!

36("

4,"

8)"

2x"

3y!

"18

4x"

8y!

12m

any

2x#

y!

"16

10.2

x"

3y!

"24

11.x

#14

y!

8412

.0.3

x"

0.2y

!0.

5x

#6y

!18

("6,

4)2x

"7y

!"

7(1

4,5)

x"

2y!

"5

(5,5

)

13.0

.5x

#4y

!"

114

.3x

"2y

!11

15.

x#

2y!

121 $ 2

Pra

ctic

e (A

vera

ge)

Sub

stitu

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-2

7-2

x#

2.5y

!3.

5(6

,"1)

x"

y!

4(5

,2)

x"

2y!

6(1

2,3)

16.

x"

y!

317

.4x

"5y

!"

718

.x"

3y!

"4

2x#

y!

25y

!5x

2x#

6y!

5

(12,

1)!

,1"

!",1

"E

MP

LO

YM

EN

TF

or E

xerc

ises

19–

21,u

se t

he

foll

owin

g in

form

atio

n.

Ken

isha

sel

ls a

thle

tic

shoe

s pa

rt-t

ime

at a

dep

artm

ent

stor

e.Sh

e ca

n ea

rn e

ithe

r $5

00 p

erm

onth

plu

s a

4% c

omm

issi

on o

n he

r to

tal s

ales

,or

$400

per

mon

th p

lus

a 5%

com

mis

sion

on

tota

l sal

es.

19.W

rite

a s

yste

m o

f eq

uati

ons

to r

epre

sent

the

sit

uati

on.

y!

0.04

x#

500

and

y!

0.05

x#

400

20.W

hat

is t

he t

otal

pri

ce o

f th

e at

hlet

ic s

hoes

Ken

isha

nee

ds t

o se

ll to

ear

n th

e sa

me

inco

me

from

eac

h pa

y sc

ale?

$10,

000

21.W

hich

is t

he b

ette

r of

fer?

the

first

off

er if

she

exp

ects

to

sell

less

tha

n$1

0,00

0 in

sho

es,a

nd t

he s

econ

d of

fer

if sh

e ex

pect

s to

sel

l mor

e th

an$1

0,00

0 in

sho

es

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

ults

and

$5.

50 f

or s

tude

nts.

A g

roup

of

frie

nds

purc

hase

d8

tick

ets

for

$52.

75.

22.W

rite

a s

yste

m o

f eq

uati

ons

to r

epre

sent

the

sit

uati

on.

x#

y!

8 an

d 7.

25x

#5.

5y!

52.7

5

23.H

ow m

any

adul

t ti

cket

s an

d st

uden

t ti

cket

s w

ere

purc

hase

d?5

adul

t an

d 3

stud

ent

1 $ 123 $ 4

2 $ 31 $ 3

1 $ 3

1 $ 2



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csS

ubst

itutio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 u

sed

to

pre

dic

t m

edia

use

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

2 at

the

top

of

page

376

in y

our

text

book

.

•W

hat

is t

he s

yste

m o

f eq

uati

ons?

y!

"2.

8x#

170,

y!

14.4

x#

2•

Bas

ed o

n th

e gr

aph,

are

ther

e 0,

1,or

infi

nite

ly m

any

solu

tion

s of

the

syst

em?

1 so

lutio

n

Read

ing

the

Less

on

1.D

escr

ibe

how

you

wou

ld u

se s

ubst

itut

ion

to s

olve

eac

h sy

stem

of

equa

tion

s.

a.y

!"

2xx

#3y

!15

b.3x

"2y

!12

x!

2y

c.x

#2y

!7

2x"

8y!

8

d."

3x#

5y!

812x

#y

!24

2.Je

ss s

olve

d a

syst

em o

f eq

uati

ons

and

her

resu

lt w

as "

8 !

"8.

All

of h

er w

ork

was

corr

ect.

Des

crib

e th

e gr

aph

of t

he s

yste

m.E

xpla

in.

It is

a li

ne.T

here

are

infin

itely

man

y so

lutio

ns,s

ince

"8

!"

8 is

alw

ays

true

.

3.M

igue

l sol

ved

a sy

stem

of

equa

tion

s an

d hi

s re

sult

was

5 !

"2.

All

of h

is w

ork

was

corr

ect.

Des

crib

e th

e gr

aph

of t

he s

yste

m.E

xpla

in.

The

grap

h ha

s tw

o pa

ralle

llin

es,s

ince

5 !

"2

is a

lway

s fa

lse.

Hel

ping

You

Rem

embe

r

4.W

hat

is u

sual

ly t

he f

irst

ste

p in

sol

ving

a s

yste

m o

f eq

uati

ons

by s

ubst

itut

ion?

Sam

ple

answ

er:S

olve

one

of

the

equa

tions

for

one

vari

able

in t

erm

s of

the

oth

er.

Sol

ve t

he s

econ

d eq

uatio

n fo

r y.

Sub

stitu

te t

heex

pres

sion

you

fin

d fo

r y

in t

he f

irst

equ

atio

n.S

impl

ifyan

d so

lve

for

x.Th

en u

se t

he v

alue

of

xto

fin

d th

eva

lue

of y

.

Sol

ve t

he f

irst

equ

atio

n fo

r x.

Sub

stitu

te t

he e

xpre

ssio

nyo

u fin

d fo

r x

in t

he s

econ

d eq

uatio

n.S

impl

ify a

nd s

olve

for

y.Th

en u

se t

he v

alue

of

yto

fin

d th

e va

lue

of x

.

Sub

stitu

te 2

yfo

r x

in t

he f

irst

equ

atio

n.S

impl

ify a

ndso

lve

for

y.Th

en u

se t

he v

alue

of

yto

fin

d th

e va

lue

of x

.

Sub

stitu

te "

2xfo

r y

in t

he s

econ

d eq

uatio

n.S

impl

ifyan

d so

lve

for

x.Th

en u

se t

he v

alue

of

xto

fin

d th

eva

lue

of y

.

©G

lenc

oe/M

cGra

w-H

ill41

4G

lenc

oe A

lgeb

ra 1

Equ

atio

ns o

f Li

nes

and

Pla

nes

in In

terc

ept

Form

One

for

m t

hat

a lin

ear

equa

tion

may

tak

e is

inte

rcep

tfo

rm.T

he c

onst

ants

aan

d b

are

the

x- a

nd y

-int

erce

pts

of t

he g

raph

.

#!

1

In t

hree

-dim

ensi

onal

spa

ce,t

he e

quat

ion

of a

pla

ne

take

s a

sim

ilar

form

.

##

!1

Her

e,th

e co

nsta

nts

a,b,

and

car

e th

e po

ints

whe

re t

hepl

ane

mee

ts t

he x

,y,a

nd z

-axe

s.

Sol

ve e

ach

pro

blem

.

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

ATE

____

____

____

PE

RIO

D__

___

7-2

7-2

1.G

raph

the

equ

atio

n #

#!

1.

5.G

raph

the

equ

atio

n #

#!

1.

2.Fo

r th

e pl

ane

in E

xerc

ise

1,w

rite

an

equa

tion

for

the

line

whe

re t

he p

lane

inte

rsec

ts t

he x

y-pl

ane.

Use

inte

rcep

tfo

rms.

#!

1

3.W

rite

an

equa

tion

for

the

line

whe

reth

e pl

ane

inte

rsec

ts t

he x

z-pl

ane.

#!

1

4.W

rite

an

equa

tion

for

the

line

whe

reth

e pl

ane

inte

rsec

ts t

he y

z-pl

ane.

#!

1

6.W

rite

an

equa

tion

for

the

xy-

plan

e.

z!

0

7.W

rite

an

equa

tion

for

the

yz-

plan

e.

x!

0

8.W

rite

an

equa

tion

for

a p

lane

par

alle

lto

the

xy-

plan

e w

ith

a z-

inte

rcep

t of

2.

z!

2

9.W

rite

an

equa

tion

for

a p

lane

par

alle

lto

the

yz-

plan

e w

ith

an x

-inte

rcep

t 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

Gu

ide

and I

nte

rven

tion

Elim

inat

ion

Usi

ng A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 A

ddit

ion

In s

yste

ms

of e

quat

ions

in w

hich

the

coe

ffic

ient

s of

the

xor

yte

rms

are

addi

tive

inve

rses

,sol

ve t

he s

yste

m b

y ad

ding

the

equ

atio

ns.B

ecau

se o

ne o

fth

e va

riab

les

is e

limin

ated

,thi

s m

etho

d is

cal

led

elim

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 e

quat

ions

in c

olum

n fo

rm a

nd a

ddto

elim

inat

e y.

x"

3y!

7(#

) 3x

#3y

!9

4x!

16So

lve

for

x.

!

x!

4Su

bsti

tute

4 f

or x

in e

ithe

r eq

uati

on a

ndso

lve

for

y.4

"3y

!7

4 "

3y"

4 !

7 "

4"

3y!

3

!

y!

"1

The

sol

utio

n is

(4,

"1)

.

3$ "

3"

3y$ "

3

16 $ 44x $ 4

Th

e su

m o

f tw

o n

um

bers

is 7

0 an

d t

hei

r d

iffe

ren

ce i

s 24

.Fin

dth

e n

um

bers

.

Let

xre

pres

ent

one

num

ber

and

yre

pres

ent

the

othe

r nu

mbe

r.x

#y

!70

(#) x

"y

!24

2x!

94

!

x!

47Su

bsti

tute

47

for

xin

eit

her

equa

tion

.47

#y

!70

47 #

y"

47 !

70 "

47y

!23

The

num

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

.The

dif

fere

nce

of t

heir

age

s is

12

and

the

sum

of

thei

r ag

es is

50.F

ind

the

age

of e

ach.

Rem

a is

31

and

Ken

is 1

9.

14.T

he s

um o

f th

e di

gits

of

a tw

o-di

git

num

ber

is 1

2.T

he d

iffe

renc

e of

the

dig

its

is 2

.Fin

dth

e nu

mbe

r if

the

uni

ts d

igit

is la

rger

tha

n th

e te

ns d

igit

.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 S

ubtr

acti

onIn

sys

tem

s of

equ

atio

ns w

here

the

coe

ffic

ient

s of

the

xor

yte

rms

are

the

sam

e,so

lve

the

syst

em b

y su

btra

ctin

g th

e eq

uati

ons.

Use

su

btra

ctio

n 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.

(") 5

x"

3y!

14"

3x!

"3

Sub

trac

t the

two

equa

tions

. 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

from

eac

h si

de.

"3y

!9

Sim

plify

.

!D

ivid

e ea

ch s

ide

by "

3.

y!

"3

Sim

plify

.

The

sol

utio

n is

(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 s

um o

f tw

o nu

mbe

rs is

70.

One

num

ber

is t

en m

ore

than

tw

ice

the

othe

r nu

mbe

r.F

ind

the

num

bers

.50

;20

14.G

EOM

ETRY

Tw

o an

gles

are

sup

plem

enta

ry.T

he m

easu

re o

f on

e an

gle

is 1

0°m

ore

than

thre

e ti

mes

the

oth

er.F

ind

the

mea

sure

of

each

ang

le.

42.5

%;13

7.5%

1 $ 21 $ 2

9$ "

3"

3y$ "

3

"3

$ "3

"3x

$ "3

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Elim

inat

ion

Usi

ng A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-3

7-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Ans

wer

s

Skil

ls P

ract

ice

Elim

inat

ion

Usi

ng A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 s

um o

f tw

o nu

mbe

rs is

28

and

thei

r di

ffer

ence

is 4

.Wha

t ar

e th

e nu

mbe

rs?

12,1

6

18.F

ind

the

two

num

bers

who

se s

um is

29

and

who

se d

iffe

renc

e is

15.

7,22

19.T

he s

um o

f tw

o nu

mbe

rs is

24

and

thei

r di

ffer

ence

is 2

.Wha

t ar

e th

e nu

mbe

rs?

13,1

1

20.F

ind

the

two

num

bers

who

se s

um is

54

and

who

se d

iffe

renc

e is

4.

25,2

9

21.T

wo

tim

es a

num

ber

adde

d to

ano

ther

num

ber

is 2

5.T

hree

tim

es t

he f

irst

num

ber

min

usth

e ot

her

num

ber

is 2

0.F

ind

the

num

bers

.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

"9y

!"

129.

"4c

"2d

!"

2"

2x#

2y!

"14

3x"

15y

!"

62c

"2d

!"

14

(3,"

4)("

7,"

1)("

2,5)

10.2

x"

6y!

611

.7x

#2y

!2

12.4

.25x

"1.

28y

!"

9.2

2x#

3y!

247x

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30x

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)("

2,8)

(1.6

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5)

13.2

x#

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1014

.2.5

x#

y!

10.7

15.6

m"

8n!

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2.5

2.5x

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!"

3

(2.5

,1.2

5)(3

.4,2

.2)

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.4a

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17."

x"

y!

"2

18.

x"

y!

84a

#3b

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x"

y!

4x

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(10,

"1)

(12,

2)

19.T

he s

um o

f tw

o nu

mbe

rs is

41

and

thei

r di

ffer

ence

is 5

.Wha

t ar

e th

e nu

mbe

rs?

18,2

3

20.F

our

tim

es o

ne n

umbe

r ad

ded

to a

noth

er n

umbe

r is

36.

Thr

ee t

imes

the

fir

st n

umbe

rm

inus

the

oth

er n

umbe

r is

20.

Fin

d th

e nu

mbe

rs.

8,4

21.O

ne n

umbe

r ad

ded

to t

hree

tim

es a

noth

er n

umbe

r is

24.

Fiv

e ti

mes

the

fir

st n

umbe

rad

ded

to t

hree

tim

es t

he o

ther

num

ber

is 3

6.F

ind

the

num

bers

.3,

7

22.L

AN

GU

AG

ESE

nglis

h is

spo

ken

as t

he f

irst

or

prim

ary

lang

uage

in 7

8 m

ore

coun

trie

sth

an F

arsi

is s

poke

n as

the

fir

st la

ngua

ge.T

oget

her,

Eng

lish

and

Fars

i are

spo

ken

as a

firs

t la

ngua

ge in

130

cou

ntri

es.I

n ho

w m

any

coun

trie

s is

Eng

lish

spok

en a

s th

e fi

rst

lang

uage

? In

how

man

y co

untr

ies

is F

arsi

spo

ken

as t

he f

irst

lang

uage

?E

nglis

h:10

4 co

untr

ies,

Fars

i:26

cou

ntri

es

23.D

ISC

OU

NTS

At

a sa

le o

n w

inte

r cl

othi

ng,C

ody

boug

ht t

wo

pair

s of

glo

ves

and

four

hat

sfo

r $4

3.00

.Tor

i bou

ght

two

pair

s of

glo

ves

and

two

hats

for

$30

.00.

Wha

t w

ere

the

pric

esfo

r th

e gl

oves

and

hat

s?gl

oves

:$8.

50,h

ats:

$6.5

0.

1 $ 23 $ 2

2 $ 31 $ 3

1 $ 23 $ 4

4 $ 31 $ 3

3 $ 41 $ 2

Pra

ctic

e (A

vera

ge)

Elim

inat

ion

Usi

ng A

dditi

on a

nd S

ubtr

actio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-3

7-3 !"

,4"

1 $ 2



Rea

din

g t

o L

earn

Math

emati

csE

limin

atio

n U

sing

Add

ition

and

Sub

trac

tion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

blem

s ab

out

wea

ther

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

3 at

the

top

of

page

382

in y

our

text

book

.

Wha

t fa

ct e

xpla

ins

why

the

var

iabl

e d

gets

elim

inat

ed f

rom

the

sys

tem

of

equa

tion

s?d

"d

!0

Read

ing

the

Less

on

1.W

rite

add

itio

nor

sub

trac

tion

to t

ell w

hich

ope

rati

on it

wou

ld b

e ea

sies

t to

use

to

elim

inat

e a

vari

able

of

the

syst

em.E

xpla

in y

our

choi

ce.

Sys

tem

of

Equ

atio

nsO

pera

tion

Exp

lana

tion

a.3x

#5y

!12

addi

tion

The

coef

ficie

nts

of t

he x

term

s ar

e "

3x#

2y!

6ad

ditiv

e in

vers

es.

b.3x

#5y

!7

subt

ract

ion

The

coef

ficie

nts

of t

he x

term

s ar

e 3x

"2y

!8

the

sam

e.

c."

x"

4y!

9su

btra

ctio

nTh

e co

effic

ient

s of

the

yte

rms

are

4x"

4y!

6th

e sa

me.

d.5x

"7y

!17

addi

tion

The

coef

ficie

nts

of t

he y

term

s ar

e 8x

#7y

!9

addi

tive

inve

rses

.

Hel

ping

You

Rem

embe

r

2.Te

ll ho

w y

ou c

an d

ecid

e w

heth

er t

o us

e ad

diti

on o

r su

btra

ctio

n to

elim

inat

e a

vari

able

ina

syst

em o

f eq

uati

ons.

Sam

ple

answ

er:L

ook

at t

he c

oeff

icie

nts

of e

ach

vari

able

.If

the

coef

ficie

nts

of o

ne o

f th

e va

riab

les

are

addi

tive

inve

rses

,yo

u ca

n us

e ad

ditio

n to

elim

inat

e th

e va

riab

le.I

f th

e co

effic

ient

s of

one

vari

able

are

the

sam

e,yo

u ca

n us

e su

btra

ctio

n to

elim

inat

e th

e va

riab

le.

©G

lenc

oe/M

cGra

w-H

ill42

0G

lenc

oe A

lgeb

ra 1

Róz

sa P

éter

Róz

sa P

éter

(190

5–19

77) w

as a

Hun

gari

an m

athe

mat

icia

n de

dica

ted

tote

achi

ng o

ther

s ab

out

mat

hem

atic

s.A

s pr

ofes

sor

of m

athe

mat

ics

at a

teac

hers

’ col

lege

in B

udap

est,

she

wro

te s

ever

al m

athe

mat

ics

text

book

san

d ch

ampi

oned

ref

orm

s in

the

tea

chin

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

whi

ch s

he a

ttem

pted

to

conv

ey t

he s

piri

t of

mat

hem

atic

s to

the

gen

eral

pub

lic.

By

far

Pét

er’s

grea

test

con

trib

utio

n to

mat

hem

atic

s w

as h

er p

ione

erin

gre

sear

ch in

the

fie

ld o

f re

curs

ive

func

tion

the

ory.

Whe

n yo

u ev

alua

te a

func

tion

rec

ursi

vely

,you

beg

in w

ith

one

init

ial v

alue

of x

.Wor

king

from

this

sin

gle

num

ber,

you

can

use

the

func

tion

to

gene

rate

an

enti

rese

quen

ce o

f nu

mbe

rs.F

or in

stan

ce,h

ere

is h

ow y

ou u

se a

n in

itia

lva

lue

of x

!1

to e

valu

ate

the

func

tion

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 ↓

The

fir

st f

ive

num

bers

of

the

sequ

ence

gen

erat

ed b

y th

is f

unct

ion

are

3,9,

27,8

1,an

d 24

3.

Wri

te t

he

firs

t fi

ve n

um

bers

of

the

sequ

ence

gen

erat

ed b

y ea

ch

fun

ctio

n,u

sin

g th

e gi

ven

nu

mbe

r as

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

;256

;65,

536;

4,29

4,96

7,29

6

5.h(

x) !

"x;

x!

36.

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

ATE

____

____

____

PE

RIO

D__

___

7-3

7-3



Ans

wer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Elim

inat

ion

Usi

ng M

ultip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 M

ulti

plic

atio

nSo

me

syst

ems

of e

quat

ions

can

not

be s

olve

dsi

mpl

y by

add

ing

or s

ubtr

acti

ng t

he e

quat

ions

.In

such

cas

es,o

ne o

r bo

th e

quat

ions

mus

tfi

rst

be m

ulti

plie

d by

a n

umbe

r be

fore

the

sys

tem

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

ulti

ply

the

seco

nd e

quat

ion

by "

2,yo

u ca

n el

imin

ate

the

yte

rms.

x#

10y

!3

(#) "

8x"

10y

!"

10"

7x!

"7

!

x!

1Su

bsti

tute

1 f

or x

in e

ithe

r eq

uati

on.

1 #

10y

!3

1 #

10y

"1

!3

"1

10y

!2

!

y!

The

sol

utio

n is

!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

ulti

ply

the

firs

t eq

uati

on b

y 2

and

the

seco

nd e

quat

ion

by "

3,yo

u ca

nel

imin

ate

the

xte

rms.

6x"

4y!

"14

(#) "

6x#

15y

!"

3011

y!

"44

!

y!

"4

Subs

titu

te "

4 fo

r y

in e

ithe

r eq

uati

on.

3x"

2("

4) !

"7

3x#

8 !

"7

3x#

8 "

8 !

"7

"8

3x!

"15

!

x!

"5

The

sol

utio

n is

("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

The

leng

th o

f Sa

lly’s

gar

den

is 4

met

ers

grea

ter

than

3 t

imes

the

wid

th.

The

per

imet

er o

f he

r ga

rden

is 7

2 m

eter

s.W

hat

are

the

dim

ensi

ons

of S

ally

’s g

arde

n?28

m b

y 8

m

14.A

nita

is 4

year

s ol

der

than

Bas

ilio.

Thr

ee t

imes

Ani

ta’s

age

add

ed t

o si

x ti

mes

Bas

ilio’

s

age

is 3

6.H

ow o

ld a

re A

nita

and

Bas

ilio?

1 $ 2

1 $ 23 $ 2

1 $ 2

5 $ 21 $ 2

Ani

ta:7

yr;

Bas

ilio:

2yr

1 $ 2

©G

lenc

oe/M

cGra

w-H

ill42

2G

lenc

oe A

lgeb

ra 1

Det

erm

ine

the

Best

Met

hod

The

met

hods

to

use

for

solv

ing

syst

ems

of li

near

equa

tion

s ar

e su

mm

ariz

ed in

the

tab

le b

elow

.

Met

hod

The

Bes

t Tim

e to

Use

Gra

phin

gto

est

imat

e th

e so

lutio

n, s

ince

gra

phin

g us

ually

doe

s no

t giv

e an

exa

ct s

olut

ion

Sub

stitu

tion

if on

e of

the

varia

bles

in e

ither

equ

atio

n ha

s a

coef

ficie

nt o

f 1 o

r "

1

Elim

inat

ion

Usi

ng A

dditi

onif

one

of th

e va

riabl

es h

as o

ppos

ite c

oeffi

cien

ts in

the

two

equa

tions

Elim

inat

ion

Usi

ng S

ubtr

actio

nif

one

of th

e va

riabl

es h

as th

e sa

me

coef

ficie

nt in

the

two

equa

tions

Elim

inat

ion

Usi

ng M

ultip

licat

ion

if no

ne o

f the

coe

ffici

ents

are

1 o

r "

1 an

d ne

ither

of t

he v

aria

bles

can

be

elim

inat

ed b

y si

mpl

y ad

ding

or

subt

ract

ing

the

equa

tions

Det

erm

ine

the

best

met

hod

to

solv

e th

e sy

stem

of

equ

atio

ns.

Th

enso

lve

the

syst

em.

6x#

2y!

20"

2x#

4y!

"16

Sinc

e th

e co

effi

cien

ts o

f xw

ill b

e ad

diti

ve in

vers

es o

f ea

ch o

ther

if y

ou m

ulti

ply

the

seco

ndeq

uati

on b

y 3,

use

elim

inat

ion.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Elim

inat

ion

Usi

ng M

ultip

licat

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-4

7-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises

6x#

2y!

20(#

) "6x

#12

y!

"48

Mul

tiply

the

seco

nd e

quat

ion

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 e

ach

side

.6x

!24

Sim

plify

.

!D

ivid

e ea

ch s

ide

by 6

.

x!

4S

impl

ify.

24 $ 66x $ 6

The

sol

utio

n is

(4,

"2)

.

Det

erm

ine

the

best

met

hod

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.T

hen

sol

ve t

he

syst

em.

1.x

#2y

!3

2.m

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!"

83.

a"

b!

6x

#y

!1

m!

2n#

8a

!2b

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elim

inat

ion

(");

("1,

2)su

bstit

utio

n;(4

,"2)

subs

titut

ion;

(5,"

1)

4.4x

#y

!15

5.3c

"d

!14

6.x

#2y

!"

9"

x"

3y!

"12

c"

d!

2y

!4x

subs

titut

ion;

(3,3

)el

imin

atio

n ("

);(6

,4)

subs

titut

ion;

("1,

"4)

7.4x

!2y

"10

8.x

!"

2y9.

2s"

3t!

42x

#2y

!5

4x#

4y!

"10

3s#

2t!

24su

bstit

utio

n;("

1,3)

subs

titut

ion;

!"5,

"el

imin

atio

n (&

);(1

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2,"

elim

inat

ion

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"su

bstit

utio

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3,"

3)1 $ 5

1 $ 2

5 $ 2



Skil

ls P

ract

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Elim

inat

ion

Usi

ng M

ultip

licat

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NA

ME

____

____

____

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____

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7-4

7-4

©G

lenc

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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

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13("

7,6)

3.2x

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!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!

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("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

num

ber

plus

thr

ee t

imes

ano

ther

num

ber

equa

ls 1

3.T

he s

um o

f th

e tw

onu

mbe

rs is

7.W

hat

are

the

num

bers

?8,

"1

14.F

our

tim

es a

num

ber

min

us t

wic

e an

othe

r nu

mbe

r is

"16

.The

sum

of

the

two

num

bers

is "

1.F

ind

the

num

bers

."

3,2

Det

erm

ine

the

best

met

hod

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.T

hen

sol

ve t

he

syst

em.

15.2

x#

3y!

10el

imin

atio

n (&

);16

.8x

"7y

!18

elim

inat

ion

(#);

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("4,

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#7y

!26

(4,2

)

17.y

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subs

titut

ion;

18.3

x#

y!

6el

imin

atio

n ("

);3x

#2y

!35

(5,1

0)3x

#y

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no s

olut

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19.3

x"

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17el

imin

atio

n (&

);20

.y!

3x#

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bstit

utio

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1in

finite

ly m

any

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tions

©G

lenc

oe/M

cGra

w-H

ill42

4G

lenc

oe A

lgeb

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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!

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(2,1

0)("

4,6)

4.2x

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225.

3x#

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11(3

,7)

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29.

2x"

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25y

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x#

y!

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,3)

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10.E

ight

tim

es a

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plus

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e ti

mes

ano

ther

num

ber

is "

13.T

he s

um o

f th

e tw

onu

mbe

rs is

1.W

hat

are

the

num

bers

?"

6,7

11.T

wo

tim

es a

num

ber

plus

thr

ee t

imes

ano

ther

num

ber

equa

ls 4

.Thr

ee t

imes

the

fir

stnu

mbe

r pl

us f

our

tim

es t

he o

ther

num

ber

is 7

.Fin

d th

e nu

mbe

rs.

5,"

2

Det

erm

ine

the

best

met

hod

to

solv

e ea

ch s

yste

m o

f eq

uat

ion

s.T

hen

sol

ve t

he

syst

em.

12.5

x#

7y!

313

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!2

14."

6x"

2y!

142x

"7y

!"

382x

"3y

!"

286x

#8y

!"

20el

imin

atio

n (#

);el

imin

atio

n (&

);el

imin

atio

n (#

);("

5,4)

("2,

8)("

2,"

1)

15.x

!2y

#6

16.4

x#

3y!

"2

17.y

!x

1 $ 21 $ 23 $ 4

Pra

ctic

e (A

vera

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Elim

inat

ion

Usi

ng M

ultip

licat

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NA

ME

____

____

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____

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7-4

7-4 x

"y

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subs

titut

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infin

itely

el

imin

atio

n ("

);su

bstit

utio

n;(6

,3)

man

y so

lutio

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sol

utio

n

18.F

INA

NC

EG

unth

er in

vest

ed $

10,0

00 in

tw

o m

utua

l fun

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ne o

f th

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6%

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ther

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e 9%

in o

ne y

ear.

If G

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inve

stm

ent

rose

a t

otal

of

$684

in o

ne y

ear,

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d he

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l fun

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in t

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% f

und

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19.C

AN

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iles

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our

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till

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1.75

mi/h

20.N

UM

BER

TH

EORY

The

sum

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two-

digi

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11.

If t

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igit

s ar

ere

vers

ed,t

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ew n

umbe

r is

45

mor

e th

an t

he o

rigi

nal n

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r.F

ind

the

num

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38

5 $ 2

1 $ 2



Ans

wer

s

Rea

din

g t

o L

earn

Math

emati

csE

limin

atio

n U

sing

Mul

tiplic

atio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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

ucti

on t

o L

esso

n 7-

4 at

the

top

of

page

387

in y

our

text

book

.

Can

the

sys

tem

of

equa

tion

s be

sol

ved

by e

limin

atio

n w

ith

addi

tion

or

subt

ract

ion?

Exp

lain

.N

o;ne

ither

var

iabl

e ha

s co

effic

ient

s th

at a

read

ditiv

e in

vers

es o

r eq

ual.

Read

ing

the

Less

on

1.C

ould

elim

inat

ion

by m

ulti

plic

atio

n be

use

d to

sol

ve t

he s

yste

m s

how

n be

low

? E

xpla

in.

3x"

5y!

15"

6x#

7y!

11Ye

s;yo

u ca

n m

ultip

ly t

he f

irst

equ

atio

n by

2 t

o m

ake

the

coef

ficie

nts

ofth

e x

term

s ad

ditiv

e in

vers

es.

2.Te

ll w

heth

er it

wou

ld b

e ea

sies

t to

use

sub

stit

utio

n,el

imin

atio

n by

add

itio

n,el

imin

atio

nby

sub

trac

tion

,or

elim

inat

ion

by m

ulti

plic

atio

n to

sol

ve t

he s

yste

m.E

xpla

in y

our

choi

ce.

Sys

tem

of

Equ

atio

nsS

olut

ion

Met

hod

Exp

lana

tion

a."

3x#

4y!

2el

imin

atio

n by

Th

e co

effic

ient

s of

the

xte

rms

are

3x#

2y!

10ad

ditio

nad

ditiv

e in

vers

es.

b.x

"2y

!0

subs

titut

ion

It is

eas

y to

sol

ve t

he f

irst

equ

atio

n 5x

"4y

!8

for

x.

c.6x

"5y

!"

18el

imin

atio

n by

S

ampl

e an

swer

:You

can

mul

tiply

the

2x

#10

y!

27m

ultip

licat

ion

first

equ

atio

n by

2 t

o el

imin

ate

yby

addi

tion.

d."

2x#

3y!

9el

imin

atio

n by

Th

e co

effic

ient

s of

the

yte

rms

are

3x#

3y!

12su

btra

ctio

nth

e sa

me.

Hel

ping

You

Rem

embe

r

3.If

you

are

goi

ng t

o so

lve

a sy

stem

by

elim

inat

ion,

how

do

you

deci

de w

heth

er y

ou w

illne

ed t

o m

ulti

ply

one

or b

oth

equa

tion

s by

a n

umbe

r?S

ampl

e an

swer

:If

both

addi

ng t

he e

quat

ions

and

sub

trac

ting

the

equa

tions

res

ult

in e

quat

ions

that

stil

l hav

e tw

o va

riab

les,

you

will

nee

d to

use

mul

tiplic

atio

n.

©G

lenc

oe/M

cGra

w-H

ill42

6G

lenc

oe A

lgeb

ra 1

Geo

rge

Was

hing

ton

Car

ver

and

Perc

y Ju

lian

In 1

990,

Geo

rge

Was

hing

ton

Car

ver

and

Perc

y Ju

lian

beca

me

the

firs

t A

fric

anA

mer

ican

s el

ecte

d to

the

Nat

iona

l Inv

ento

rs H

all o

f Fam

e.C

arve

r (1

864–

1943

)w

as a

n ag

ricu

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al s

cien

tist

kno

wn

wor

ldw

ide

for

deve

lopi

ng h

undr

eds

of u

ses

for

the

pean

ut a

nd t

he s

wee

t po

tato

.His

wor

k re

vita

lized

the

eco

nom

y of

the

sout

hern

Uni

ted

Stat

es b

ecau

se it

was

no

long

er d

epen

dent

sol

ely

upon

cot

ton.

Julia

n (1

898–

1975

) w

as a

res

earc

h ch

emis

t w

ho b

ecam

e fa

mou

s fo

r in

vent

ing

a m

etho

d of

mak

ing

a sy

nthe

tic

cort

ison

e fr

om s

oybe

ans.

His

dis

cove

ry h

asha

d m

any

med

ical

app

licat

ions

,par

ticu

larl

y in

the

tre

atm

ent

of a

rthr

itis

.

The

re a

re d

ozen

s of

oth

er A

fric

an A

mer

ican

inve

ntor

s w

hose

acc

ompl

ishm

ents

are

not

as w

ell k

now

n.T

heir

inve

ntio

ns r

ange

fro

m c

omm

on h

ouse

hold

item

slik

e th

e ir

onin

g bo

ard

to c

ompl

ex d

evic

es t

hat

have

rev

olut

ioni

zed

man

ufac

turi

ng.T

he e

xerc

ises

tha

t fo

llow

will

hel

p yo

u id

enti

fy ju

st a

few

of

thes

e in

vent

ors

and

thei

r in

vent

ions

.

Mat

ch t

he

inve

nto

rs w

ith

th

eir

inve

nti

ons

by m

atch

ing

each

sys

tem

wit

h i

ts s

olu

tion

.(N

ot a

ll t

he

solu

tion

s w

ill

be u

sed

.)

1.Sa

ra B

oone

x#

y!

2E

A.(

1,4)

auto

mat

ic t

raff

ic s

igna

l x

"y

!10

2.Sa

rah

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

Jone

sy

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x"

3

4.J.

L.L

ove

2x#

3y !

8F

D.(

"5,

7)fo

ldin

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bine

t be

d 2x

"y

!"

8

5.T.

J.M

arsh

all

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3x!

9C

E.(

6,"

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onin

g bo

ard

2y#

x!

4

6.Ja

n M

atze

liger

y#

4 !

2xJ

F.("

2,4)

penc

il sh

arpe

ner

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12

7.G

arre

tt A

.3x

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!"

5A

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"3,

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-ray

mac

hine

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orga

n3y

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orbe

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illie

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IH

.(2,

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play

er p

iano

y

"3x

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I.no

sol

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apor

atin

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n fo

r re

fini

ng s

ugar

J.in

fini

tely

last

ing

(sha

ping

) m

any

mac

hine

for

so

luti

ons

man

ufac

turi

ng s

hoes

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-4

7-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Gra

phin

g S

yste

ms

of In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

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 In

equa

litie

sT

he s

olut

ion

of a

sys

tem

of

ineq

ual

itie

sis

the

set

of

all

orde

red

pair

s th

at s

atis

fy b

oth

ineq

ualit

ies.

If y

ou g

raph

the

ineq

ualit

ies

in t

he s

ame

coor

dina

te p

lane

,the

sol

utio

n is

the

reg

ion

whe

re t

he g

raph

s ov

erla

p.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by g

rap

hin

g.y

'x

#2

y(

"2x

"1

The

sol

utio

n in

clud

es t

he o

rder

ed p

airs

in t

he in

ters

ecti

on o

f th

egr

aphs

.Thi

s re

gion

is s

hade

d at

the

rig

ht.T

he g

raph

s of

y!

x#

2an

d y

!"

2x"

1 ar

e bo

unda

ries

of

this

reg

ion.

The

gra

ph o

f y

!x

#2

is d

ashe

d an

d is

not

incl

uded

in t

he g

raph

of y

%x

#2.

Sol

ve t

he

syst

em o

f in

equ

alit

ies

by g

rap

hin

g.x

#y

'4

x#

y)

"1

The

gra

phs

of x

#y

!4

and

x#

y!

"1

are

para

llel.

Bec

ause

the

tw

o re

gion

s ha

ve n

o po

ints

in c

omm

on,t

he s

yste

m o

f in

equa

litie

s ha

s no

sol

utio

n.

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.y

%"

12.

y%

"2x

#2

3.y

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2y

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y !

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xx

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O

2x #

y !

1

x "

y !

"2

x

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y !

x #

1

3x #

4y

! 1

2

x

y

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y !

x #

1

y !

"2x

# 2

x

y

O

x !

0 y !

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x

y

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x

y O

x #

y !

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x #

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x #

2

y !

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Exam

ple1

Exam

ple1

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ple2

Exam

ple2

Exer

cises

Exer

cises

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lenc

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cGra

w-H

ill42

8G

lenc

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lgeb

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Real

-Wor

ld P

robl

ems

In r

eal-

wor

ld p

robl

ems,

som

etim

es o

nly

who

le n

umbe

rs m

ake

sens

e fo

r th

e so

luti

on,a

nd o

ften

onl

y po

siti

ve v

alue

s of

xan

d y

mak

e se

nse.

BU

SIN

ESS

AA

A G

em C

omp

any

pro

du

ces

nec

kla

ces

and

bra

cele

ts.I

n a

40-

hou

r w

eek

,th

e co

mp

any

has

400

gem

s to

use

.A n

eck

lace

req

uir

es 4

0 ge

ms

and

a

brac

elet

req

uir

es 1

0 ge

ms.

It t

akes

2 h

ours

to

pro

du

ce a

nec

kla

ce a

nd

a b

race

let

requ

ires

on

e h

our.

How

man

y of

ea

ch t

ype

can

be

pro

du

ced

in

a w

eek

?

Let

n!

the

num

ber

of n

eckl

aces

tha

t w

ill b

e pr

oduc

ed a

nd b

!th

enu

mbe

r of

bra

cele

ts t

hat

will

be

prod

uced

.Nei

ther

nor

bca

n be

ane

gati

ve n

umbe

r,so

the

fol

low

ing

syst

em o

f in

equa

litie

s re

pres

ents

th

e co

ndit

ions

of

the

prob

lem

s.

n(

0b

(0

b#

2n'

4010

b#

40n

'40

0

The

sol

utio

n is

the

set

ord

ered

pai

rs in

the

inte

rsec

tion

of

the

grap

hs.T

his

regi

on is

sha

ded

at t

he r

ight

.Onl

y w

hole

-num

ber

solu

tion

s,su

ch a

s (5

,20)

mak

e se

nse

in t

his

prob

lem

.

For

eac

h e

xerc

ise,

grap

h t

he

solu

tion

set

.Lis

t th

ree

pos

sibl

e so

luti

ons

to t

he

pro

blem

.

1.H

EALT

HM

r.F

low

ers

is o

n a

rest

rict

ed

2.R

ECR

EATI

ON

Mar

ia h

ad $

150

in g

ift

diet

tha

t al

low

s hi

m t

o ha

ve b

etw

een

cert

ific

ates

to

use

at a

rec

ord

stor

e.Sh

e 16

00 a

nd 2

000

Cal

orie

s pe

r da

y.H

is

boug

ht f

ewer

tha

n 20

rec

ordi

ngs.

Eac

h da

ily f

at in

take

is r

estr

icte

d to

bet

wee

nta

pe c

ost

$5.9

5 an

d ea

ch C

D c

ost

$8.9

5.45

and

55

gram

s.W

hat

daily

Cal

orie

H

ow m

any

of e

ach

type

of r

ecor

ding

mig

ht

and

fat

inta

kes

are

acce

ptab

le?

she

have

bou

ght?

Sam

ple

answ

ers:

1600

Cal

orie

s,S

ampl

e an

swer

s:10

tap

es,9

CD

s;45

fat

gra

ms;

1800

Cal

orie

s,50

fat

0 ta

pes,

16 C

Ds;

14 t

apes

,5 C

Ds

gram

s;20

00 C

alor

ies,

55 f

at g

ram

s

t # c

! 2

0 5.95

t # 8

.95c

! 1

50

30 25 20 15 10 5

Tape

s

Compact Discs

510

1520

2530

0

60 50 40 30 20 10

Calo

ries

Fat Grams

1000

2000

3000

0

Nec

klac

es

Bracelets

1020

3040

500

50 40 30 20 10

10b

# 4

0n !

400

b #

2n

! 4

0

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Gra

phin

g S

yste

ms

of In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-5

7-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Ans

wer

s

Skil

ls P

ract

ice

Gra

phin

g S

yste

ms

of In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-5

7-5

©G

lenc

oe/M

cGra

w-H

ill42

9G

lenc

oe A

lgeb

ra 1

Lesson 7-5

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.x

%"

12.

y%

23.

y%

x#

3y

'"

3x

&"

2y

'"

1

4.x

&2

5.x

#y

'"

16.

y"

x%

4y

"x

'2

x#

y(

3*

x#

y%

2

7.y

%x

#1

8.y

("

x#

29.

y&

2x#

4y

("

x#

1y

&2x

"2

y(

x#

1

Wri

te a

sys

tem

of

ineq

ual

itie

s fo

r ea

ch g

rap

h.

10.

11.

12.

y(

x#

2,y

+x

"3

y'

"x,

y'

xy

+x

#1,

y)

1x

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

# 1

y !

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 !

"2

x

y

O

x !

"1

y !

"3

x

y

O

©G

lenc

oe/M

cGra

w-H

ill43

0G

lenc

oe A

lgeb

ra 1

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s by

gra

ph

ing.

1.y

%x

"2

2.y

(x

#2

3.x

#y

(1

y'

xy

%2x

#3

x#

2y%

1

4.y

&2x

"1

5.y

%x

"4

6.2x

"y

(2

y%

2 "

x2x

#y

'2

x"

2y(

2

FITN

ESS

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.D

iego

sta

rted

an

exer

cise

pro

gram

in w

hich

eac

h w

eek

he

wor

ks o

ut a

t th

e gy

m b

etw

een

4.5

and

6 ho

urs

and

wal

ksbe

twee

n 9

and

12 m

iles.

7.M

ake

a gr

aph

to s

how

the

num

ber

of h

ours

Die

go w

orks

ou

t at

the

gym

and

the

num

ber

of m

iles

he w

alks

per

wee

k.

8.L

ist

thre

e po

ssib

le c

ombi

nati

ons

of w

orki

ng o

ut a

nd

wal

king

tha

t m

eet

Die

go’s

goa

ls.

Sam

ple

answ

ers:

gym

5 h

,wal

k 9

mi;

gym

6 h

,wal

k 10

mi,

gym

5.5

h,w

alk

11 m

i

SOU

VEN

IRS

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

Em

ily w

ants

to

buy

turq

uois

e st

ones

on

her

trip

to

New

Mex

ico

to g

ive

to a

t le

ast

4 of

her

fri

ends

.The

gif

t sh

op s

ells

sto

nes

for

eith

er $

4 or

$6

per

ston

e.E

mily

has

no

mor

e th

an $

30 t

o sp

end.

9.M

ake

a gr

aph

show

ing

the

num

bers

of

each

pri

ce o

f st

one

Em

ily c

an p

urch

ase.

10.L

ist

thre

e po

ssib

le s

olut

ions

.S

ampl

e an

swer

:one

$4

ston

e an

d fo

ur $

6 st

ones

;thr

ee $

4 st

ones

and

th

ree

$6 s

tone

s;fiv

e $4

sto

nes

and

one

$6 s

tone

$4 S

tone

s

Turq

uo

ise

Sto

nes

$6 Stones

13

24

56

78

06 5 4 3 2 1

Gym

(hou

rs)

Die

go

’s R

ou

tin

e

Waking (miles)

13

24

56

78

0

16 14 12 10 8 6 4 2

2x "

y !

2

x "

2y

! 2

x

y

O

y !

x "

4

2x #

y !

2

x

y

O

y !

2x

" 1

y !

2 "

x

x

y

O

x #

y !

1

x #

2y

! 1

x

y

O

y !

2x

# 3

y !

x #

2 x

y

O

y !

x "

2y

! x

x

y

OPra

ctic

e (A

vera

ge)

Gra

phin

g S

yste

ms

of In

equa

litie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-5

7-5



Rea

din

g t

o L

earn

Math

emati

csG

raph

ing

Sys

tem

s of

Ineq

ualit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-5

7-5

©G

lenc

oe/M

cGra

w-H

ill43

1G

lenc

oe A

lgeb

ra 1

Lesson 7-5

Pre-

Act

ivit

yH

ow c

an y

ou u

se a

sys

tem

of

ineq

ual

itie

s to

pla

n a

sen

sibl

e d

iet?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 7-

5 at

the

top

of

page

394

in y

our

text

book

.

The

gre

en s

ecti

on o

n th

e gr

aph

repr

esen

ts a

ran

ge o

f

Cal

orie

s a

day

and

gram

s of

fat

per

day

.

Read

ing

the

Less

onW

rite

th

e in

equ

alit

y sy

mbo

ls t

hat

you

nee

d t

o ge

t a

syst

em w

hos

e gr

aph

loo

ks

lik

eth

e on

e sh

own

.Use

),(

,',o

r +

.

1.2.

yx

#2

yx

#2

y"

2x"

1y

"2x

"1

3.4.

yx

#2

yx

#2

y"

2x"

1y

"2x

"1

Hel

ping

You

Rem

embe

r

5.D

escr

ibe

how

you

wou

ld e

xpla

in t

he p

roce

ss o

f us

ing

a gr

aph

to s

olve

a s

yste

m o

fin

equa

litie

s to

a f

rien

d w

ho m

isse

d L

esso

n 7-

5.G

raph

eac

h in

equa

lity

on t

hesa

me

coor

dina

te p

lane

.The

sol

utio

ns a

re t

he o

rder

ed p

airs

for

the

poin

tsin

bot

h gr

aphs

.

'(

('

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

60 t

o 75

2000

to

2400

©G

lenc

oe/M

cGra

w-H

ill43

2G

lenc

oe A

lgeb

ra 1

Des

crib

ing

Reg

ions

The

sha

ded

regi

on in

side

the

tri

angl

e ca

n be

des

crib

ed

wit

h a

syst

em o

f th

ree

ineq

ualit

ies.

y&

2x#

1

y%

x"

3

y%

29x

"31

Wri

te s

yste

ms

of i

neq

ual

itie

s to

des

crib

e ea

ch r

egio

n.Y

ou m

ay

firs

t n

eed

to

div

ide

a re

gion

in

to t

rian

gles

or

quad

rila

tera

ls.

1.2.

y)

x#

12y

'"

x"

2y

)x

#2

y)

"x

#y

'x

"2

y)

"x

#2

y)

2y

'"

3

y)

5y

'"

5

3.to

p:y

)x

#7,

y)

"x

#7,

y'

4

mid

dle:

y)

4,y

'0,

y'

"x

"4,

y'

x"

4

bott

om le

ft:y

)4x

#12

,y'

x"

2,y

)0,

x)

0

bott

om r

ight

:y)

"4x

#12

,y

'"

x"

2,y

)0,

x,

01 $ 2

1 $ 2

4 $ 3

4 $ 3

3 $ 23 $ 2

x

y

O19 $ 25 $ 2

5 $ 2

x

y

Ox

y

O

1 $ 3x

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

7-5

7-5


Documents

Chapter 7 Resource Masters - Math Classmrsstuckeysmathclass.weebly.com/uploads/6/1/1/2/61126813/alg1_… · ©Glencoe/McGraw-Hill iv Glencoe Algebra 1 Teacher’s Guide to Using the