Chapter 7: Solving Systems of Linear Equations and ...manualmath.info/downloads/chap07_bk.pdf · 366C Chapter 7 Solving Systems of Linear Equations and ... Chapter 7 Solving Systems

366A Chapter 7 Solving Systems of Linear Equations and Inequalities

Pacing suggestions for the entire year can be found on pages T20–T21.

Solving Systems of LinearEquations and InequalitiesChapter Overview and Pacing

Solving Systems of LinearEquations and InequalitiesChapter Overview and Pacing

PACING (days)Regular Block

Basic/ Basic/ Average Advanced Average Advanced

Graphing Systems of Equations (pp. 368–375) 2 2 1.5 1.5Preview: Use a spreadsheet to investigate when two quantities will be equal. (with 7-1 (with 7-1 (with 7-1 (with 7-1• Determine whether a system of linear equations has 0, 1, or infinitely many solutions. Follow-Up) Preview and Preview and Preview and• Solve systems of equations by graphing. Follow-Up) Follow-Up) Follow-Up)Follow-Up: Use a graphing calculator to solve a system of equations.

Substitution (pp. 376–381) 2 2 1 1• Solve systems of equations by using substitution.• Solve real-world problems involving systems of equations.

Elimination Using Addition and Subtraction (pp. 382–386) 2 2 1 1• Solve systems of equations by using elimination with addition.• Solve systems of equations by using elimination with subtraction.

Elimination Using Multiplication (pp. 387–392) 2 2 1 1• Solve systems of equations by using elimination with multiplication.• Determine the best method for solving systems of equations.

Graphing Systems of Inequalities (pp. 394–398) 1 1 0.5 0.5• Solve systems of inequalities by graphing.• Solve real-world problems involving systems of inequalities.

Study Guide and Practice Test (pp. 399–403) 1 1 0.5 0.5Standardized Test Practice (pp. 404–405)

Chapter Assessment 1 1 0.5 0.5

TOTAL 11 11 6 6

LESSON OBJECTIVES

*Key to Abbreviations: GCS � Graphing Calculator and Speadsheet Masters,SC � School-to-Career Masters, SM � Science and Mathematics Lab Manual

Study Guide and Intervention, Skills Practice, Practice, and Parent and Student Study Guide Workbooks are also available in Spanish.

ELL

Chapter 7 Solving Systems of Linear Equations and Inequalities 366B

Materials

403–404 405–406 407 408 SC 13 53 7-1 7-1 (Follow-Up: graphing calculator)

409–410 411–412 413 414 447 27–28 SM 63–66 54 7-2 7-2 16 algebra tiles, equation mat

415–416 417–418 419 420 447, 449 GCS 36 55 7-3 7-3

421–422 423–424 425 426 448 SC 14 56 7-4 7-4

427–428 429–430 431 432 448 GCS 35 57 7-5 7-5 17 graphing calculator

433–446, 58450–452

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See pages T12–T13.

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366C Chapter 7 Solving Systems of Linear Equations and Inequalities

Mathematical Connections and BackgroundMathematical Connections and Background

Graphing Systems of EquationsA solution of a system of equations is the set of

points that satisfy each equation in the system.Carefully graph each equation on the same coordinateplane. There is only one solution if the graphs of thelines intersect, since the intersection is at only one point.There is no solution if the lines are parallel. The linesnever intersect, so no one point is common to bothgraphs. If the graphs are the same line, the system hasan infinite number of solutions.

If there is one solution or infinitely many solu-tions, the system of equations is described as consistent.Systems with one solution are said to be independent,while those with infinitely many solutions are said to bedependent. If there is no solution, the system isdescribed as inconsistent.

SubstitutionIt is sometimes difficult to determine the exact

solution of a system of equations from a graph.Therefore, an algebraic method may be used to find theexact solution. Substitution is an algebraic method. Firstsolve one equation for one variable in terms of theother. Substitute the expression into the other equationso that one variable is eliminated. Solve for the remain-ing variable. Substitute this value into either equationand solve for the other variable. The two values makeup the solution of the system. They are written in theform (x, y) to represent the point where the two linesintersect if graphed.

If a solution results in an identity, for example 2 � 2, the system has an infinite number of solutions. Ifthe result is a false statement, such as 5 � 3, there is nosolution. If you incur either of these situations duringany part of the solution process, you may stop solvingand write either infinitely many solutions or no solution.

Prior KnowledgePrior KnowledgeIn Chapter 2, students learned that when youadd additive inverses, the sum is always 0. Theyused the Addition, Subtraction, Multiplication,and Division Properties of Equality to solveequations throughout Chapter 3. Students

graph linear equations in Chapter 5 and graph linear inequal-

ities in Chapter 6.

This Chapter

Future ConnectionsFuture ConnectionsBusiness analysts use systems of linear equa-tions to determine where break-even pointsare and to analyze trends for predicting futureevents. There are not only systems of linearequations and inequalities, but also systems ofall types of functions including quadratic, ab-solute value, and sine. These systems can mixany types of functions. The solutions of thesesystems are not only used in business, butalso in science and other fields.

Continuity of InstructionContinuity of Instruction

This ChapterChapter 7 introduces students to systems oflinear equations. They first solve the systemsby graphing and then classify the systems asconsistent or inconsistent, and as independentor dependent. Students also learn to apply thealgebraic methods to solving the systems.These methods include substitution, elimina-tion using addition or subtraction, and elimi-nation using multiplication first. Students mustdetermine which method is best for different

systems. The chapter ends with students solving systems of inequalities

by graphing.

Chapter 7 Solving Systems of Linear Equations and Inequalities 366D

Elimination Using Addition and SubtractionElimination is another algebraic method used

to solve systems of equations. The objective is to com-bine the two equations to eliminate one of the vari-ables. If the coefficients of one variable are additiveinverses of each other, use addition. If the coefficientsof one variable are the same, use subtraction. Becausethe Addition and Subtraction Properties of Equalitystate that equal amounts can be added to or subtract-ed from each side of an equation, you can add or sub-tract one equation with the other. This step eliminatesone variable. Then solve for the other variable.Substitute this value into either original equation tofind the value of the variable that was eliminated.The two values are the solution of the system.

Elimination UsingMultiplicationIf the coefficients of one variable are neither

additive inverses nor equal, one or both equationsmust be changed so that the elimination method canbe applied to solve the system of equations. Changeeither one, or both, of the equations by applying theMultiplication Property of Equality. Every term of theequation is multiplied by the same number, or bothequations are multiplied by different numbers, inorder to make one pair of coefficients of a variableeither additive inverses or the same. Then follow thesteps for solving the system using the eliminationmethod.

Five methods for solving systems of equationshave been seen in this chapter. They are graphing,substitution, elimination using addition, eliminationusing subtraction, and elimination using multiplication.Graphing is used only if an estimation is needed, sinceit is difficult to get an exact solution. Use substitutionif one of the variables has a coefficient of 1 or �1.Elimination using addition is used when one of thevariables has coefficients that are additive inverses ofeach other, and elimination using subtraction is bestused when one of the variables has the same coeffi-cient. Apply elimination using multiplication if noneof the above situations occur.

Graphing Systems ofInequalitiesA solution of a system of inequalities is all the

points that satisfy both inequalities. Use the methodslearned in Lesson 6-6 to graph each inequality. Thepoints that are solutions of both inequalities lie in theregion where the graphs overlap, or intersect. A sys-tem of inequalities has no solution if the boundarylines are parallel and the shaded regions do not over-lap. Otherwise, there are infinitely many solutionssince the overlapping shaded region extends onindefinitely.

Additional mathematical information and teaching notesare available in Glencoe’s Algebra 1 Key Concepts:Mathematical Background and Teaching Notes, which is available at www.algebra1.com/key_concepts. The lessons appropriate for this chapter are as follows.• Graphing Systems of Equations (Lesson 18)• Substitution (Lesson 19)• Elimination Using Addition and Subtraction (Lesson 20)• Elimination Using Multiplication (Lesson 21)

www.algebra1.com/key_concepts

http://www.algebra1.com/key_concepts

366E Chapter 7 Solving Systems of Linear Equations and Inequalities

TestCheck and Worksheet BuilderThis networkable software has three modules for interventionand assessment flexibility:• Worksheet Builder to make worksheet and tests• Student Module to take tests on screen (optional)• Management System to keep student records (optional)

Special banks are included for SAT, ACT, TIMSS, NAEP, and End-of-Course tests.

Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters

Ongoing Prerequisite Skills, pp. 367, 374,381, 386, 392

Practice Quiz 1, p. 381Practice Quiz 2, p. 392

AlgePASS: Tutorial Pluswww.algebra1.com/self_check_quizwww.algebra1.com/extra_examples

5-Minute Check TransparenciesPrerequisite Skills Workbook, pp. 27–28Quizzes, CRM pp. 447–448Mid-Chapter Test, CRM p. 449Study Guide and Intervention, CRM pp. 403–404,

409–410, 415–416, 421–422, 427–428

MixedReview

Cumulative Review, CRM p. 450 pp. 374, 381, 386, 392, 398

ErrorAnalysis

Find the Error, TWE pp. 384, 396Unlocking Misconceptions, TWE p. 372Tips for New Teachers, TWE pp. 367, 384

Find the Error, pp. 384, 396

StandardizedTest Practice

TWE pp. 404–405Standardized Test Practice, CRM pp. 451–452

Standardized Test Practice CD-ROM

www.algebra1.com/standardized_test

pp. 374, 381, 384, 385, 386,392, 398, 403, 404–405

Open-EndedAssessment

Modeling: TWE pp. 374Speaking: TWE pp. 381, 392Writing: TWE pp. 386, 398Open-Ended Assessment, CRM p. 445

Writing in Math, pp. 374, 381,386, 392, 398

Open Ended, pp. 371, 379, 384,390, 396

Standardized Test, p. 405

ChapterAssessment

Multiple-Choice Tests (Forms 1, 2A, 2B), CRM pp. 433–438

Free-Response Tests (Forms 2C, 2D, 3), CRM pp. 439–444

Vocabulary Test/Review, CRM p. 446

TestCheck and Worksheet Builder(see below)

MindJogger Videoquizzes www.algebra1.com/

vocabulary_reviewwww.algebra1.com/chapter_test

Study Guide, pp. 399–402Practice Test, p. 403

Additional Intervention ResourcesThe Princeton Review’s Cracking the SAT & PSATThe Princeton Review’s Cracking the ACTALEKS

and Assessmentand AssessmentA

SSES

SMEN

TIN

TER

VEN

TIO

N

Type Student Edition Teacher Resources Technology/Internet

http://www.algebra1.com/self_check_quiz

http://www.algebra1.com/extra_examples

http://www.algebra1.com/standardized_test

http://www.algebra1.com/vocabulary_review

http://www.algebra1.com/chapter_test

Chapter 7 Solving Systems of Linear Equations and Inequalities 366F

Algebra 1Lesson

AlgePASS Lesson

7-2 16 Solving a System of SimultaneousEquations

7-5 17 Graphing Linear Inequalities on theCoordinate Plane

ALEKS is an online mathematics learning system thatadapts assessment and tutoring to the student’s needs.Subscribe at www.k12aleks.com.

For more information on Reading and Writing inMathematics, see pp. T6–T7.

Intervention at HomeParent and Student Study Guide Parents and students may work together to reinforce theconcepts and skills of this chapter. (Workbook, pp. 53–58 or log on to www.algebra1.com/parent_student)

Intervention TechnologyAlgePASS: Tutorial Plus CD-ROM offers a complete, self-paced algebra curriculum.

Reading and Writingin Mathematics

Reading and Writingin Mathematics

Glencoe Algebra 1 provides numerous opportunities toincorporate reading and writing into the mathematics classroom.

Student Edition

• Foldables Study Organizer, p. 367• Concept Check questions require students to verbalize

and write about what they have learned in the lesson.(pp. 371, 379, 384, 390, 396)

• Reading Mathematics, p. 393 • Writing in Math questions in every lesson, pp. 374, 381,

386, 392, 398• WebQuest, pp. 373, 398

Teacher Wraparound Edition

• Foldables Study Organizer, pp. 367, 399• Study Notebook suggestions, pp. 372, 379, 384, 390,

393, 396 • Modeling activities, p. 374• Speaking activities, pp. 381, 392• Writing activities, pp. 386, 398• Differentiated Instruction, (Verbal/Linguistic), p. 389• Resources, pp. 366, 373, 380, 385, 389, 391,

393, 397, 399

Additional Resources

• Vocabulary Builder worksheets require students todefine and give examples for key vocabulary terms asthey progress through the chapter. (Chapter 7 ResourceMasters, pp. vii-viii)

• Reading to Learn Mathematics master for each lesson(Chapter 7 Resource Masters, pp. 407, 413, 419, 425,431)

• Vocabulary PuzzleMaker software creates crossword,jumble, and word search puzzles using vocabulary liststhat you can customize.

• Teaching Mathematics with Foldables provides suggestions for promoting cognition and language.

• Reading and Writing in the Mathematics Classroom• WebQuest and Project Resources• Hot Words/Hot Topics Sections 6.2–6.4, 6.6–6.8, 9.4

ELL

For more information on Intervention andAssessment, see pp. T8–T11.

Log on for student study help.• For each lesson in the Student Edition, there are Extra

Examples and Self-Check Quizzes.www.algebra1.com/extra_exampleswww.algebra1.com/self_check_quiz

• For chapter review, there is vocabulary review, test practice, and standardized test practice.www.algebra1.com/vocabulary_reviewwww.algebra1.com/chapter_testwww.algebra1.com/standardized_test

http://www.k12aleks.com

http://www.algebra1.com/parent_student






Have students read over the listof objectives and make a list ofany words with which they arenot familiar.

Point out to students that this isonly one of many reasons whyeach objective is important.Others are provided in theintroduction to each lesson.

Solving Systems ofLinear Equationsand Inequalities

• system of equations (p. 369)• substitution (p. 376)• elimination (p. 382)• system of inequalities (p. 394)

Key Vocabulary

Business decision makers often use systems of linear equations to model a real-world situation in order to predict future events. Being able to make an accurate prediction helps them plan andmanage their businesses.

Trends in the travel industry change with time. For example, in recent years, the number of tourists traveling to South America, theCaribbean, and the Middle East is on the rise. You will use a system of

linear equations to model the trends in tourism in Lesson 7-2.

• Lesson 7-1 Solve systems of linear equations bygraphing.

• Lessons 7-2 through 7-4 Solve systems of linearequations algebraically.

• Lesson 7-5 Solve systems of linear inequalitiesby graphing.

366 Chapter 7 Solving Systems of Linear Equations and Inequalities366 Chapter 7 Solving Systems of Linear Equations and Inequalities

366 Chapter 7 Solving Systems of Linear Equations and Inequalities

NotesNotes

NCTM LocalLesson Standards Objectives

7-1 1, 2, 6, 9, 10Preview

7-1 2, 6, 8, 9, 10

7-1 2, 6Follow-Up

7-2 2, 6, 8, 9, 10

7-3 2, 6, 8, 9, 10

7-4 2, 6, 8, 9, 10

7-5 2, 6, 8, 9, 10

Key to NCTM Standards: 1=Number & Operations, 2=Algebra,3=Geometry, 4=Measurement, 5=Data Analysis & Probability, 6=ProblemSolving, 7=Reasoning & Proof,8=Communication, 9=Connections,10=Representation

Vocabulary BuilderThe Key Vocabulary list introduces students to some of the main vocabulary termsincluded in this chapter. For a more thorough vocabulary list with pronunciations ofnew words, give students the Vocabulary Builder worksheets found on pages vii andviii of the Chapter 7 Resource Masters. Encourage them to complete the definition of each term as they progress through the chapter. You may suggest that they addthese sheets to their study notebooks for future reference when studying for theChapter 7 test.

ELL

This section provides a review ofthe basic concepts needed beforebeginning Chapter 7. Pagereferences are included foradditional student help.

Additional review is provided inthe Prerequisite Skills Workbook,pp. 27–28.

Prerequisite Skills in the GettingReady for the Next Lesson sectionat the end of each exercise setreview a skill needed in the nextlesson.

Chapter 7 Solving Systems of Linear Equations and Inequalities 367

Make this Foldable to record information about solving systemsof equations and inequalities. Begin with five sheets of gridpaper.

Solvi

ng Sy

stem

s of

Equa

tions

and

Ineq

uatio

ns

7-1 GraphingSystems ofEquationsLabel each page

with a lesson number and title.

Stack the sheets and staple to

form a booklet.

Unfold and cut four rows from left side of each sheet, from the

top to the crease.

Fold each sheet in half along the

width.

Reading and Writing As you read and study the chapter, unfold each page and fill the journal with notes, graphs, and examples for systems of equations and inequalities.

Prerequisite Skills To be successful in this chapter, you’ll need to masterthese skills and be able to apply them in problem-solving situations. Reviewthese skills before beginning Chapter 7.

For Lesson 7-1 Graph Linear Equations

Graph each equation. (For review, see Lesson 4-5.) 1–6. See pp. 405A–405D.1. y � 1 2. y � �2x 3. y � 4 � x

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

�x � 2

For Lesson 7-2 Solve for a Given Variable

Solve each equation or formula for the variable specified. (For review, see Lesson 3-8.)

7. 4x � a � 6x, for x x � �a2

� 8. 8a � y � 16, for a a � �16

8� y�

9. �7bc

10� d� � 12, for b b � �

1207c

� d� 10. �

7mq� n� � 2m, for q q � �7m

2m� n�

For Lessons 7-3 and 7-4 Simplify Expressions

Simplify each expression. If not possible, write simplified. (For review, see Lesson 1-5.)

11. (3x � y) � (2x � y) x 12. (7x � 2y) � (7x � 4y) �6y 13. (16x � 3y) � (11x � 3y)

14. (8x � 4y) � (�8x � 5y) y 15. 4(2x � 3y) � (8x � y) 13y 16. 3(x � 4y) � (x � 12y) 4x17. 2(x � 2y) � (3x � 4y) 5x 18. 5(2x � y) � 2(5x � 3y) 19. 3(x � 4y) � 2(2x � 6y) 7x

�11y

27y

Fold

Stack and Staple

Cut

Label

Chapter 7 Solving Systems of Linear Equations and Inequalities 367

For PrerequisiteLesson Skill

7-2 Solving Equations for aSpecified Value, p. 374

7-3 Simplifying Expressions, p. 381

7-4 Distributive Property, p. 386

7-5 Graphing Inequalities, p. 392

Organization of Data: Visualization Journal After studentsmake their visualization journal, have them label the top of eachfront page with a lesson number. Under the tabs of their Foldable,students take notes and define terms presented in each lesson. At the end of each lesson, ask students to design a visual (graph,diagram, picture, chart) that presents the lesson information in aconcise, easy-to-study format. Encourage students to clearly labeltheir visuals.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

In addition toreviewinggraphing as inExercises 1-6,you may want

to review slope-interceptform. These are two skillsessential for students’ successin this chapter.

New

SpreadsheetInvestigation

TeachTeach

Getting StartedGetting StartedA Preview of Lesson 7-1

Copying Formulas Once thespreadsheet formulas are enteredfor one sales amount, thoseformulas can be dragged to copyfor all the other sales amounts.Students must take their timecreating the first formulas as anyerrors will be duplicated to allthe other cells.

• In Excel, you create a graph byusing the Chart Wizard buttonon the standard toolbar. Ifusing another spreadsheetsoftware program, consult theHelp feature for instructionson creating graphs (charts).

• Remind students that spread-sheet software cannot interpretan expression such as 0.1x.Students must type themultiplication sign implied.

• Urge students not to assumethat the software will apply theorder of operations. Studentsshould use parentheses toinsure that operations areperformed in the correct order.

• You may wish to extend theactivity by discussing the prosand cons of different salescommissions. Students who canimagine themselves as sales-people can relate personally tothe amounts calculated and tothe process of using equationsto explore pay rates.

Exercises 1–2 Students shouldwrite the correct expressions totranslate the real-world scenariointo an algebraic representation.

Exercise 4 Students should beable to translate between themathematical solution and itsreal-world meaning.


A Preview of Lesson 7-1

Systems of EquationsYou can use a spreadsheet to investigate when two quantities will be equal. Entereach formula into the spreadsheet and look for the time when both formulas havethe same result.

ExampleBill Winters is considering two job offers in telemarketing departments. The salary at the first job is $400 per week plus 10% commission on Mr. Winters’ sales. At the second job, the salary is $375 per week plus 15% commission. For what amount of sales would the weekly salary be the same at either job?

Enter different amounts for Mr. Winters’ weekly sales in column A. Then enter the formula for the salary at the first job in each cell in column B. In each cell of column C, enter the formula for the salary at the second job.

The spreadsheet shows that for sales of $500 the total weekly salary for each job is $450.

4. (500, 450); If Mr. Winters makes $500 in sales, he will make $450 for either job.ExercisesFor Exercises 1– 4, use the spreadsheet of weekly salaries above.1. If x is the amount of Mr. Winters’ weekly sales and y is his total weekly salary,

write a linear equation for the salary at the first job. y � 400 � 0.1x

2. Write a linear equation for the salary at the second job. y � 375 � 0.15x

3. Which ordered pair is a solution for both of the equations you wrote for Exercises 1 and 2? ca. (100, 410) b. (300, 420) c. (500, 450) d. (900, 510)

4. Use the graphing capability of the spreadsheet program to graph the salary data using a line graph. At what point do the two lines intersect? What is the significance of that point in the real-world situation?

5. How could you find the sales for which Mr. Winters’ salary will be equal without using a spreadsheet? Sample answer: Write and graph two linear equations. Find the point where the graphs intersect.


AssessAssess

5-Minute CheckTransparency 7-1 Use as a

quiz or review of Chapter 6.

Mathematical Background notesare available for this lesson on p. 366C.

can you use graphs tocompare the sales of

two products?Ask students:• The graph that is sloping down-

ward represents sales of whichproduct? cassette singles

• The graph that is slopingupward represents sales ofwhich product? CD singles

• In what year were the sales ofcassette and CD singles equal?1996; Note: The x-value of 0represents the year 1991.

• You are told to assume that thesales are linear functions. Whatdoes this mean? It means thatsales either increase or decreaseby the same amount every year.Also, the graphs of linear functionsare straight lines.

Cassette and CD Singles Sales

1 2 3 4 5 6 7 8

10

0

20

30

40Sa

les

(mill

ion

s o

f d

olla

rs)

50

60

70

Years Since 1991

y � 5.7 � 6.3x

y � 69 � 6.9x

Sales of cassette singlesequals sales of CD singles.

x

y

Systems of Equations

Graph of a System

Number of Solutions exactly one solution infinitely many no solutions

Terminology consistent and independent consistent and dependent inconsistent

O

y

x

Parallel Lines

O

y

x

Same Line

O

y

x

Intersecting Lines

NUMBER OF SOLUTIONS Two equations, such as y � 69 � 6.9x and y � 5.7 � 6.3x, together are called a . A solution of a system ofequations is an ordered pair of numbers that satisfies both equations. A system oftwo linear equations can have 0, 1, or an infinite number of solutions.

• If the graphs intersect or coincide, the system of equations is said to be .That is, it has at least one ordered pair that satisfies both equations.

• If the graphs are parallel, the system of equations is said to be . Thereare no ordered pairs that satisfy both equations.

• Consistent equations can be or . If a system has exactlyone solution, it is independent. If the system has an infinite number of solutions, it is dependent.

dependentindependent

inconsistent

consistent

system of equations

Graphing Systems of Equations

Lesson 7-1 Graphing Systems of Equations 369

Vocabulary• system of equations• consistent• inconsistent• independent• dependent

• Determine whether a system of linear equations has 0, 1, or infinitely many solutions.

• Solve systems of equations by graphing.

During the 1990s, sales of cassette singles decreased, and sales of CD singlesincreased. Assume that the sales of these singles were linear functions. If x represents the years since 1991 and y represents the sales in millions of dollars, the following equations represent the sales of these singles.

Cassette singles: y � 69 � 6.9xCD singles: y � 5.7 � 6.3x

These equations are graphed at the right.

The point at which the two graphs intersect represents the time when the sales of cassette singles equaled the sales of CD singles. The ordered pair of this point is a solution of both equations.

can you use graphs to compare the sales of two products?can you use graphs to compare the sales of two products?

Lesson x-x Lesson Title 369

Chapter 7 Resource Masters• Study Guide and Intervention, pp. 403–404• Skills Practice, p. 405• Practice, p. 406• Reading to Learn Mathematics, p. 407• Enrichment, p. 408

Parent and Student Study GuideWorkbook, p. 53

School-to-Career Masters, p. 13

5-Minute Check Transparency 7-1Real-World Transparency 7Answer Key Transparencies

TechnologyInteractive Chalkboard

Workbook and Reproducible Masters

Resource ManagerResource Manager

Transparencies

LessonNotes

1 Focus1 Focus

11

In-Class ExampleIn-Class Example

Building on PriorKnowledge

In Chapter 4, students learned tograph linear equations. In thislesson, they should recognize thatgraphing systems of equationssimply involves graphing morethan one linear equation on thesame coordinate grid.

NUMBER OF SOLUTIONS

Reading Tip Tell students topay close attention to the labelson the graph. If any lines arelabeled with more than oneequation, then the twoequations will have infinitelymany solutions.

Use the graph to determinewhether each system has nosolution, one solution, orinfinitely many solutions.

a. y � �x � 1y � �x � 4

The graphs are parallel, so thereare no solutions.

b. 3x � 3y � 9y � �x � 1

The graphs intersect, so there isone solution.

c. x � y � 33x � 3y � 9

The graphs coincide, so there areinfinitely many solutions.

x

y

O

y � �x � 1

3x � 3y � 9

x � y � 3

y � �x � 4

SOLVE BY GRAPHING One method of solving systems of equations is tocarefully graph the equations on the same coordinate plane.


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

a. y � �x � 5y � x � 3

Since the graphs of y � �x � 5 and y � x � 3 intersect, there is one solution.

b. y � �x � 52x � 2y � �8

Since the graphs of y � �x � 5 and 2x � 2y � �8 are parallel, there are nosolutions.

c. 2x � 2y � �8y � �x � 4

Since the graphs of 2x � 2y � �8 and y � �x � 4 coincide, there are infinitelymany solutions.

y

xO

y � 2x � 14

y � �x � 4 y � x � 3

y � �x � 5

2x � 2y � �8

Example 1Example 1

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

a. y � �x � 8y � 4x � 7

The graphs appear to intersect at the point with coordinates (3, 5). Check this estimate by replacing x with 3 and y with 5 in eachequation.

CHECK y � �x � 8 y � 4x � 7

5 � �3 � 8 5 � 4(3) � 7

5 � 5 � 5 � 12 � 7

5 � 5 �

The solution is (3, 5).

b. x � 2y � 52x � 4y � 2

The graphs of the equations are parallel lines. Since they do not intersect, there are no solutions to this system of equations. Notice that the lines have the same slope but differenty-intercepts. Recall that a system of equations that

has no solution is said to be inconsistent.

y

xO

x � 2y � 5

2x � 4y � 2

y

xO

y � �x � 8

y � 4x � 7

(3, 5)

Example 2Example 2

Look BackTo review graphing linearequations, see Lesson 4-5.

Study Tip


2 Teach2 Teach

Logical Before students graph a system of equations, have them writethe two equations in slope-intercept form and compare slopes andintercepts. Different slopes mean the lines intersect and there is onesolution. Same slope, same intercept means they are the same lineand there are infinite solutions. Same slope, different interceptsindicates parallel lines and no solution.

Differentiated Instruction

PowerPoint®


Guided Practice

GUIDED PRACTICE KEYExercises Examples

4–7 18–13 2

14 3

1. OPEN ENDED Draw the graph of a system of equations that has one solution at (�2, 3).

2. Determine whether a system of equations with (0, 0) and (2, 2) as solutionssometimes, always, or never has other solutions. Explain.

3. Find a counterexample for the following statement.

If the graphs of two linear equations have the same slope, then the system of equations has no solution.

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

4. y � x � 4 5. y � �13

�x � 2 noy � �

13

�x � 2 one y � �13

�x � 2solution

6. x � y � 4 7. x � y � 4y � x � 4 y � ��

13

�x � 4 oneinfinitely many

Concept Check1–3. See pp.

405A–405D.

www.algebra1.com/extra_examples

Write and Solve a System of EquationsWORLD RECORDS Use the information on Guy Delage’s swim at the left. IfGuy can swim 3 miles per hour for an extended period and the raft drifts about1 mile per hour, how many hours did he spend swimming each day?

Words You have information about the amount of time spent swimming and floating. You also know the rates and the total distance traveled.

Variables Let s � the number of hours Guy swam, and let f � the number ofhours he floated each day. Write a system of equations to represent the situation.

EquationsThe number of the number of the total number of

hours swimming plus hours floating equals hours in a day.

s � f � 24

The daily miles the daily miles the total miles traveled swimming plus traveled floating equals traveled in a day.

3s � 1f � 44

Graph the equations s � f � 24 and 3s � f � 44. The graphs appear to intersect at the point with coordinates (10, 14). Check this estimate by replacing s with 10 and f with 14 in each equation.

CHECK s � f � 24 3s � f � 44

10 � 14 � 24 3(10) � 14 � 44

24 � 24 � 30 � 14 � 44

44 � 44 �

Guy Delage spent about 10 hours swimming each day.

f

sO

(10, 14)

2428

201612

4

4�4 8 12 16 20 24 28

8

3s � f � 44

s � f � 24

��

��

Example 3Example 3

World RecordsIn 1994, Guy Delage swam2400 miles across theAtlantic Ocean from CapeVerde to Barbados.Everyday he would swimawhile and then rest whilefloating with the current ona huge raft. He averaged44 miles per day.Source: Banner Aerospace, Inc.

y

xO

y � � x � 413

y � x � 213

y � x � 213

x � y � 4

y �x � 4


33

22

In-Class ExamplesIn-Class ExamplesSOLVE BY GRAPHING

Graph each system of equa-tions. Then determine whetherthe system has no solution,one solution, or infinitelymany solutions. If the systemhas one solution, name it.

a. 2x � y � �3 infinitely many 8x � 4y � �12 solutions

b. x � 2y � 4x � 2y � �2 no solution

BICYCLING Tyler and Pearlwent on a 20-kilometer bikeride that lasted 3 hours.Because there were manysteep hills on the bike ride,they had to walk for most ofthe trip. Their walking speedwas 4 kilometers per hour.Their riding speed was 12 kilometers per hour. Howmuch time did they spendwalking?Let r � the number of hours theyrode and w � the number ofhours they walked.r � w � 312r � 4w � 20

They walked for 2 hours.

r

w

O

12r � 4w � 20

r � w � 3

x

y

O

x � 2y � �2

x � 2y � 4

x

y

O

2x � y � �3

8x � 4y � �12

PowerPoint®

InteractiveChalkboard

PowerPoint®

Presentations

This CD-ROM is a customizable Microsoft® PowerPoint®presentation that includes:• Step-by-step, dynamic solutions of each In-Class Example

from the Teacher Wraparound Edition• Additional, Your Turn exercises for each example• The 5-Minute Check Transparencies• Hot links to Glencoe Online Study Tools


3 Practice/Apply3 Practice/Apply

Study NotebookStudy NotebookHave students—• add the definitions/examples of

the vocabulary terms to theirVocabulary Builder worksheets forChapter 7.

• include examples of graphs thathave exactly one solution, infinitelymany solutions, and no solution.

• include any other item(s) that theyfind helpful in mastering the skillsin this lesson.


Application

Practice and ApplyPractice and Apply

indicates increased difficulty�

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

15. x � �3 16. y � �x � 2 y � 2x � 1 one y � 2x � 4 one

17. y � x � �2 inf. 18. y � 2x � 1y � �x � 2 many y � 2x � 4 no solution

19. y � �3x � 6 20. 2y � 4x � 2y � 2x � 4 one y � 2x � 4 no solution

21. 2y � 4x � 2 22. 2y � 4x � 2y � �3x � 6 one y � 2x � 1 infinitely many

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. 23–40. See pp. 405A–405D for graphs.23. y � �6 24. x � 2 25. y � �

12

�x4x � y � 2 one; (2, �6) 3x � y � 8 one; (2, �2) 2x � y � 10 one; (4, 2)

26. y � �x 27. y � 3x � 4 28. y � 2x � 6y � 2x � 6 one; (2, �2) y � �3x � 4 one; (0, �4) y � �x � 3 one; (�3, 0)

29. x � 2y � 2 30. x � y � 2 31. 3x � 2y � 123x � y � 6 one; (2, 0) 2y � x � 10 one; (�2, 4) 3x � 2y � 6 no solution

32. 2x � 3y � 4 infinitely 33. 2x � y � �4 34. 4x � 3y � 24 one; (3, 4)�4x � 6y � �8 many 5x � 3y � �6 one; (�6, 8) 5x � 8y � �17

35. 3x � y � 3 infinitely 36. y � x � 3 37. 2x � 3y � �172y � �6x � 6 many 3y � x � 5 one; (�1, 2) y � x � 4 one; (�1, �5)

38. y � �23

�x � 5 39. 6 � �38

�y � x infinitely 40. �12

�x � �13

�y � 6

3y � 2x no solution �23

�x � �14

�y � 4many

y � �12

�x � 2 one; (8, 6)

41. GEOMETRY The length of the rectangle at the right is 1 meter less than twice its width. What are the dimensions of the rectangle?13 m by 7 m

�

wPerimeter � 40 m

y

xO

2y � 4x � 2

y � x � �2

x � �3

y � 2x � 1

y � 2x � 4

y � �x � 2

y � �3x � 6

Homework HelpFor See

Exercises Examples15–22 123–40 241–54 3

Extra PracticeSee page 835.

�

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. 8–13. See pp. 405A–405D for graphs.8. y � �x 9. x � y � 8 10. 2x � 4y � 2 infinitely

y � 2x one; (0, 0) x � y � 2 one; (5, 3) 3x � 6y � 3 many

11. x � y � 4 12. x � y � 2 13. x � y � 2x � y � 1 no solution 3y � 2x � 9 one; (3, 1) y � 4x � 7 one; (�1, 3)

14. RESTAURANTS The Rodriguez family and the Wong family went to a brunchbuffet. The restaurant charges one price for adults and another price for children.The Rodriguez family has two adults and three children, and their bill was$40.50. The Wong family has three adults and one child, and their bill was$38.00. Determine the price of the buffet for an adult and the price for a child.$10.50; $6.50


About the Exercises…Organization by Objective• Number of Solutions: 15–22• Solve by Graphing: 23–54

Odd/Even AssignmentsExercises 15–40 are structuredso that students practice thesame concepts whether theyare assigned odd or evenproblems.

Assignment GuideBasic: 15–41 odd, 42–47,55–68

Average: 15–41 odd, 42–47,55–68

Advanced: 16–40 even, 48–64(optional: 65–68)

Infinitely Many Solutions Though two equations may look differentbecause variables are on different sides of the equals sign, or becauseall terms are multiplied by a common factor, any two equations thathave the same graph are equivalent. The easiest way to tell whethertwo equations have the same graph, and thus infinitely many solutions,is to rewrite both equations in slope-intercept form.

Unlocking Misconceptions

Study Guide and Intervention


NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

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.


1. y � �x � 3 2. 2x � 2y � �6y � x � 1 y � �x � 3one infinitely many

3. y � �x � 3 4. 2x � 2y � �62x � 2y � 4 3x � y � 3none one

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

Study Guide and Intervention, p. 403 (shown) and p. 404


1. x � y � 3 2. 2x � y � �3x � y � �3 4x � 2y � �6no solution infinitely many

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

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.

5. 3x � y � �2 no 6. y � 2x � 3 infinitely 7. x � 2y � 3 one;3x � y � 0 solution 4x � 2y � 6 many 3x � y � �5 (�1, 2)

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? 20

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? 30 CDs and 10 videos

c � v � 404c � 6v � 180

4c � 6v � 180

c � v � 40

(30, 10)

CD Sales ($)

CD and Video Sales

Vid

eo S

ales

($)

5 1510 20 25 30 35 40 450

40

35

30

25

20

15

10

5

y � 1.5x

y � 0.5x � 20(20, 30)

Sales ($)

Dog Treats

Co

st (

$)

5 1510 20 25 30 35 40 450

40

35

30

25

20

15

10

5

3x � y � �5

x � 2y � 3

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

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1Skills Practice, p. 405 and Practice, p. 406 (shown)

Reading to Learn Mathematics


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? Sample answer:a function whose graph is a line

• What does it mean to say that two lines intersect? Sample answer:The lines cross.

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. D

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

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

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

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

Helping You Remember

2. Describe how you can solve a system of equations by graphing. Sample answer:Graph the equations on the same coordinate plane. Locate any points ofintersection.

x

y

Ox

y

O

x

y

Ox

y

O

Reading to Learn Mathematics, p. 407

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?

about 80 miles

t

d

O

slope is 50

slope is –25

2

50

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1Enrichment, p. 408

You can graph a system

of equations to predict

when men’s and

women’s Olympic times

will be the same. Visit

www.algebra1.com/webquest to continue

work on your WebQuest

project.


GEOMETRY For Exercises 42 and 43, use the graphs of y � 2x � 6, 3x � 2y � 19, and y � 2, which contain the sides of a triangle.

42. Find the coordinates of the vertices of the triangle. (�2, 2), (5, 2), (1, 8)43. Find the area of the triangle. 21 units2

BALLOONING For Exercises 44 and 45, use the information in the graphic at the right.

44. In how many minutes will the balloons beat the same height? 4 min

45. How high will the balloons be at thattime? 70 m

SAVINGS For Exercises 46 and 47, use thefollowing information. Monica and Michael Gordon both want tobuy a scooter. Monica has already saved $25and plans to save $5 per week until she canbuy the scooter. Michael has $16 and plans tosave $8 per week.

46. In how many weeks will Monica andMichael have saved the same amount ofmoney? 3 weeks

47. How much will each person have saved at that time? $40

BUSINESS For Exercises 48–50, use the graph at the right.

48. Which company had the greater profit during the ten years? Widget Company

49. Which company had a greater rate of growth? neither

50. If the profit patterns continue, will the profits of the two companies ever be equal? Explain. No; the graphs are parallel so the lines will never meet and there is no year when the profits will be equal.

POPULATION For Exercises 51–54, use the following information.The U.S. Census Bureau divides the country into four sections. They are theNortheast, the Midwest, the South, and the West.

51. In 1990, the population of the Midwest was about 60 million. During the 1990s, thepopulation of this area increased an average of about 0.4 million per year. Write anequation to represent the population of the Midwest for the years since 1990.

52. The population of the West was about 53 million in 1990. The population of thisarea increased an average of about 1 million per year during the 1990s. Write anequation to represent the population of the West for the years since 1990.

53. Graph the population equations. See pp. 405A–405D.54. Assume that the rate of growth of each of these areas remains the same.

Estimate when the population of the West would be equal to the population of the Midwest. in about 11.7 years or sometime in 2001

55. CRITICAL THINKING The solution of the system of equations Ax � y � 5 and Ax � By � 20 is (2, �3). What are the values of A and B? A � 4, B � �4

Yearly Profits

2 4 6 8

2

4

Pro

fit

(m

illio

ns

of

do

llars

)

6

8

Year

0

WidgetCompany

GadgetCompany

Balloon 1is 10 metersabove theground, rising15 meters per minute.

Balloon 2is 150 metersabove theground,descending 20 meters per minute.

51. p � 60 � 0.4t

52. p � 53 � 1t or p � 53 � t

www.algebra1.com/self_check_quiz

�

�

�

�


ELL

http://www.algebra1.com/webquest


Open-Ended Assessment

Modeling Model a line on acoordinate plane with string,spaghetti, or a similar item. Thenask volunteers to come up andmodel another line that wouldrepresent a system of equationswith one solution. Do the samefor systems of equations with nosolutions and with infinitelymany solutions.

Getting Ready for Lesson 7-2PREREQUISITE SKILL Studentswill learn to solve systems ofequations by substitution inLesson 7-2. The process ofsubstitution involves solvingequations for a specific variable.Use Exercises 65–68 to determineyour students’ familiarity withsolving equations for a specificvariable.

Answer

56. Graphs can show when the salesof one item is greater than thesales of the other item and whenthe sales of the items are equal.Answers should include thefollowing.

• The sales of cassette singlesequaled the sales of CD singlesin about 5 years or by the end of1995.

• The graph of each equationcontains all of the points whosecoordinates satisfy theequation. If a point is containedin both lines, then itscoordinates satisfy bothequations.

56. Answer the question that was posed at the beginning of thelesson. See margin.

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

Include the following in your answer:

• an estimate of the year in which the sales of cassette singles equaled the salesof CD singles, and

• an explanation of why graphing works.

57. Which graph represents a system of equations with no solution? B

58. How many solutions exist for the system of equations below? B4x � y � 73x � y � 0

no solution one solution

infinitely many solutions cannot be determinedDC

BA

y

xO

y � �3x � 5

y � 2x � 5

Dy

xO

x � y � 1

y � �x � 1

C

y

xO

y � � x � 213

y � � x � 113

By

xO

y � x � 2

y � �x

A

WRITING IN MATH



Maintain Your SkillsMaintain Your Skills

Determine which ordered pairs are part of the solution set for each inequality.(Lesson 6-6)

59. y � 2x, {(1, 4), (�1, 5), (5, �6), (�7, 0)} {(5, �6)}60. y � 8 � 3x, {(�4, 2), (�3, 0), (1, 4), (1, 8)} {(�4, 2), (�3, 0), (1, 4)}

61. MANUFACTURING The inspector at a perfume manufacturer accepts a bottle if it is less than 0.05 ounce above or below 2 ounces. What are the acceptablenumbers of ounces for a perfume bottle? (Lesson 6-5) {n1.95 � n � 2.05}

Write each equation in standard form. (Lesson 5-5)

62. y � 1 � 4(x � 5) 63. y � 2 � �13

�(x � 3) 64. y � 4 � �6(x � 2)

4x � y � 19 x � 3y � 3 6x � y � �8PREREQUISITE SKILL Solve each equation for the variable specified.(To review solving equations for a specified variable, see Lesson 3-8.)

65. 12x � y � 10x, for y y � 2x 66. 6a � b � 2a, for a a � ��14

�b

67. �7m

q� n� � 10, for q q � �

7m1�0

n� 68. �

5tz2� s� � 6, for z z � �

125�t

s�

Mixed Review

Getting Ready forthe Next Lesson


4 Assess4 Assess

Entering Equations The equa-tions must be solved for y beforethey are entered in the calculator.

Investigating Slope-Intercept Form 375Graphing Calculator Investigation Systems of Equations 375

A Follow-Up of Lesson 7-1

You can use a TI-83 Plus graphing calculator to solve a system of equations.

Systems of Equations

ExampleSolve the system of equations. State the decimal solution to the nearest hundredth.2.93x � y � 6.088.32x � y � 4.11

Solve each equation for y to enter them into the calculator.

2.93x � y � 6.08 First equation

2.93x � y � 2.93x � 6.08 � 2.93x Subtract 2.93x from each side.

y � 6.08 � 2.93x Simplify.

8.32x � y � 4.11 Second equation

8.32x � y � 8.32x � 4.11 � 8.32x Subtract 8.32x from each side.

�y � 4.11 � 8.32x Simplify.

(�1) � y � (�1)(4.11 � 8.32x) Multiply each side by �1.

y � �4.11 � 8.32x Simplify.

Enter these equations in the Y= list and graph.

KEYSTROKES: Review on pages 224–225.

Use the CALC menu to find the point of intersection.

KEYSTROKES: [CALC] 5

The solution is approximately (0.91, 3.43).

ENTERENTERENTER2nd

[10, 10] scl: 1 by [�10, 10] scl: 1

www.algebra1.com/other_calculator_keystrokes

ExercisesUse a graphing calculator to solve each system of equations. Write decimal solutions to the nearest hundredth.1. y � 3x � 4 2. y � 2x � 5

y � �0.5x � 6 (2.86, 4.57) y � �0.2x � 4 (�4.09, �3.18)

3. x � y � 5.35 4. 0.35x � y � 1.123x � y � 3.75 (2.28, 3.08) 2.25x � y � �4.05 (�1.13, �1.51)

5. 1.5x � y � 6.7 6. 5.4x � y � 1.85.2x � y � 4.1 (1.61, 4.28) 6.2x � y � �3.8 (�0.17, �2.73)

7. 5x � 4y � 26 8. 2x � 3y � 114x � 2y � 53.3 (10.2, 6.25) 4x � y � �6 (�2.9, 5.6)

9. 0.22x � 0.15y � 0.30 10. 125x � 200y � 800�0.33x � y � 6.22 (�2.35, 5.44) 65x � 20y � 140 (1.14, �3.29)

Graphing Calculator Investigation Systems of Equations 375

GraphingCalculatorInvestigation

TeachTeach

Getting StartedGetting StartedA Follow-Up of Lesson 7-1

AssessAssess

• Step 2 Remind students toclear all previous equationsfrom the Y= list first.

Have students graph eachsystem using the standardviewing window. If theintersection is not visible, havestudents adjust the window toan area suggested by thedirections of the lines.

• Step 3 The GUESS feature thatappears after the second gives students an opportunity touse the arrow keys to estimatethe solution and then checktheir estimate by pressing

the third time.

Ask students how they canverify that their solution iscorrect. They should respondthat substitution of values intothe original equations willconfirm solutions.

ENTER

ENTER

http://www.algebra1.com/other_calculator_keystrokes


quiz or review of Lesson 7-1.

Mathematical Background notesare available for this lesson on p. 366C.

can a system ofequations be used to

predict media use?Ask students:• Which is changing at a greater

rate: the number of newspaperreaders, or the number ofpeople online? the number ofpeople online

• According to the graph, inabout what year will thenumber of hours online perperson equal time spent readingnewspapers? in about 2003

• If the two equations weren’tlabeled, how would you knowwhich was which? The one withnegative slope representsnewspaper readers because thetime spent reading newspapers isdeclining.

Vocabulary• substitution

Substitution

Using Substitution

Use algebra tiles and an equation mat to solve the system of equations.3x � y � 8 and y � x � 4

Model and Analyze

Since y � x � 4, use 1 positive x tile and 4 negative 1 tiles to represent y. Use algebra tiles to represent 3x � y � 8. 1. Use what you know about

equation mats to solve for x. What is the value of x? 3

2. Use the y � x � 4 to solve for y. �13. What is the solution of the system of equations? (3, �1)

Make a Conjecture

4. Explain how to solve the following system of equations using algebra tiles. 4x � 3y � 10 and y � x � 1 5. You substitute a representation of y for y.

5. Why do you think this method is called substitution?

�

y

3x

x

x x x1

1

1

1

1

1

1

1

�1

�1

�1

�1

4. On an equationmat, use algebra tiles to model 4x �3y � 10 using 1 positive x tile and 1 positive 1 tile torepresent each y. Usewhat you know aboutequation mats to solvefor x. Use the value ofx and y � x � 2 tosolve for y. Thesolution is (1, 2).

SUBSTITUTION The exact solution of a system of equations can be found byusing algebraic methods. One such method is called .substitution

• Solve systems of equations by using substitution.

• Solve real-world problems involving systems of equations.

can a system of equations be usedto predict media use?can a system of equations be usedto predict media use?

Americans spend more time online than they spend reading daily newspapers. Ifx represents the number of years since 1993 and y represents the average numberof hours per person per year, the following system represents the situation.

reading daily newspapers: y � �2.8x � 170online: y � 14.4x � 2

The solution of the system represents the year that the number of hoursspent on each activity will be the same.To solve this system, you could graphthe equations and find the point ofintersection. However, the exactcoordinates of the point would be verydifficult to determine from the graph.You could find a more accuratesolution by using algebraic methods.

Ho

urs

per

Per

son

per

Yea

r

20 1 3 5 7 94 6 8 10 1211 1314Years Since 1993

250200150100

50

Media Usage

Daily Newspapers

Online


LessonNotes

1 Focus1 Focus

Chapter 7 Resource Masters• Study Guide and Intervention, pp. 409–410• Skills Practice, p. 411• Practice, p. 412• Reading to Learn Mathematics, p. 413• Enrichment, p. 414• Assessment, p. 447


Prerequisite Skills Workbook, pp. 27–28

Science and Mathematics Lab Manual, pp. 63–66

Teaching Algebra With ManipulativesMasters, pp. 10–11, 16, 125

5-Minute Check Transparency 7-2Answer Key Transparencies

TechnologyAlgePASS: Tutorial Plus, Lesson 16Interactive Chalkboard



Transparencies

Lesson 7-2 Substitution 377www.algebra1.com/extra_examples

Solve Using SubstitutionUse substitution to solve the system of equations.

y � 3xx � 2y � �21

Since y � 3x, substitute 3x for y in the second equation.

x � 2y � �21 Second equation

x � 2(3x) � �21 y � 3x

x � 6x � �21 Simplify.

7x � �21 Combine like terms.

�77x� � �

�721� Divide each side by 7.

x � �3 Simplify.

Use y � 3x to find the value of y.

y � 3x First equation

y � 3(�3) x � �3

y � �9 The solution is (�3, �9).

Example 1Example 1

Solve for One Variable, Then SubstituteUse substitution to solve the system of equations.

x � 5y � �33x � 2y � 8

Solve the first equation for x since the coefficient of x is 1.

x � 5y � �3 First equation

x � 5y � 5y � �3 � 5y Subtract 5y from each side.

x � �3 � 5y Simplify.

Find the value of y by substituting �3 � 5y for x in the second equation.

3x � 2y � 8 Second equation

3(�3 � 5y) � 2y � 8 x � �3 � 5y

�9 � 15y � 2y � 8 Distributive Property

�9 � 17y � 8 Combine like terms.

�9 � 17y � 9 � 8 � 9 Add 9 to each side.

�17y � 17 Simplify.

��

�

1

1

7

7

y� � �

�1177

� Divide each side by �17.

y � �1 Simplify.

Substitute �1 for y in either equation to find the value of x.Choose the equation that is easier to solve.

x � 5y � �3 First equation

x � 5(�1) � �3 y � �1

x � 5 � �3 Simplify.

x � 2 Add 5 to each side.

The solution is (2, �1). The graph verifies the solution.

xO

(2, �1)

y

x � 5y � �3

3x � 2y � 8

Example 2Example 2

Look BackTo review solving linearequations, see Lesson 3-5.

Study Tip

Lesson 7-2 Substitution 377

2 Teach2 Teach

11

22

In-Class ExamplesIn-Class ExamplesSUBSTITUTION

Teaching Tip Explain that thepurpose of the substitutionmethod is that once you findone of the values (either x or y),you can then substitute it intoeither of the original equationsto find the other value. InExample 1, the value of y is givenin terms of x because y � 3x.So, substitute 3x for y in thesecond equation and solve.

Use substitution to solve thesystem of equations.

x � 4y4x � y � 75 (20, 5)

Teaching Tip You must solvefor one variable first becauseneither equation gives one vari-able in terms of the other as inExample 1. The easiest choice inExample 2 is to solve the firstequation for x by subtracting 5yfrom both sides.


4x � y � 12�2x � 3y � 14 (5, �8)

Algebra Activity

Materials: algebra tiles, equation mat• Remind students that anything they add to one side of the equation mat

must also be added to the other side of the mat.• Remind students that the purpose is to eliminate zero pairs.• Once a value is found for x, have students use that value to find y.

PowerPoint®


44


33


Teaching Tip If there areinfinitely many solutions, thenthe two equations represent thesame line. If you solve bothequations for y, then you willsee this is true. If there are nosolutions, then the equationsrepresent two parallel lines. Ifyou solve both equations for y,the equations will have thesame slope but a different y-intercept.


2x � 2y � 8x � y � �2

no solution

REAL-WORLD PROBLEMS

GOLD Gold is alloyed withdifferent metals to make ithard enough to be used injewelry. The amount of goldpresent in a gold alloy is mea-sured in 24ths called karats.

24-karat gold is or 100%

gold. Similarly, 18-karat gold

is or 75% gold. How many

ounces of 18-karat gold shouldbe added to an amount of 12-karat gold to make 4 ounces of 14-karat gold?

1 ounces of 18-karat gold and

2 ounces of 12-karat gold2�3

1�3

18�24

24�24

In general, if you solve a system of linear equations and the result is a truestatement (an identity such as �4 � �4), the system has an infinite number ofsolutions. However, if the result is a false statement (for example, �4 � 5), thesystem has no solution.

REAL-WORLD PROBLEMS Sometimes it is helpful to organize data beforesolving a problem. Some ways to organize data are to use tables, charts, differenttypes of graphs, or diagrams.


Dependent SystemUse substitution to solve the system of equations.

6x � 2y � �4y � 3x � 2

Since y � 3x � 2, substitute 3x � 2 for y in the first equation.

6x � 2y � �4 First equation

6x � 2(3x � 2) � �4 y � 3x � 2

6x � 6x � 4 � �4 Distributive Property

�4 � �4 Simplify.

The statement �4 � �4 is true. This means that there are infinitely many solutions ofthe system of equations. This is true because the slope-intercept form of both equationsis y � 3x � 2. That is, the equations are equivalent, and they have the same graph.

Example 3Example 3

Write and Solve a System of EquationsMETAL ALLOYS A metal alloy is 25% copper. Another metal alloy is 50%copper. How much of each alloy should be used to make 1000 grams of a metalalloy that is 45% copper?

Let a � the number of grams of the 25% copper alloy and b � the number ofgrams of the 50% copper alloy. Use a table to organize the information.

The system of equations is a � b � 1000 and 0.25a � 0.50b � 0.45(1000). Usesubstitution to solve this system.

a � b � 1000 First equation

a � b � b � 1000 � b Subtract b from each side.

a � 1000 � b Simplify.

0.25a � 0.50b � 0.45(1000) Second equation

0.25(1000 � b) � 0.50b � 0.45(1000) a � 1000 � b

250 � 0.25b � 0.50b � 450 Distributive Property

250 � 0.25b � 450 Combine like terms.

250 � 0.25b � 250 � 450 � 250 Subtract 250 from each side.

0.25b � 200 Simplify.

�00..2255b

� � �02.0205

� Divide each side by 0.25.

b � 800 Simplify.

Example 4Example 4

AlternativeMethodUsing a system ofequations is an alternativemethod for solving theweighted averageproblems that you studied in Lesson 3-9.

Study Tip

a b 1000

0.25a 0.50b 0.45(1000)

25% Copper 50% Copper 45% Copper

Total GramsGrams of Copper


Intrapersonal If students have difficulty solving systems of equationsby using substitution, ask them to indicate where they get confused inthe process before doing another example. Encourage students to thinkthrough each step.


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4–9 1–310 4

1. Explain why you might choose to use substitution rather than graphing to solvea system of equations. Substitution may result in a more accurate solution.

2. Describe the graphs of two equations if the solution of the system of equationsyields the equation 4 � 2. They are parallel lines.

3. OPEN-ENDED Write a system of equations that has infinitely many solutions.Sample answer: y � x � 3, 2y � 2x � 6

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.

4. x � 2y 5. y � 3x � 8 6. 2x � 7y � 34x � 2y � 15 (3, 1.5) y � 4 � x (3, 1) x � 1 � 4y (5, �1)

7. 6x � 2y � �4 8. x � 3y � 12 9. y � �35

�x no solutiony � 3x � 2 infinitely many x � y � 8 (9, 1) 3x � 5y � 15

10. TRANSPORTATION The Thrust SSCis the world’s fastest land vehicle.Suppose the driver of a car whose topspeed is 200 miles per hour requests arace against the SSC. The car gets a head start of one-half hour. If there isunlimited space to race, at what distancewill the SSC pass the car?about 135.5 mi

Thrust SSC top speed is 763 mph.

Guided Practice

Application


11. y � 5x 12. x � 4y 13. x � 4y � 52x � 3y � 34 (2, 10) 2x � 3y � 44 (16, 4) x � 3y � 2 (�23, �7)

14. y � 2x � 3 15. 4c � 3d � 3 16. 4x � 5y � 11y � 4x � 1 (2, 7) c � d � 1 (6, 7) y � 3x � 13 (4, �1)

17. 8x � 2y � 13 18. 2x � y � �4 19. 3x � 5y � 114x � y � 11 no solution �3x � y � �9 (13, 30) x � 3y � 1 (7, 2)

20. 2x � 3y � 1 21. c � 5d � 2 22. 5r � s � 5�3x � y � 15 (�4, 3) 2c � d � 4 (2, 0) �4r � 5s � 17 (2, 5)

23. 3x � 2y � 12 24. x � 3y � 0 25. �0.3x � y � 0.5x � 2y � 6 �4�

12

�, �34

�� 3x � y � 7 �2�110�, �

170�� 0.5x � 0.3y � 1.9 (5, 2)

26. 0.5x � 2y � 17 27. y � �12

�x � 3 28. x � �12

�y � 32x � y � 104 (50, 4) y � 2x � 1 �2�

23

�, 4�13

�� 2x � y � 6 infinitelymany

Concept Check


Exercises Examples11–28 1–329–37 4


a � b � 1000 First equation

a � 800 � 1000 b � 800

a � 800 � 800 � 1000 � 800 Subtract 800 from each side.

a � 200 Simplify.

200 grams of the 25% copper alloy and 800 grams of the 50% copper alloy shouldbe used.




Study NotebookStudy NotebookHave students—• add the definitions/examples of


• include examples of how to solve asystem of equations by usingsubstitution.


About the Exercises…Organization by Objective• Substitution: 11–28• Real–World Problems: 29–37


Alert! Exercise 38 requires theInternet or other researchmaterials.

Assignment GuideBasic: 11–33 odd, 34, 35, 39–53

Average: 11–33 odd, 34, 35,37, 39–53


All: Practice Quiz 1 (1–5)



Substitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

Less

on

7-2

Substitution One method of solving systems of equations is substitution.

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

Substitute 2x for y in the secondequation.

4x � y � �4 Second equation

4x � 2x � �4 y � 2x

2x � �4 Combine like terms.

� Divide each side by 2.


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

y � 2(�2) x � �2


The solution is (�2, �4).

�4�2

2x�2

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

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

x � 3y � 7 First equation

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

x � 7 � 3y Simplify.

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

2x � 4y � �6 Second equation

2(7 � 3y) � 4y � �6 x � 7 � 3y

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

14 � 10y � �6 Combine like terms.

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

�10y � �20 Simplify.


y � 2 Simplify.

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


�20��10

�10y��10

Example 1Example 1 Example 2Example 2

ExercisesExercises


1. y � 4x 2. x � 2y 3. x � 2y � 33x � y � 1 (�1, �4) y � x � 2 (4, 2) x � 2y � 4 no solution

4. x � 2y � �1 5. c � 4d � 1 infinitely 6. x � 2y � 03y � x � 4 (5, 3) 2c � 8d � 2 many 3x � 4y � 4 (4, �2)

7. 2b � 6a � 14 infinitely 8. x � y � 16 9. y � �x � 3 no3a � b � 7 many 2y � �2x � 2 no solution 2y � 2x � 4 solution

10. x � 2y (20, 10) 11. x � 2y � �5 12. �0.2x � y � 0.50.25x � 0.5y � 10 x � 2y � �1 (�3, 1) 0.4x � y � 1.1 (1, 0.7)



1. y � 6x 2. x � 3y 3. x � 2y � 72x � 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 � 12y � x � 2 (4, 6) 2x � y � 2 no solution y � �x � 2 (7, �9)

7. x � 2y � 13 (�3, 8) 8. x � 2y � 3 infinitely 9. x � 5y � 36 (�4, �8)�2x � 3y � �18 4x � 8y � 12 many 2x � y � �16

10. 2x � 3y � �24 11. x � 14y � 84 12. 0.3x � 0.2y � 0.5x � 6y � 18 (�6, 4) 2x � 7y � �7 (14, 5) x � 2y � �5 (5, 5)

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

Practice (Average)

Substitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

x � 2.5y � 3.5 (6, �1) x � y � 4 (5, 2) x � 2y � 6 (12, 3)

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

2x � y � 25 y � 5x 2x � 6y � 5

(12, 1) � , 1 � �� , 1 �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.

19. Write a system of equations to represent the situation.y � 0.04x � 500 and y � 0.05x � 400

20. What is the total price of the athletic shoes Kenisha needs to sell to earn the sameincome from each pay scale? $10,000

21. Which is the better offer? the first offer if she expects to sell less than$10,000 in shoes, and the second offer if she expects to sell more than$10,000 in shoes

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.

22. Write a system of equations to represent the situation.x � y � 8 and 7.25x � 5.5y � 52.75

23. How many adult tickets and student tickets were purchased? 5 adult and 3 student

1�12

3�4

2�3

1�3

1�3

1�2

Skills Practice, p. 411 and Practice, p. 412 (shown)


Substitution

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

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


• What is the system of equations? y � �2.8x � 170, y � 14.4x � 2• Based on the graph, are there 0, 1, or infinitely many solutions of the

system? 1 solution

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. It is a line. There are infinitelymany solutions, since �8 � �8 is always true.

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. The graph has two parallellines, since 5 � �2 is always false.


4. What is usually the first step in solving a system of equations by substitution? Sampleanswer: Solve one of the equations for one variable in terms of the other.

Solve the second equation for y. Substitute theexpression you find for y in the first equation. Simplifyand solve for x. Then use the value of x to find thevalue of y.

Solve the first equation for x. Substitute the expressionyou find for x in the second equation. Simplify and solvefor y. Then use the value of y to find the value of x.

Substitute 2y for x in the first equation. Simplify andsolve for y. Then use the value of y to find the value of x.

Substitute �2x for y in the second equation. Simplifyand solve for x. Then use the value of x to find thevalue of y.


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 � � � 1.2. For the plane in Exercise 1, write an

equation for the line where the planeintersects the xy-plane. Use interceptforms.

yx

z

z�1

y�2

x�3

Enrichment, p. 414


TourismEvery year, multitudes ofvisitors make their way toSouth America to stand inawe of Machu Picchu, thespectacular ruins of theLost City of the Incas. Source: www.about.com

�

29. GEOMETRY The base of the triangle is 4 incheslonger than the length of one of the other sides. Use asystem of equations to find the length of each side ofthe triangle. 14 in., 14 in., 18 in.

30. FUND-RAISING The Future Teachers of America Club at Paint Branch HighSchool is making a healthy trail mix to sell to students during lunch. The mix willhave three times the number of pounds of raisins as sunflower seeds. Sunflowerseeds cost $4.00 per pound, and raisins cost $1.50 per pound. If the group has$34.00 to spend on the raisins and sunflower seeds, how many pounds of eachshould they buy? 4 lb of sunflower seeds, 12 lb of raisins

31. CHEMISTRY MX Labs needs to make 500 gallons of a 34% acid solution. Theonly solutions available are a 25% acid solution and a 50% acid solution. Howmany gallons of each solution should be mixed to make the 34% solution?320 gal of 25% acid, 180 gal of 50% acid

32. GEOMETRY Supplementary angles are two angles whose measures have thesum of 180 degrees. Angles X and Y are supplementary, and the measure ofangle X is 24 degrees greater than the measure of angle Y. Find the measures of angles X and Y. m �X � 102, m �Y � 78

33. SPORTS At the end of the 2000 baseball season, the New York Yankees and the Cincinnati Reds had won a total of 31 World Series. The Yankees had won5.2 times as many World Series as the Reds. How many World Series did eachteam win? Yankees: 26, Reds: 5

JOBS For Exercises 34 and 35, use the following information. Shantel Jones has two job offers as a car salesperson. At one dealership, she will receive $600 per month plus a commission of 2% of the total price of theautomobiles she sells. At the other dealership, she will receive $1000 per month plus a commission of 1.5% of her total sales.

34. What is the total price of the automobiles that Ms. Jones must sell each month to make the same income from either dealership? $80,000

35. Explain which job offer is better. The second offer is better if she sells lessthan $80,000. The first offer is better if she sells more than $80,000.

36. LANDSCAPING A blue spruce grows an average of 6 inches per year. A hemlockgrows an average of 4 inches per year. If a blue spruce is 4 feet tall and a hemlockis 6 feet tall, when would you expect the trees to be the same height? in 12 years

37. TOURISM In 2000, approximately 40.3 million tourists visited South Americaand the Caribbean. The number of tourists to that area had been increasing at an average rate of 0.8 million tourists per year. In the same year, 17.0 milliontourists visited the Middle East. The number of tourists to the Middle East hadbeen increasing at an average rate of 1.8 million tourists per year. If the trendcontinues, when would you expect the number of tourists to South America and the Caribbean to equal the number of tourists to the Middle East?during the year 2023

38. RESEARCH Use the Internet or other resources to find the pricing plans forvarious cell phones. Determine the number of minutes you would need to usethe phone for two plans to cost the same amount of money. Support youranswer with a table, a graph, and/or an equation. See students’ work.

x in. x in.

y in.

Perimeter � 46 in.

�


ELL

http://www.about.com


Speaking Write a system ofequations on the chalkboard.Have students explain how theywould check this system ofequations to find out whether ithas no solution, one solution, orinfinitely many solutions. Thenuse student suggestions to solvethe system.

Getting Ready for Lesson 7-3PREREQUISITE SKILL Studentswill learn to solve systems ofequations by elimination usingaddition and subtraction inLesson 7-3. This method includessimplifying expressions. UseExercises 50–53 to determineyour students’ familiarity withsimplifying expressions.

Assessment Options

Practice Quiz 1 The quiz pro-vides students with a brief reviewof the concepts and skills inLessons 7-1 and 7-2. Lessonnumbers are given to the right ofexercises or instruction lines sostudents can review concepts notyet mastered.

Quiz (Lessons 7-1 and 7-2) isavailable on p. 447 of the Chapter 7Resource Masters.

Answer

40. When problems about technologyinvolve a system of equations, theproblem can be solved by substi-tution. Answers should includethe following.

• To solve a system of equationsusing substitution, solve oneequation for one unknown. Sub-stitute this value for the unknownin the other equation and solvethe equation. Use this numberto find the other unknown.

• The number of hours will be thesame about 9.8 years after 1993.That represents the year 2002.

Practice Quiz 1Practice Quiz 1

Graph each system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it. (Lesson 7-1)

1. x � y � 3 2. 3x � 2y � �6x � y � 1 one; (2, 1) 3x � 2y � 6 no solution

Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. (Lesson 7-2)

3. x � y � 0 4. x � 2y � 5 5. x � y � 23x � y � �8 (�4, 4) 3x � 5y � 8 (�9, �7) y � 2 � x infinitely many

Lessons 7-1 and 7-2



39. CRITICAL THINKING Solve the system of equations. Write the solution as anordered triple of the form (x, y, z).

2x � 3y � z � 17y � �3z � 72x � z � 2 (�1, 5, �4)

40. Answer the question that was posed at the beginning ofthe lesson. See margin.

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


• an explanation of solving a system of equations by using substitution, and

• the year when the number of hours spent reading daily newspapers is thesame as the hours spent online.

41. When solving the following system, which expression could be substituted for x?

x � 4y � 12x � 3y � �9 B

4y � 1 1 � 4y 3y � 9 �9 � 3y

42. If x � 3y � �9 and 5x � 2y � 7, what is the value of x? C1 2 3 4DCBA

DCBA

WRITING IN MATH


Mixed Review


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. (Lesson 7-1) 43–45. See pp. 405A–405D for graphs.43. x � y � 3 44. x � 2y � 1 45. 2x � y � 3 infinitely

x � y � 4 no solution 2x � y � 5 one; (3, �1) 4x � 2y � 6 many

Graph each inequality. (Lesson 6-6) 46–48. See pp. 405A–405D.46. y � �5 47. x � 4 48. 2x � y 6

49. RECYCLING When a pair of blue jeans is made, the leftover denim scraps can berecycled. One pound of denim is left after making every five pair of jeans. Howmany pounds of denim would be left from 250 pairs of jeans? (Lesson 3-6) 50 lb

PREREQUISITE SKILL Simplify each expression.(To review simplifying expressions, see Lesson 1-5.)

50. 6a � 9a �3a 51. 8t � 4t 12t 52. �7g � 8g �15g 53. 7d � (2d � b)5d � b

1–2. See margin for graphs.


4 Assess4 Assess

1. 2. y

xO

3x � 2y � �6

3x � 2y � 6

y

xO

(2, 1)x � y � 3

x � y � 1



Mathematical Background notesare available for this lesson on p. 366D.

can you use a system ofequations to solve

problems about weather?Ask students:• If 2n � 36, then how many

hours of darkness are there inSeward, AK, on the wintersolstice? 18 hours

• Since you know the value of nis 18, how can you find thevalue of d? Use substitution.

• How many hours of daylightare there in Seward, AK, on thewinter solstice? 6

• Geography The Arctic Circleis at latitude 66.5° North andpasses through northern Alaska.Because of the Earth’s tilt on itsaxis, the sun does not rise atthe Arctic Circle on the wintersolstice. Use this fact to write asystem of equations for thenumber of hours of daylight atthe Arctic Circle on the wintersolstice. Use the problem in thetext as an example. n � d � 24

n � d � 24

ELIMINATION USING ADDITION Sometimes adding two equationstogether will eliminate one variable. Using this step to solve a system of equations is called .elimination

Vocabulary• elimination

Elimination Using Addition and Subtraction


• Solve systems of equations by using elimination with addition.

• Solve systems of equations by using elimination with subtraction.

can you use a system of equations to solve problems about weather?can you use a system of equations to solve problems about weather?

On the winter solstice, there are fewer hours of daylight in the NorthernHemisphere than on any other day. On that day in Seward, Alaska, thedifference between the number of hours of darkness n and the number of hoursof daylight d is 12. The following system of equations represents the situation.

n � d � 24n � d � 12

Notice that if you add these equations, the variable d is eliminated.

n � d � 24(�)n � d � 12

2n � 36

Elimination Using AdditionUse elimination to solve each system of equations.

3x � 5y � �162x � 5y � 31

Since the coefficients of the y terms, �5 and 5, are additive inverses, you caneliminate the y terms by adding the equations.

3x � 5y � �16 Write the equations in column form and add.

5x � 15 Notice that the y variable is eliminated.

�55x� � �

155� Divide each side by 5.

x � 3 Simplify.

Now substitute 3 for x in either equation to find the value of y.

3x � 5y � �16 First equation

3(3) � 5y � �16 Replace x with 3.

9 � 5y � �16 Simplify.

9 � 5y � 9 � �16 – 9 Subtract 9 from each side.


��

�

5

5

y� � �

��

255


y � 5 Simplify.


(�) 2x � 5y � 31

Example 1Example 1

LessonNotes

1 Focus1 Focus

Chapter 7 Resource Masters• Study Guide and Intervention, pp. 415–416• Skills Practice, p. 417• Practice, p. 418• Reading to Learn Mathematics, p. 419• Enrichment, p. 420• Assessment, pp. 447, 449

Graphing Calculator and Spreadsheet Masters, p. 36






Transparencies

33


Lesson 7-3 Elimination Using Addition and Subtraction 383

ELIMINATION USING SUBTRACTION Sometimes subtracting oneequation from another will eliminate one variable.


Write and Solve a System of EquationsTwice one number added to another number is 18. Four times the first numberminus the other number is 12. Find the numbers.

Let x represent the first number and y represent the second number.

Twice one number added to another number is 18.

2x � y � 18

Four times the first number minus the other number is 12.

4x � y � 12

Use elimination to solve the system.

2x � y � 18 Write the equations in column form and add.

6x � 30 Notice that the variable y is eliminated.

�66x� � �


x � 5 Simplify.

Now substitute 5 for x in either equation to find the value of y.

4x � y � 12 Second equation

4(5) � y � 12 Replace x with 5.

20 � y � 12 Simplify.

20 � y � 20 � 12 � 20 Subtract 20 from each side.

�y � �8 Simplify.

��

�

y1� � �

��

81� Divide each side by �1.

y � 8 The numbers are 5 and 8.

(�) 4x � y � 12

� � ��

Example 2Example 2

Elimination Using SubtractionUse elimination to solve the system of equations.

5s � 2t � 69s � 2t � 22

Since the coefficients of the t terms, 2 and 2, are the same, you can eliminate the t terms by subtracting the equations.

5s � 2t � 6 Write the equations in column form and subtract.

�4s � �16 Notice that the variable t is eliminated.

��

44s

� � ��

146


s � 4 Simplify.

Now substitute 4 for s in either equation to find the value of t.5s � 2t � 6 First equation

5(4) � 2t � 6 s � 4

20 � 2t � 6 Simplify.

20 � 2t � 20 � 6 � 20 Subtract 20 from each side.

2t � �14 Simplify.

�22t� � �


t � �7 The solution is (4, �7).

(�) 9s � 2t � 22

Example 3Example 3

Look BackTo review translatingverbal sentences into equations, see Lesson 3-1.

Study Tip

Lesson 7-3 Elimination Using Addition and Substitution 383

2 Teach2 Teach

11

22

In-Class ExamplesIn-Class Examples


In Lesson 7-2 students learned tosolve a system of equations byusing substitution. Even thoughthey are learning differentmethods for solving systems ofequations in this lesson, they stillmust use substitution as part ofthose methods.

ELIMINATION USINGADDITION

Teaching Tip Explain that,when using the eliminationmethod, either the x or ycoefficients must be the same.

Use elimination to solve thesystem of equations.

�3x � 4y � 123x � 6y � 18 (�24, �15)

Four times one number minusthree times another numberis 12. Two times the firstnumber added to three timesthe second number is 6. Findthe numbers. The numbers are3 and 0.

Concept CheckAsk students to explain why thesystem, 2x � y � 7 and 7x � y � 32, cannot be solvedwith elimination using addition.Adding these two equations will noteliminate a variable.

ELIMINATION USINGSUBTRACTION


4x � 2y � 284x � 3y � 18 (6, 2)

Kinesthetic If students have difficulty solving systems of equations byusing elimination, have them write the terms of the equations on piecesof paper. Then, when they eliminate one of the variables, have themremove the papers with eliminated terms. The act of removing theterms should help cement the concept in students’ minds.


PowerPoint®

PowerPoint®



Study NotebookStudy Notebook

44



1. OPEN ENDED Write a system of equations that can be solved by using additionto eliminate one variable. Sample answer: 2a � b � 5; a � b � 4

2. Describe a system of equations that can be solved by using subtraction toeliminate one variable.

3. FIND THE ERROR Michael and Yoomee are solving a system of equations.

Who is correct? Explain your reasoning.

Concept Check

2. a system in whichone variable has thesame coefficient3. Michael; in order toeliminate the s-terms,you must add the twoequations.

Elimination Using SubtractionMultiple-Choice Test Item

Read the Test Item

You are given a system of equations, and you are asked to find the value of x.

Solve the Test Item

You can eliminate the x terms by subtracting one equation from the other.

x � 3y � 7 Write the equations in column form and subtract.

�5y � 5 Notice the x variable is eliminated.

��

�

5

5

y� � �

�55� Divide each side by �5.


Now substitute �1 for y in either equation to find the value of x.

x � 2y � 2 Second equation

x � 2(�1) � 2 y � �1

x � 2 � 2 Simplify.

x � 2 � 2 � 2 � 2 Add 2 to each side.

x � 4 Simplify.

Notice that B is the value of y and D is the solution of the system of equations.However, the question asks for the value of x. The answer is A.

(�) x � 2y � 2

Example 4Example 4

If x � 3y � 7 and x � 2y � 2, what is the value of x?

4 �1 (�1, 4) (4, �1)DCBA

Test-Taking TipAlways read the questioncarefully. Ask yourself,“What does the questionask?” Then answer thatquestion.


Michael

2r + s = 5

3r = 6

r = 2

2r + s = 5

2(2) + s = 5

4 + s = 5

s = 1


(+) r – s= 1

Yoomee

2r + s = 5

r = 4

r – s = 1

4 – s = 1

–s = -3

s = 3

The so lut ion is (4 , 3) .

(-) r – s= 1


InterventionSince subtrac-tion is the sameas adding theinverse, you

might suggest that studentschange the signs of theterms and then add toeliminate the variable.

New

MULTIPLE CHOICE TESTITEM If 8x � y � 16 and �6x � y � �26, what is thevalue of y? CA (3, �8) B 3C �8 D (�8, 3)

Have students—• add the definitions/examples of


• include examples of how to solve asystem of equations by usingelimination with addition orsubtraction.


FIND THE ERRORTell students to

look at the originalsystem of equations before

they evaluate the students’ work.Do they need to use addition orsubtraction to eliminate s?

Example 4 is an example of a test item in whichtoo much information is given in some of theanswer choices. Always carefully compare theanswer choices with the question before marking

your final answer. By doing so, you can eliminate choices C and D because thequestion asks for the value of x, and choices C and D are solution sets.

PowerPoint®




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 (�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 � �1 9. x � y � 23x � 2y � 12 (2, �3) �4x � 4y � �2 ��1, � � x � y � �3 �� , � �

10. 2x � 3y � 12 11. �0.2x � y � 0.5 12. 0.1x � 0.3y � 0.94x � 3y � 24 (6, 0) 0.2x � 2y � 1.6 (1, 0.7) 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. Rema is 31 and Ken is 19.

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. 57

7�6

11�2

5�2

1�2

3�2

3�2



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

(�4, �5) (3, �5) (5, 3)

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

(6, �3) (�5, 7) (2, 4)

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

(3, �4) (�7, �1) (�2, 5)

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

(9, 2) (�2, 8) (1.6, 12.5)

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

(2.5, 1.25) (3.4, 2.2) �1 , �16. 4a � b � 2 17. � x � y � �2 18. x � y � 8

4a � 3b � 10x � y � 4 x � y � 19

(10, �1) (12, 2)

19. The sum of two numbers is 41 and their difference is 5. What are the numbers? 18, 23

20. Four times one number added to another number is 36. Three times the first numberminus the other number is 20. Find the numbers. 8, 4

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. 3, 7

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?English: 104 countries, Farsi: 26 countries

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? gloves: $8.50, hats: $6.50.

1�2

3�2

2�3

1�3

1�2

3�4

4�3

1�3

3�4

1�2

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

�� , 4�1�2




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? d � d � 0

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 addition The coefficients of the x terms are �3x � 2y � 6 additive inverses.

b. 3x � 5y � 7 subtraction The coefficients of the x terms are 3x � 2y � 8 the same.

c. �x � 4y � 9 subtraction The coefficients of the y terms are 4x � 4y � 6 the same.

d. 5x � 7y � 17 addition The coefficients of the y terms are 8x � 7y � 9 additive inverses.


2. Tell how you can decide whether to use addition or subtraction to eliminate a variable ina system of equations. Sample answer: Look at the coefficients of eachvariable. If the coefficients of one of the variables are additive inverses,you can use addition to eliminate the variable. If the coefficients of onevariable are the same, you can use subtraction to eliminate the variable.


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

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____


Lesson 7-3 Elimination Using Addition and Subtraction 385

Guided Practice


Use elimination to solve each system of equations. 9. ��2�12

�, �2�4. x � y � 14 5. 2a � 3b � �11 6. 4x � y � �9 (�2, �1)

x � y � 20 (17, 3) a � 3b � 8 (�1, 3) 4x � 2y � �10

7. 6x � 2y � �10 8. 2a � 4b � 30 (6.5, 4.25) 9. �4m � 2n � 62x � 2y � �10 (0, �5) �2a � 2b � �21.5 �4m � n � 8

10. The sum of two numbers is 24. Five times the first number minus the secondnumber is 12. What are the two numbers? 6, 18

11. If 2x � 7y � 17 and 2x � 5y � 11, what is the value of 2y? D�4 �2 3 6DCBA


4–9 1, 310 211 4


Exercises Examples 12–29 1, 330–39 242, 43 4


ParksMammoth Cave inKentucky was declared anational park in 1941. Ithas more than 336 miles of explored caves, makingit the longest recordedcave system in the world.Source: National Park Service




Use elimination to solve each system of equations. 26. (1.75, 2.5) 27. (15.8, 3.4)12. x � y � �3 13. s � t � 4 14. 3m � 2n � 13

x � y � 1 (�1, �2) s � t � 2 (3, �1) m � 2n � 7 (5, 1)

15. �4x � 2y � 8 16. 3a � b � 5 17. 2m � 5n � �64x � 3y � �10 (�1, 2) 2a � b � 10 (�5, 20) 2m � 7n � �14 (7, 4)

18. 3r � 5s � �35 19. 13a � 5b � �11 20. 3x � 5y � 162r � 5s � �30 (�5, 4) 13a � 11b � 7 (�2, 3) �3x � 2y � �10(2, �2)

21. 6s � 5t � 1 22. 4x � 3y � 12 23. a � 2b � 56s � 5t � 11 (1, �1) 4x � 3y � 24 �4�

12

�, 2� 3a � 2b � 9 �2, �1�12

��24. 4x � 5y � 7 25. 8a � b � 1 26. 1.44 x � 3.24y � �5.58

8x � 5y � 9 ��12

�, 1� 8a � 3b � 3 ��136�, ��

12

�� 1.08x � 3.24y � 9.99

27. 7.2m � 4.5n � 129.06 28. �35

�c � �15

�d � 9 29. �23

�x � �12

�y � 147.2m � 6.7n � 136.54

�75

�c � �15

�d � 11 (10, �15) �56

�x � �12

�y � 18 (24, 4)

30. The sum of two numbers is 48, and their difference is 24. What are the numbers?

31. Find the two numbers whose sum is 51 and whose difference is 13. 32, 19

32. Three times one number added to another number is 18. Twice the first numberminus the other number is 12. Find the numbers. 6, 0

33. One number added to twice another number is 23. Four times the first numberadded to twice the other number is 38. What are the numbers? 5, 9

34. BUSINESS In 1999, the United States produced about 2 million more motorvehicles than Japan. Together, the two countries produced about 22 millionmotor vehicles. How many vehicles were produced in each country?U.S.: about 12 million vehicles, Japan: about 10 million vehicles

35. PARKS A youth group and their leaders visited Mammoth Cave. Two adultsand 5 students in one van paid $77 for the Grand Avenue Tour of the cave. Twoadults and 7 students in a second van paid $95 for the same tour. Find the adultprice and the student price of the tour. adult: $16, student: $9

36. FOOTBALL During the National Football League’s 1999 season, Troy Aikman,the quarterback for the Dallas Cowboys, earned $0.467 million more than Deion Sanders, the Cowboys cornerback. Together they cost the Cowboys$12.867 million. How much did each player make? Aikman: $6.667 million,

Sanders: $6.200 million

�

�

36, 12

Lesson 7-3 Elimination Using Addition and Substitution 385

ELL



Writing Have students write areal-world problem that can besolved with elimination usingaddition and subtraction.

Getting Ready for Lesson 7-4PREREQUISITE SKILL Studentswill learn to solve systems ofequations by elimination usingmultiplication in Lesson 7-4. Thismethod will include multiplyingwhole equations and simplifyingthem using the DistributiveProperty. Use Exercises 51–54 todetermine your students’ famili-arity with rewriting expressionsusing the Distributive Property.

Assessment Options

Quiz (Lesson 7-3) is availableon p. 447 of the Chapter 7Resource Masters.

Mid-Chapter Test (Lessons 7-1through 7-3) is available on p. 449 of the Chapter 7 ResourceMasters.



POPULATIONS For Exercises 37–39, use the information in thegraph at the right.

37. Let x represent the number of years since 2000 and yrepresent population inbillions. Write an equation torepresent the population ofChina. y � 0.0048x � 1.28

38. Write an equation to representthe population of India.

39. Use elimination to find the yearwhen the populations of Chinaand India are predicted to bethe same. What is the predictedpopulation at that time? 2048;1.51 billion

40. CRITICAL THINKING The graphs of Ax � By � 15 and Ax � By � 9 intersect at(2, 1). Find A and B. A � 6, B � 3

41. Answer the question that was posed at the beginning ofthe lesson. See pp. 405A–405D.

How can you use a system of equations to solve problems about weather?


• an explanation of how to use elimination to solve a system of equations, and

• a step-by-step solution of the Seward daylight problem.

42. If 2x � 3y � �9 and 3x � 3y � �12, what is the value of y? B�3 1 (�3, 1) (1, �3)

43. What is the solution of 4x � 2y � 8 and 2x � 2y � 2? C(�2, 3) (3, 2) (3, �2) (12, �3)DCBA

DCBA

WRITING IN MATH

38. y � 0.0104x �1.01

India’s exploding populationIndia is expected to pass China as the world’s mostpopulous nation within 50 years. Biggest populationstoday vs. predicted in 2050 (in billions):

Source: Population Reference Bureau, United Nations Population Division

By Anne R. Carey and Kay Worthington, USA TODAY

1.28

1.01

0.280.21

1.53

1.52

0.36

0.39

2050

2000China

India

USA

Indonesia

India

China

USA

Pakistan

USA TODAY Snapshots®


Use substitution to solve each system of equations. If the system does nothave exactly one solution, state whether it has no solution or infinitely manysolutions. (Lesson 7-2)

44. y � 5x 45. x � 2y � 3 46. 2y � x � �5x � 2y � 22 (2, 10) 3x � 4y � �1 (1, �1) 4y � 3x � �1 (�9, �7)

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. (Lesson 7-1) 47–49. See pp. 405A–405D for graphs. 49. infinitely many47. x � y � 3 one; (1, �2) 48. 2x � 3y � 7 no solution 49. 4x � y � 12

3x � y � 1 3y � 7 � 2x x � 3 � �14

�y

50. Write an equation of a line that is parallel to the graph of y � �54

�x � 3 and passesthrough the origin. (Lesson 5-6) y � �5

4�x

PREREQUISITE SKILL Use the Distributive Property to rewrite each expressionwithout parentheses. (To review the Distributive Property, see Lesson 1-5.)

51. 2(3x � 4y) 52. 6(2a � 5b) 53. �3(�2m � 3n) 54. �5(4t � 2s)6x � 8y 12a � 30b 6m � 9n �20t � 10s

Mixed Review


�

�

�


About the Exercises…Organization by Objective• Elimination Using

Addition: 12–15, 20–22, 26,28, 30–32, 34, 36

• Elimination UsingSubtraction: 16–19, 23–25, 27,29, 33, 35


Assignment GuideBasic: 13–25 odd, 29–35 odd,40–54

Average: 13–35 odd, 40–54


4 Assess4 Assess

Lou Jane Tynan Sacred Heart Model School, Louisville, KY

"Many students will forget to distribute the negative sign over the entireequation being subtracted in Example 3. I require my students to change thesigns of the equation being subtracted and then add the two equations."

5s + 2t = 6 5s + 2t = 6(–) 9s + 2t = 22 ⇒ (+) –9s – 2t = -22

Teacher to TeacherTeacher to Teacher




can a manager use asystem of equations to

plan employee time?Ask students:• Explain what the first equation

in the system of equationsrepresents. The first equationrepresents preparation time becausethe cookies take 20 minutes to pre-pare, the bread takes 10 minutes toprepare, and the bakery has allotted800 minutes of employee time forpreparation.

• Explain what the second equa-tion in the system of equationsrepresents. The second equationrepresents baking time because thecookies take 10 minutes to bake,the bread takes 30 minutes to bake,and the bakery has allotted 900 min-utes of oven time for preparation.

ELIMINATION USING MULTIPLICATION Neither variable in the systemabove can be eliminated by simply adding or subtracting the equations. However,you can use the Multiplication Property of Equality so that adding or subtractingeliminates one of the variables.

Elimination Using Multiplication

Lesson 7-4 Elimination Using Multiplication 387

can a manager use a system of equations to plan employee time?can a manager use a system of equations to plan employee time?

Multiply One Equation to EliminateUse elimination to solve the system of equations.

3x � 4y � 65x � 2y � �4

Multiply the second equation by �2 so the coefficients of the y terms are additiveinverses. Then add the equations.

3x � 4y � 6 3x � 4y � 65x � 2y � �4 Multiply by �2. (�) �10x � 4y � 8

�7x � 14 Add the equations.

��

77x

� � ��14



Now substitute �2 for x in either equation to find the value of y.

3x � 4y � 6 First equation

3(�2) � 4y � 6 x � �2

�6 � 4y � 6 Simplify.

�6 � 4y � 6 � 6 � 6 Add 6 to each side.

4y � 12 Simplify.

�4

4

y� � �


y � 3 The solution is (�2, 3).

Example 1Example 1

• Solve systems of equations by using elimination with multiplication.

• Determine the best method for solving systems of equations.

The Finneytown Bakery is making peanut butter cookies and loaves of quickbread. The preparation and baking times for each are given in the table below.

For these two items, the management has allotted 800 minutes of employee timeand 900 minutes of oven time. If c represents the number of batches of cookiesand b represents the number of loaves of bread, the following system ofequations can be used to determine how many of each to bake.

20c � 10b � 80010c � 30b � 900

Cookies(per batch)

Bread(per loaf)

PreparationBaking

10 min30 min

20 min10 min

Lesson x-x Lesson Title 387

Chapter 7 Resource Masters• Study Guide and Intervention, pp. 421–422• Skills Practice, p. 423• Practice, p. 424• Reading to Learn Mathematics, p. 425• Enrichment, p. 426• Assessment, p. 448


School-to-Career Masters, p. 14





Transparencies

LessonNotes

1 Focus1 Focus

11


ELIMINATION USINGMULTIPLICATION


2x � y � 233x � 2y � 37 (9, 5)


4x � 3y � 83x � 5y � �23 (�1, 4)

For some systems of equations, it is necessary to multiply each equation by adifferent number in order to solve the system by elimination. You can choose toeliminate either variable.


Multiply Both Equations to EliminateUse elimination to solve the system of equations.

3x � 4y � �252x � 3y � 6

Method 1 Eliminate x.

3x � 4y � �25 Multiply by 2. 6x � 8y � �502x � 3y � 6 Multiply by �3. (�) �6x � 9y � �18

17y � �68 Add the equations.

�1

1

7

7

y� � �

�1678



Now substitute �4 for y in either equation to find the value of x.


2x � 3(�4) � 6 y � �4

2x � 12 � 6 Simplify.

2x � 12 � 12 � 6 � 12 Subtract 12 from each side.

2x � �6 Simplify.

�22x� � �



The solution is (�3, �4).

Method 2 Eliminate y.

3x � 4y � �25 Multiply by 3. 9x � 12y � �752x � 3y � 6 Multiply by 4. (�) 8x � 12y � 24

17x � �51 Add the equations.

�1177x

� � ��1571



Now substitute �3 for x in either equation to find the value of y.


2(�3) � 3y � 6 x � �3

�6 � 3y � 6 Simplify.

�6 � 3y � 6 � 6 � 6 Add 6 to each side.

�3y � 12 Simplify.

��

�

3

3

y� � �

�12



The solution is (�3, �4), which matches the result obtained with Method 1.

Example 2Example 2

Study TipUsingMultiplicationThere are many othercombinations ofmultipliers that could beused to solve the systemin Example 2. Forinstance, the first equationcould be multiplied by �2and the second by 3.


2 Teach2 Teach

22

PowerPoint®

Solving Systems of Equations


DETERMINE THE BEST METHOD You have learned five methods forsolving systems of linear equations.


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

4x � 3y � 12x � 2y � 14

• For an exact solution, an algebraic method is best.

• Since neither the coefficients of x nor the coefficients of y are the same oradditive inverses, you cannot use elimination using addition or subtraction.

• Since the coefficient of x in the second equation is 1, you can use the substitutionmethod. You could also use elimination using multiplication.

The following solution uses substitution. Which method would you prefer?


x � 2y � 2y � 14 � 2y Subtract 2y from each side.

x � 14 � 2y Simplify.

4x � 3y � 12 First equation

4(14 � 2y) � 3y � 12 x � 14 � 2y

56 � 8y � 3y � 12 Distributive Property

56 � 11y � 12 Combine like terms.

56 � 11y � 56 � 12 � 56 Subtract 56 from each side.


��

�

1

1

1

1

y� � �

��

4141


y � 4 Simplify.


x � 2(4) � 14 y � 4

x � 8 � 14 Simplify.

x � 8 � 8 � 14 � 8 Subtract 8 from each side.

x � 6 Simplify.


Example 3Example 3

Method The Best Time to Use

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

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

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

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

Elimination Using if none of the coefficients are 1 or �1 and neither of the variablesMultiplication can be eliminated by simply adding or subtracting the equations

Study TipAlternativeMethodThis system could also besolved easily bymultiplying the secondequation by 4 and thensubtracting the equations.


Verbal/Linguistic Place students in pairs or small groups and assignsystems of equations for them to solve. Tell students to use theConcept Summary on p. 389 to discuss which method is the best touse to solve the system of equations they have been assigned. Makesure all group members participate in the discussion.

Differentiated Instruction ELL

33


DETERMINE THE BEST METHOD

Determine the best methodto solve the system of equa-tions. Then solve the system.

x � 5y � 43x � 7y � �10

The best method to use is substi-tution because the coefficient of xin the first equation is 1. (�1, 1)

PowerPoint®




44


TRANSPORTATION A fishingboat travels 10 miles down-stream in 30 minutes. Thereturn trip takes the boat 40minutes. Find the rate of theboat in still water. 17.5 mi/h

Have students—• add the definitions/examples of


• include examples of how to solve asystem of equations by usingelimination with multiplication.


Downstream

Upstream

b � c 2 24 2b � 2c � 24

b � c 3 24 3b � 3c � 24

r t d r t � d


Concept Check

Guided Practice

Example 4Example 4More About . . .

TransportationAbout 203 million tons offreight are transported onthe Ohio River each yearmaking it the second mostused commercial river inthe United States.Source: World Book

Encyclopedia

1. Explain why multiplication is sometimes needed to solve a system of equationsby elimination. See margin.

2. OPEN ENDED Write a system of equations that could be solved by multiplyingone equation by 5 and then adding the two equations together to eliminate onevariable. Sample answer: 3x � 2y � 5, 4x � 10y � �6

3. Describe two methods that could be used to solve the following system ofequations. Which method do you prefer? Explain. See pp. 405A–405D.a � b � 52a � 3b � 15


4. 2x � y � 6 5. x � 5y � 43x � 4y � �2 (2, �2) 3x � 7y � �10 (�1, 1)

6. 4x � 7y � 6 7. 4x � 2y � 10.56x � 5y � 20 (5, �2) 2x � 3y � 10.75 (1.25, 2.75)

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

8. 4x � 3y � 19 9. 3x � 7y � 63x � 4y � 8 elimination (�); (4, 1) 2x � 7y � 4 elimination (�); (2, 0)

10. y � 4x � 11 11. 5x � 2y � 12 elimination (�);3x � 2y � �7 substitution; (�3, �1) 3x � 2y � �2 (7, 11.5)


4–7 1–28–11 3

12 4

Write and Solve a System of EquationsTRANSPORTATION A coal barge on the Ohio River travels 24 miles upstream in3 hours. The return trip takes the barge only 2 hours. Find the rate of the bargein still water.

Let b � the rate of the barge in still water and c � the rate of the current. Use theformula rate time � distance, or rt � d.

This system cannot easily be solved using substitution. It cannot be solved by justadding or subtracting the equations.

The best way to solve this system is to use elimination using multiplication. Sincethe problem asks for b, eliminate c.

2b � 2c � 24 Multiply by 3. 6b � 6c � 723b � 3c � 24 Multiply by 2.

12b � 120 Add the equations.

�1122b

� � �11220


b � 10 Simplify.

The rate of the barge in still water is 10 miles per hour.

(�) 6b � 6c � 48


About the Exercises…Organization by Objective• Elimination Using

Multiplication: 13–26• Determine the Best

Method: 27–43


Assignment GuideBasic: 13–39 odd, 40, 44–57

Average: 13–39 odd, 40, 41,43–57


All: Practice Quiz 2 (1–5)

Answer

1. If one of the variables cannot beeliminated by adding or subtractingthe equations, you must multiplyone or both of the equations bynumbers so that a variable will beeliminated when the equations areadded or subtracted.

PowerPoint®



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 (�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 � �8 9. 2x � 2y � 55s � 2t � 8 (2, �1) 2a � 2b � 3 �� , 2� 4x � 4y � 10 � , 0�

10. 6x � 4y � �8 11. 4x � 2y � �5 12. 2x � y � 3.5 (0.9, 1.7)4x � 2y � �3 ��1, � �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?28 m by 8 m

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

1�2

3�2

1�2

5�2

1�2

Anita: 7 yr; Basilio: 2 yr1�2



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

4. 2x � 4y � �22 5. 3x � 2y � �9 6. 4x � 2y � 323x � 3y � 30 5x � 3y � 4 �3x � 5y � �11(3, 7) (�1, �3) (7, �2)

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

x � y � 0(5, 3) (4, �8)(�2, 4)

10. Eight times a number plus five times another number is �13. The sum of the twonumbers is 1. What are the numbers? �6, 7

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. 5, �2

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 � �20elimination (�); elimination (�); elimination (�);(�5, 4) (�2, 8) (�2, �1)

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

1�2

3�4

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

x � y � 3 4x � 3y � 3x � 2y � 9

substitution; infinitely elimination (�); substitution; (6, 3)many solutions no solution

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?$7200 in the 6% fund and $2800 in the 9% 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. 1.75 mi/h

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. 38

5�2

1�2




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. No; neither variable has coefficients that areadditive inverses or equal.

Reading the Lesson

1. Could elimination by multiplication be used to solve the system shown below? Explain.3x � 5y � 15�6x � 7y � 11Yes; you can multiply the first equation by 2 to make the coefficients ofthe x terms additive inverses.

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 � 2 elimination by The coefficients of the x terms are 3x � 2y � 10 addition additive inverses.

b. x � 2y � 0 substitution It is easy to solve the first equation 5x � 4y � 8 for x.

c. 6x � 5y � �18 elimination by Sample answer: You can multiply the 2x � 10y � 27 multiplication first equation by 2 to eliminate y by

addition.

d. �2x � 3y � 9 elimination by The coefficients of the y terms are 3x � 3y � 12 subtraction the same.


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? Sample answer: If bothadding the equations and subtracting the equations result in equationsthat still have two variables, you will need to use multiplication.


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 E A. (1, 4) automatic traffic signal

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



29. substitution; (2, 6)30. substitution; nosolution31. elimination (�);

�8, �43

��33. elimination (�) orsubstitution; (3, 1) 34. substitution; infinitely many solutions35. elimination (�);no solution


Application

BasketballIn the 2000–2001 season,Kobe Bryant ranked 18thin the NBA in free-throwpercentage. He made 475of the 557 free throws thathe attempted.Source: NBA




Exercises Examples13–26 1, 227–38 339–43 4



13. �5x � 3y � 6 14. x � y � 3 15. 2x � y � 5x � y � 4 (�9, �13) 2x � 3y � 16 (5, �2) 3x � 2y � 4 (2, 1)

16. 4x � 3y � 12 17. 5x � 2y � �15 18. 8x � 3y � �11x � 2y � 14 (6, 4) 3x � 8y � 37 (�1, 5) 2x � 5y � 27 (�4, �7)

19. 4x � 7y � 10 20. 2x � 3y � 2 21. 1.8x � 0.3y � 14.43x � 2y � �7 (�1, �2) 5x � 4y � 28 (4, 2) x � 0.6y � 2.8 (10, 12)

22. 0.4x � 0.5y � 2.5 23. 3x � �12

�y � 10 24. 2x � �23

�y � 41.2x � 3.5y � 2.5 (5, 1)

5x � �14

�y � 8 (2, �8) x � �12

�y � 7 (4, �6)

25. Seven times a number plus three times another number equals negative one.The sum of the two numbers is negative three. What are the numbers? 2, �5

26. Five times a number minus twice another number equals twenty-two. The sumof the numbers is three. Find the numbers. 4, �1

Determine the best method to solve each system of equations. Then solve thesystem. 27. elimination (�); (�2, 1) 28. elimination (�); (2, �3)27. 3x � 4y � �10 28. 9x � 8y � 42 29. y � 3x

5x � 8y � �2 4x � 8y � �16 3x � 4y � 30

30. x � 4y � 8 31. 2x � 3y � 12 32. 4x � 2y � 142x � 8y � �3 x � 3y � 12 y � x substitution; (7, 7)

33. x � y � 2 34. y � 2x � 9 35. 6x � y � 95x � 3y � 18 2x � y � �9 6x � y � 11

36. x � 8y subst.; (16, 2) 37. �23

�x � �12

�y � 14 38. �12

�x � �23

�y � �73

�

2x � 3y � 38�56

�x � �12

�y � 18 �32

�x � 2y � �25

elimin. (�); (24, 4) elimin. (�); (�6, �8)39. BASKETBALL In basketball, a free throw is 1 point and a field goal is either

2 points or 3 points. In the 2000–2001 season, Kobe Bryant scored a total of 1938points. The total number of 2-point field goals and 3-point field goals was 701.Use the information at the left to find the number of Kobe Bryant’s 2-point fieldgoals and 3-point field goals that season. 640 2-point field goals,

61 3-point field goalsOnline Research Data Update What are the current statistics for KobeBryant and other players? Visit www.algebra1.com/data_update to learn more.

40. CRITICAL THINKING The solution of the system 4x � 5y � 2 and 6x � 2y � bis (3, a). Find the values of a and b. a � �2, b � 22

41. CAREERS Mrs. Henderson discovered that she had accidentally reversed thedigits of a test and shorted a student 36 points. Mrs. Henderson told the studentthat the sum of the digits was 14 and agreed to give the student his correct scoreplus extra credit if he could determine his actual score without looking at histest. What was his actual score on the test? 95

�

12. BUSINESS The owners of the River View Restaurant have hired enough serversto handle 17 tables of customers, and the fire marshal has approved the restaurantfor a limit of 56 customers. How many two-seat tables and how many four-seattables should the owners purchase? 6 two-seat tables, 11 four-seat tables


ELL

http://www.algebra1.com/data_update



Speaking Using the ConceptSummary from p. 389, write onthe board a list of the differentmethods for solving systems ofequations. For each method, havea volunteer explain in his or herown words when it is appropriateto use the method.

Getting Ready for Lesson 7-5PREREQUISITE SKILL Studentswill learn to graph systems of in-equalities in Lesson 7-5. Studentsmust be comfortable graphinginequalities before beginning thislesson because otherwise theywill find it very confusing tohave different shaded regions.Use Exercises 54–57 to determineyour students’ familiarity withgraphing inequalities.

Assessment Options

Practice Quiz 2 The quiz pro-vides students with a brief reviewof the concepts and skills inLessons 7-3 and 7-4. Lessonnumbers are given to the right ofexercises or instruction lines sostudents can review concepts notyet mastered.

Quiz (Lesson 7-4) is availableon p. 448 of the Chapter 7Resource Masters.

Answers

54.

xO

y

y � x � 7


42. NUMBER THEORY The sum of the digits of a two-digit number is 14. If thedigits are reversed, the new number is 18 less than the original number. Find theoriginal number. 86

43. TRANSPORTATION Traveling against the wind, a plane flies 2100 miles fromChicago to San Diego in 4 hours and 40 minutes. The return trip, traveling witha wind that is twice as fast, takes 4 hours. Find the rate of the plane in still air.475 mph


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


• a demonstration of how to solve the system of equations concerning thecookies and bread, and

• an explanation of how a restaurant manager would schedule oven andemployee time.

45. If 5x � 3y � 12 and 4x � 5y � 17, what is the value of y? A�1 3 (�1, 3) (3, �1)

46. Determine the number of solutions of the system x � 2y � �1 and 2x � 4y � �2. D0 1 2 infinitely manyDCBA

DCBA

WRITING IN MATH



Mixed Review


Practice Quiz 2Practice Quiz 2

Use elimination to solve each system of equations. (Lessons 7-3 and 7-4)

1. 5x � 4y � 2 2. 2x � 3y � 13 3. 6x � 2y � 24 4. 5x � 2y � 4 3x � 4y � 14 2x � 2y � �2 3x � 4y � 27 10x � 4y � 9(2, �2) (2, �3) (5, 3) no solution

5. The price of a cellular telephone plan is based on peak and nonpeak service. Kelsey used 45 peak minutes and 50 nonpeak minutes and was charged $27.75. That same month, Mitch used 70 peak minutes and 30 nonpeak minutes for a total charge of $36. What are the rates per minute for peak and nonpeak time? (Lesson 7-4) $0.45; $0.15

Lessons 7-3 and 7-4

Use elimination to solve each system of equations. (Lesson 7-3)

47. x � y � 8 48. 2r � s � 5 49. x � y � 18x � y � 4 (6, 2) r � s � 1 (2, 1) x � 2y � 25 (11, 7)

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.(Lesson 7-2)

50. 2x � 3y � 3 51. x � y � 0 52. x � 2y � 7x � �3y (3, �1) 3x � y � �8 (�4, 4) �3x � 6y � �21

53. CAREERS A store manager is paid $32,000 a year plus 4% of the revenue thestore makes above quota. What is the amount of revenue above quota neededfor the manager to have an annual income greater than $45,000? (Lesson 6-3)

more than $325,000PREREQUISITE SKILL Graph each inequality.(To review graphing inequalities, see Lesson 6-6.) 54–57. See margin.54. y � x � 7 55. x � 3y � 9 56. �y � x 57. �3x � y � �1

�

�

infinitely many


4 Assess4 Assess

55. 56. 57.

xO

y

�3x � y � �1

xO

y

�y � x

864

�2�4�6�8

�4�2 2 4 6 8xO

y

2

10

x � 3y � 9

ReadingMathematics

Getting StartedGetting Started

TeachTeach

AssessAssess


Ask students how they organizeinformation such as vocabularywords when they study for tests.Write down several examples.Then ask students to commenton whether visual arrangementsof information (as opposed tolists or paragraphs) helps themremember the information better.

Creating Concept Maps Afterstudents examine the sampleconcept map shown, have themcreate their own concept map.Have them create the map overan unrelated task such as theschool lunch menu or beginningof class procedures. Ask studentsto share their maps when theyare complete.

Ask students to summarize whatthey have learned about conceptmaps.

Investigating Slope-Intercept Form 393Reading Mathematics Making Concept Maps 393

After completing a chapter, it is wise to review each lesson’s main topics and vocabulary.In Lesson 7-1, the new vocabulary words were system of equations, consistent,inconsistent, independent, and dependent. They are all related in that they explainhow many and what kind of solutions a system of equations has.

A graphic organizer called a concept map is a convenient way to show these relationships.A concept map is shown below for the vocabulary words for Lesson 7-1. The mainideas are placed in boxes. Any information that describes how to move from one boxto the next is placed along the arrows.

Concept maps are used to organize information. They clearly show how ideas arerelated to one another. They also show the flow of mental processes needed to solveproblems.

Reading to LearnReview Lessons 7-2, 7-3, and 7-4. 1–4. See margin.1. Write a couple of sentences describing the information in the concept map above.

2. How do you decide whether to use substitution or elimination? Give an exampleof a system that you would solve using each method.

3. How do you decide whether to multiply an equation by a factor?

4. How do you decide whether to add or subtract two equations?

5. Copy and complete the concept map below for solving systems of equations byusing either substitution or elimination.

Making Concept Maps

no solutionsolution exists

System of Equations

Consistent Inconsistent

infinite solutionsone solution

Independent Dependent

variables have any coefficients

one variable has a coefficient of 1 or -1

System of Equations

same signopposite signs

SubtractAdd

Multiply by a Factor

Variables with Different Coefficients

EliminationSubstitution

Variables with Same Coefficients

Reading Mathematics Making Concept Maps 393

English LanguageLearners may benefit fromwriting key concepts from thisactivity in their Study Notebooksin their native language and thenin English.

ELL

Answers

1. The are two types of systems ofequations, consistent and inconsistent.Consistent equations have one or moresolutions and inconsistent equations haveno solutions. If consistent equations haveone solution, they are called independent.If consistent equations have infinitesolutions, they are called dependent.

3. Multiply by a factor if neithervariable has the same or oppositecoefficients in the two equations.

4. Add if one of the variables hasopposite coefficients in the twoequations. Subtract if one of thevariables has the samecoefficients in the two equations.

2. Use substitution if an expression of onevariable is given or if the coefficient of avariable is �1. Use elimination if bothequations are written in standard form.

Sample answers:system to solve system to solve using substitution using eliminationy � 3x � 3 4x � 3y � 95x � 2y � 6 6x � y � 10





In Chapter 6, students learned tograph linear inequalities. In thislesson, they should recognize thatgraphing systems of inequalitiessimply involves graphing morethan one inequality on the samecoordinate grid and then shadingthe regions that overlap.

can you use a system ofinequalities to plan a

sensible diet?Ask students:• Does Joshua need to eat the

exact same amount of Caloriesand fat each day? Explain. No,but his Calorie and fat intake shouldbe within the green area of thegraph.

• Suppose Joshua eats 2600Calories one day, but only 55 grams of fat. Is this anappropriate intake of Caloriesand fat? How do you know?It is not appropriate because itdoes not fall within the greensection of the graph.

SYSTEMS OF INEQUALITIES To solve a , you need tofind the ordered pairs that satisfy all the inequalities involved. One way to do this isto graph the inequalities on the same coordinate plane. The solution set is representedby the intersection, or overlap, of the graphs.

system of inequalities

Vocabulary• system of inequalities

Graphing Systems of Inequalities


• Solve systems of inequalities by graphing.

• Solve real-world problems involving systems of inequalities.

can you use a system ofinequalities to plan a sensible diet?can you use a system ofinequalities to plan a sensible diet?

Solve by GraphingSolve the system of inequalities by graphing.

y � �x � 1y 2x � 3

The solution includes the ordered pairs in theintersection of the graphs of y � �x �1 and y � 2x � 3. This region is shaded in green at the right.The graphs of y � �x � 1 and y � 2x � 3 areboundaries of this region. The graph of y � �x � 1 isdashed and is not included in the graph of y � �x � 1.The graph of y � 2x � 3 is included in the graph of y � 2x � 3.

y

xO

y � �x � 1

y � 2x � 3

Example 1Example 1

Joshua watches what he eats. His doctor told him to eatbetween 2000 and 2400 Caloriesper day. The doctor also wantshim to keep his daily fat intakebetween 60 and 75 grams. Thegraph indicates the appropriateamounts of Calories and fat forJoshua. The graph is of a systemof inequalities. He should try tokeep his Calorie and fat intake to amounts represented in thegreen section.

Joshua’s Diet

500 1000 1500 2000 2500

10

20

30

40

Fat

Gra

ms

per

Day

50

60

70

80

Calories per Day

0

AppropriateEating Habits

Look BackTo review graphing linear inequalities, seeLesson 6-6.

Study Tip

No SolutionSolve the system of inequalities by graphing.

x � y � �1x � y 3

The graphs of x � y � �1 and x � y � 3 are parallellines. Because the two regions have no points incommon, the system of inequalities has no solution.

y

xO

x � y � �1

x � y � 3

�Example 2Example 2

LessonNotes

1 Focus1 Focus

Chapter 7 Resource Masters• Study Guide and Intervention, pp. 427–428• Skills Practice, p. 429• Practice, p. 430• Reading to Learn Mathematics, p. 431• Enrichment, p. 432• Assessment, pp. 448, 453–454

Graphing Calculator and Spreadsheet Masters, p. 35



TechnologyAlgePASS: Tutorial Plus, Lesson 17Interactive Chalkboard



Transparencies


To graph the system y � 4x � 3 and y � �2x � 9 on a TI-83 Plus, select the SHADEfeature in the DRAW menu. Enter the functionthat is the lower boundary of the region to beshaded, followed by the upper boundary. (Notethat inequalities that have or � are lowerboundaries and inequalities that have � or� are upper boundaries.)

Think and Discuss

1. To graph the system y � 3x � 1 and y � �2x � 5 on a graphing calculator, which function should you enter first? y � �2x � 5

2. Use a graphing calculator to graph the system y � 3x � 1 and y � �2x � 5.

3. Explain how you could use a graphing calculator to graph the system 2x � y � 7 and x � 2y � 5.

4. Use a graphing calculator to graph the system 2x � y � 7 and x � 2y � 5.

[�10, 10] scl: 1 by [�10, 10] scl: 1

Lesson 7-5 Graphing Systems of Inequalities 395

You can use a TI-83 Plus to solve systems of inequalities.

3. Solve each inequality for y. Thenenter the function thatis the lower bound (y � 7 � 2x), acomma, and the function that is theupper bound (y 0.5x � 2.5).


Use a System of Inequalities to Solve a ProblemCOLLEGE The middle 50% of first-year students attending Florida StateUniversity score between 520 and 620, inclusive, on the verbal portion of theSAT and between 530 and 630, inclusive, on the math portion. Graph the scoresthat a student would need to be in the middle 50% of FSU freshmen.

Words The verbal score is between 520 and 620, inclusive. The math scoreis between 530 and 630, inclusive.

Variables If v � the verbal score and m � the math score, the following inequalities represent the middle 50% of Florida State University freshmen.

Inequalities The verbal score is between 520 and 620, inclusive.

520 � v � 620

The math score is between 530 and 630, inclusive.

530 � m � 630

The solution is the set of all ordered pairs whose graphs are in the intersection ofthe graphs of these inequalities. However, since SAT scores are whole numbers,only whole-number solutions make sense in this problem.

m

v

200 400 800600

50% ofFSUFreshmen

Mat

h S

core

Verbal Score

200

0

400

800

600

��Example 3Example 3

REAL-WORLD PROBLEMS In real-life problems involving systems ofinequalities, sometimes only whole-number solutions make sense.

CollegeFSU is the most wiredcampus in Florida and hasbeen recently ranked the18th most technologicallyconnected university in thenation.Source: www.fsu.edu

2, 4. See pp. 405A–405D.


2 Teach2 Teach

11

22


33


SYSTEMS OFINEQUALITIES

Solve the system ofinequalities by graphing.y � 2x � 2y � �x � 3

Solve the system ofinequalities by graphing.y � �3x � 1y � �3x � 2 �

REAL-WORLD PROBLEMS

SERVICE A college serviceorganization requires that itsmembers maintain at least a3.0 grade point average, andvolunteer at least 10 hours a week. Graph theserequirements.

Grade Point Average

Vo

lun

teer

Ho

urs

1.0 2.0 3.0 4.00

1412108642

Requirementsfor Membership

x

y

O

y � �3x � 1

y � �3x � 2

x

y

O

y � 2x � 2

y � �x � 3

Graphing Systems of Inequalities Extra applications can be downloadedfrom the Internet for the TI-83 Plus calculator, including an inequalityapplication that automates and simplifies inequality graphing.

PowerPoint®

PowerPoint®

http://www.fsu.edu




44


EMPLOYMENT Jamail mowsgrass after school but his jobpays only $3 per hour. He hasbeen offered another job as alibrary assistant for $6 perhour. Because of school, hisparents allow him to work atmost 15 hours per week.How many hours can Jamailmow grass and work in thelibrary and still make at least$60 per week?

Jamail can work any combinationof hours in the darker shadedarea of the graph.

Have students—• complete the definitions/examples

for the remaining terms on theirVocabulary Builder worksheets forChapter 7.


FIND THE ERRORFirst have

students identify thedifferences between the twographs. Once they have foundthat one graph is shaded abovethe line representing x � 2y � 2,and the other is shaded belowthe line, they can determinewhich is correct.

4 8 12 16 200

20

16

12

8

4

g

�

3g � 6� � 60

g � � � 15

1. OPEN ENDED Draw the graph of a system of inequalities that has no solution.

2. Determine which of the following ordered pairsrepresent a solution of the system of inequalitiesgraphed at the right.

a. (3, 1) yes b. (�1, �3) noc. (2, 3) yes d. (4, �2) yese. (3, �2) no f. (1, 4) no

3. FIND THE ERROR Kayla and Sonia are solving the system of inequalities x � 2y � �2 and x � y 1.

Who is correct? Explain your reasoning.

y

xO


Concept Check1. See margin forsample answer.

3. Kayla; the graph ofx � 2y � �2 is theline representing x � 2y � 2 and theregion above it.

Use a System of InequalitiesAGRICULTURE To ensure a growing season of sufficient length, Mr. Hobsonhas at most 16 days left to plant his corn and soybean crops. He can plant corn ata rate of 250 acres per day and soybeans at a rate of 200 acres per day. If he has atmost 3500 acres available, how many acres of each type of crop can he plant?

Let c � the number of days that corn will be planted and s � the number of daysthat soybeans will be planted. Since both c and s represent a number of days,neither can be a negative number. The following system of inequalities can be usedto represent the conditions of this problem.

c � 0

s � 0

c � s � 16

250c � 200s � 3500

The solution is the set of all ordered pairs whosegraphs are in the intersection of the graphs of theseinequalities. This region is shown in green at theright. Only the portion of the region in the firstquadrant is used since c � 0 and s � 0.

Any point in this region is a possible solution. For example, since (7, 8) is a pointin the region, Mr. Hobson could plant corn for 7 days and soybeans for 8 days. Inthis case, he would use 15 days to plant 250(7) or 1750 acres of corn and 200(8) or1600 acres of soybeans.

s

cO

250c � 200s � 3500

2468

1012141618

2 4 6 8 10 12 14 16 18

c � s � 16

Example 4Example 4

Kayla

y

xO

Sonia

y

xO


Interpersonal If students have difficulty graphing systems ofinequalities, place them in pairs. Have one student graph one inequalitywhile the other student graphs the other inequality on a separate pieceof paper. Then, when each student has finished, have them togethertrace each graph onto a third piece of paper for the final graph.


PowerPoint®



NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Less

on

7-5

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

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

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

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

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

Solve each system of inequalities by graphing.

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

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

5x � 2y � 6

y � �x � 1

x

y

O

y � 2x � 3

y � �1 � 2xx

y

O

2x � y � 1

x � y � �2

x

y

O

y � x � 1

3x � 4y � 12

x

y

O

y � x � 1

y � �2x � 2

x

y

O

x � 0

y � �1

x

y

O

x

y

O

x � y � �1

x � y � 4

x

y

O

y � x � 2

y � �2x � 1

Example 1Example 1

Example 2Example 2

ExercisesExercises


Solve each system of inequalities by graphing.

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. Sample answers:gym 5 h, walk 9 mi; gym 6 h, walk 10 mi,gym 5.5 h, walk 11 mi

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. Sample answer: one $4stone and four $6 stones; three $4 stones and three $6 stones; five $4 stones and one $6 stone

$4 Stones

Turquoise Stones

$6 S

ton

es

1 32 4 5 6 7 80

6

5

4

3

2

1

Gym (hours)

Diego’s Routine

Wak

ing

(m

iles)

1 32 4 5 6 7 80

16

14

12

10

8

6

4

2

2x � y � 2

x � 2y � 2 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

O

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5Skills Practice, p. 429 and Practice, p. 430 (shown)



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. Graph each inequality on thesame coordinate plane. The solutions are the ordered pairs for the pointsin both graphs.

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 to 752000 to 2400


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. yy

1�3 x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____



Solve each system of inequalities by graphing. 4–9. See pp. 405A–405D.4. x 5 5. y 3 6. y � �x � 3

y � 4 y �x � 4 y � x � 3

7. 2x � y � 4 8. 2y � x � 6 9. x � 2y � 2y � �2x � 1 3x � y 4 3x � 4y � 12

x � 0

HEALTH For Exercises 10 and 11, use the following information. Natasha walks and jogs at least 3 miles every day. Natasha walks 4 miles per hourand jogs 8 miles per hour. She only has a half-hour to exercise.

10. Draw a graph of the possible amounts of time she can spend walking and jogging.

11. List three possible solutions. Sample answers: walk: 15 min, jog: 15 min;walk: 10 min, jog: 20 min; walk: 5 min, jog: 25 min

Guided Practice

Application



4–9 1, 210, 11 3, 4




Exercises Examples 12–28 1–229–31, 3–433–35


29. See pp. 405A–405D.30. Sample answers:2 light, 8 dark; 6 light,8 dark; 7 light, 4 dark

�

Solve each system of inequalities by graphing. 12–26. See pp. 405A–405D.12. y � 0 13. x �4 14. y � �2

x � 0 y � �1 y � x � 1

15. x � 2 16. x � 3 17. y � 2x � 1y � x � 5 x � y 2 y � �x � 1

18. y � 2x � 1 19. y � x � 1 20. y � x � 3y � �x � 3 y � x 3 y � x � 2

21. 2x � y � 4 22. 3x � 4y � 1 23. x � y 43x � y � 6 x � 2y � 7 �2x � 3y � �12

24. 2x � y � �4 25. y � x � 3 26. x � 2�5x � 2y � 1 2x � 7y � 4 4y x

3x � 2y � 6 2x � y �9x � 3y � 9

Write a system of inequalities for each graph.

27. y x, 28. y �23

�x � 2,y x � 3

ART For Exercises 29 and 30, use the following information. A painter has exactly 32 units of yellow dye and 54 units of blue dye. She plans tomix the dyes to make two shades of green. Each gallon of the lighter shade of greenrequires 4 units of yellow dye and 1 unit of blue dye. Each gallon of the darkershade of green requires 1 unit of yellow dye and 6 units of blue dye.

29. Make a graph showing the numbers of gallons of the two greens she can make.

30. List three possible solutions.

31. HEALTH The LDL or “bad” cholesterol of a teenager should be less than 110.The HDL or “good” cholesterol of a teenager should be between 35 and 59.Make a graph showing appropriate levels of cholesterol for a teenager.See pp. 405A–405D.

32. CRITICAL THINKING Write a system of inequalities that is equivalent to x � 4. x � �4 and x 4

y

xO

y

xO

��

y �x � 3

Visual ArtistVisual artists create art tocommunicate ideas. Thework of fine artists is madefor display. Illustrators andgraphic designers produceart for clients, such asadvertising and publishingcompanies.

Online ResearchFor information about a career as a visualartist, visit:www.algebra1.com/careers

10. See pp. 405A–405D.


ELL

Answer1. y

xO

y � 2 � x

y � �2 � x

�

http://www.algebra1.com/careers



Writing Have students describein words the method for deter-mining whether to shade aboveor below a line when graphingan inequality, and how to deter-mine the common solutions whengraphing a system of inequalities.

Assessment Options

Quiz (Lesson 7-5) is availableon p. 448 of the Chapter 7 ResourceMasters.

Answers

33.

xO 10

10

20

20

y

Tab

les

Desks

Furniture Manufacturing

2x � 1.5y � 31

1.5x � y � 22

The Spirit of the Games

It’s time to complete your project. Use the information and datayou have gathered about the Olympics to prepare a portfolio orWeb page. Be sure to include graphs and/or tables in your project.

www.algebra1.com/webquest



MANUFACTURING For Exercises 33 and 34, use the following information. The Natural Wood Company has machines that sand and varnishdesks and tables. The table givesthe time requirements of the machines.

33. Make a graph showing the number of desks and the number of tables that canbe made in a week. See margin.

34. List three possible solutions.


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


• two appropriate Calorie and fat intakes for a day, and

• the system of inequalities that is represented by the graph.

GRAPHING SYSTEMS OF INEQUALITIES Use a graphing calculator to solve eachsystem of inequalities. Sketch the results. 36–38. See margin.36. y � x � 9 37. y � 2x � 10 38. 3x � y � 6

y � �x � 4 y � 7x � 15 x � y � �1

39. Which ordered pair does not satisfy the system x � 2y 5 and 3x � y � �2? D(�3, 7) (0, 5) (�1, 4) (0, 2.5)

40. Which system of inequalities is represented by the graph? Ay � 2x � 2 y � 2x � 2y �x � 1 y � �x � 1

y � 2x � 2 y 2x � 2y � �x � 1 y � �x � 1

DC

BA

y

xO

DCBA

WRITING IN MATH


Use elimination to solve each system of equations. (Lessons 7-3 and 7-4)

41. 2x � 3y � 1 42. 5x � 2y � �3 43. �3x � 2y � 124x � 5y � 13 (2, �1) 3x � 6y � �9 (�1, �1) 2x � 3y � �13 (�2, 3)

44. 6x � 2y � 4 45. 2x � 5y � 13 46. 3x � y � 65x � 3y � �2 (2, 4) 3x � 5y � �18 (�1, 3) 3x � 2y � 15 (3, 3)

Write an equation of the line that passes through each point with the given slope.(Lesson 5-4)

47. (4, �1), m � 2 48. (1, 0), m � �6 49. (5, �2), m � �13

�

47. y � 2x � 9 48. y � �6x � 6 49. y � �13

�x � �131�

Mixed Review

GraphingCalculator

Hours HoursTotal Hours

Machineper Desk per Table

AvailableEach Week

Sanding 2 1.5 31

Varnishing 1.5 1 22

Sample answers: 8 desks, 10 tables; 6 desks, 12 tables; 4 desks, 14 tables


About the Exercises…Organization by Objective• Systems of Inequalities:

12–28• Real-World Problems: 29–31,

33–35


Alert! Exercises 36–38 require agraphing calculator.

Assignment GuideBasic: 13–23 odd, 29–32, 35,39–49

Average: 13–27 odd, 29–32, 35,39–49 (optional: 36–38)

Advanced: 12–28 even, 32–49

4 Assess4 Assess

36. 37. 38.

[�10, 10] scl: 1 by [�10, 10] scl: 1 by [�10, 10] scl: 1 by[�10, 10] scl: 1 [�10, 10] scl: 1 [�10, 10] scl: 1

http://www.algebra1.com/webquest

Study Guide and Review

Chapter 7 Study Guide and Review 399

Choose the correct term to complete each statement.

1. If a system of equations has exactly one solution, it is (dependent, ).2. If the graph of a system of equations is parallel lines, the system is (consistent, ).3. A system of equations that has infinitely many solutions is ( , independent).4. If the equations in a system have the same slope and different intercepts, the graph of the

system is (intersecting lines, ).5. If a system of equations has the same slope and intercepts, the system has (exactly one,

) solution(s).

6. The solution of a system of equations is (3, �5). The system is ( , inconsistent).consistentinfinitely many

parallel lines

dependentinconsistent

independent

See pages369–374.

Graphing Systems of InequalitiesConcept Summary

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

3x � y � �46x � 2y � �8

When the lines are graphed, they coincide. There are infinitely many solutions.

Exercises Graph each system of equations. Then determine whether the system ofequations has one solution, no solution, or infinitely many solutions. If the systemhas one solution, name it. See Example 2 on page 370. 7–10. See margin for graphs.7. x � y � 9 8. 9x � 2 � 3y 9. 2x � 3y � 4 10. 3x � y � 8

x � y � 11 y � 3x � 8 6y � 4x � 8 3x � 4 � yone; (10, 1) no solution infinitely many one; (2, �2)

yxO

7-17-1

ExampleExample

consistent (p. 369)dependent (p. 369)

elimination (p. 382)inconsistent (p. 369)

independent (p. 369)substitution (p. 376)

Vocabulary and Concept CheckVocabulary and Concept Check

Intersecting Lines Same Line Parallel Lines

exactly one solution infinitely many no solutions

consistent and consistent and inconsistentindependent dependent

y

xO

y

xO

y

xO

www.algebra1.com/vocabulary_review

Graph of a System

Number of Solutions

Terminology

system of equations (p. 369)system of inequalities (p. 394)


Have students look through the chapter to make sure they haveincluded examples in their Foldables for each method of solvingsystems of equations they learned.

Encourage students to refer to their Foldables while completingthe Study Guide and Review and to use them in preparing for theChapter Test.

TM

For more informationabout Foldables, seeTeaching Mathematicswith Foldables.

Lesson-by-LessonReviewLesson-by-LessonReview

Vocabulary and Concept CheckVocabulary and Concept Check

• This alphabetical list ofvocabulary terms in Chapter 7includes a page referencewhere each term wasintroduced.

• Assessment A vocabularytest/review for Chapter 7 isavailable on p. 446 of theChapter 7 Resource Masters.

For each lesson,

• the main ideas aresummarized,

• additional examples reviewconcepts, and

• practice exercises are provided.

The Vocabulary PuzzleMakersoftware improves students’ mathematicsvocabulary using four puzzle formats—crossword, scramble, word search using aword list, and word search using clues.Students can work on a computer screenor from a printed handout.

Vocabulary PuzzleMaker

ELL

MindJogger Videoquizzesprovide an alternative review of conceptspresented in this chapter. Students workin teams in a game show format to gainpoints for correct answers. The questionsare presented in three rounds.

Round 1 Concepts (5 questions)Round 2 Skills (4 questions)Round 3 Problem Solving (4 questions)

MindJogger Videoquizzes

ELL

See p. 400 for the graphs forExercises 7–10.



Answers

7.

8.

9.

10.

xO

y

3x � 4 � y

3x � y � 8

(2, �2)

xO

y

6y � 4x � 82x � 3y � 4

xO

yy � 3x � 8

9x � 2 � 3y

O

8642

�2 �4�6�8

�2 2 4 6 8 10 12 14x

y

(10, 1)

x � y � 11

x � y � 9


See pages376–381.

7-27-2

ExampleExample

Chapter 7 Study Guide and ReviewChapter 7 Study Guide and Review

ExampleExample

See pages382–386.

7-37-3 Elimination Using Addition and Subtraction Concept Summary

• Sometimes adding or subtracting two equations will eliminate one variable.

Use elimination to solve the system of equations.

2m � n � 4m � n � 2

You can eliminate the n terms by adding the equations.

2m � n � 4 Write the equations in column form and add.

(�) m � n � 2

3m � 6 Notice the variable n is eliminated.

m � 2 Divide each side by 3.

SubstitutionConcept Summary

• In a system of equations, solve one equation for a variable, and thensubstitute that expression into the second equation to solve.

Use substitution to solve the system of equations.y � x � 14x � y � 19

Since y � x � 1, substitute x � 1 for y in the second equation.

4x � y � 19 Second equation

4x � (x � 1) � 19 y = x � 1

4x � x � 1 � 19 Distributive Property

3x � 1 � 19 Combine like terms.

3x � 18 Subtract 1 from each side.

x � 6 Divide each side by 3.

Use y � x � 1 to find the value of y.

y � x � 1 First equation

y � 6 � 1 x = 6

y � 5 The solution is (6, 5).

Exercises Use substitution to solve each system of equations. If the systemdoes not have exactly one solution, state whether it has no solutions orinfinitely many solutions. See Examples 1–3 on pages 377 and 378.

11. 2m � n � 1 12. x � 3 � 2y 13. 3x � y � 1 14. 0.6m � 0.2n � 0.9m � n � 8 2x � 4y � 6 2x � 4y � 3 n � 4.5 � 3m(3, �5) infinitely many ��1

2�, �1

2�� (1.5, 0)

solutions




Chapter 7 Study Guide and ReviewChapter 7 Study Guide and Review

See pages387–392.

7-47-4

Now substitute 2 for m in either equation to find n.

m � n � 2 Second equation

2 � n � 2 m � 2

2 � n � 2 � 2 � 2 Subtract 2 from each side.

n � 0 Simplify.


Exercises Use elimination to solve each system of equations.See Examples 1–3 on pages 382 and 383.

15. x � 2y � 6 16. 2m � n � 5 17. 3x � y � 11 18. 3x � 1 � �7yx � 3y � �4 2m � n � 3 x � y � 5 6x � 7y � 0(2, 2) (2, �1) (4, 1) ��1

3�, ��2

7��

Elimination Using Multiplication Concept Summary

• Multiplying one equation by a number or multiplying each equation by a different number is a strategy that can be used to solve a system ofequations by elimination.

• There are five methods for solving systems of equations.

Use elimination to solve the system of equations.

x � 2y � 83x � y � 1.5

Multiply the second equation by �2 so the coefficients of the y terms are additiveinverses. Then add the equations.

x � 2y � 8 x � 2y � 83x � y � 1.5 Multiply by �2. (�) �6x � 2y � �3

�5x � 5 Add the equations.

��

55x

� � ��55� Divide each side by �5.


ExampleExample

Graphingto estimate the solution, since graphing usually does not give anexact solution

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

Elimination Usingif one of the variables has opposite coefficients in the two equations

Addition

Elimination Usingif one of the variables has the same coefficient in the two equations

Subtraction

Elimination Using if none of the coefficients are 1 or �1 and neither of the variables Multiplication can be eliminated by simply adding or subtracting the equations

Method The Best Time to Use

(continued on the next page)



Answers

27.

28.

29.

30.

Answers (page 403)

4.

xO

y

y � x � 2

y � 2x � 7

(�5, �3)

xO

y

x � 1

y � x � 3

xO

y2x � y � 9

x � 11y � �6

xO

y

y � 2x � 1

y � �x � 1

2

4O�2�4�6�8

�10�12�14

�12 �8 �4 x

y

x � 2y � �21

y � 3x


Graphing Systems of InequalitiesConcept Summary

• Graph each inequality on a coordinate plane to determine the intersectionof the graphs.

Solve the system of inequalities.x � �3y x � 2

The solution includes the ordered pairs in the intersection of the graphs x � �3 and y � x � 2. This region is shaded in green. The graphs of x � �3 and y � x � 2 areboundaries of this region.

Exercises Solve each system of inequalities by graphing.See Examples 1 and 2 on page 394. 27–30. See margin.27. y � 3x 28. y �x � 1

x � 2y � �21 y � 2x � 1

29. 2x � y � 9 30. x � 1x � 11y � �6 y � x � 3

y

xO

x � �3 y �x � 2

• Extra Practice, see pages 835–836.• Mixed Problem Solving, see page 859.

See pages394–398.

7-57-5

ExampleExample

x � 2y � 8 First equation

�1 � 2y � 8 x � �1

�1 � 2y � 1 � 8 � 1 Add 1 to each side.

2y � 9 Simplify.

�2

2

y� � �

92


y � 4.5 Simplify.

The solution is (�1, 4.5).

Exercises Use elimination to solve each system of equations.See Examples 1 and 2 on pages 387 and 388.

19. x � 5y � 0 20. x � 2y � 52x � 3y � 7 (5, 1) 3x � 5y � 8 (�9, �7)

21. 2x � 3y � 8 �2�45

�, �45

�� 22. �5x � 8y � 21 ��35

�, 3�x � y � 2 10x � 3y � 15

Determine the best method to solve each system of equations. Then solve thesystem. See Example 3 on page 389.

23. y � 2x 24. 9x � 8y � 7x � 2y � 8 substitution; �1�

35

�, 3�15

�� 18x � 15y � 14 elimination (�); ��79

�, 0�25. 3x � 5y � 2x 26. 2x � y � 3x � 15

x � 3y � y substitution; (0, 0) x � 5 � 4y � 2x substitution; (13, �2)


5. 6.

xO

y

2y � 10 � �6x

3x � y � 5

7654321

�1�2 2 4 6 8 10 12 14

xO

y

x � 2y � 11

x � 14 � 2y

Practice Test

Chapter 7 Practice Test 403

Vocabulary and ConceptsVocabulary and Concepts

Skills and ApplicationsSkills and Applications

Choose the letter that best matches each description.

1. a system of equations with two parallel lines c2. a system of equations with at least one ordered

pair that satisfies both equations a3. a system of equations may be solved using this method b

Graph each system of equations. Then determine whether the system hasno solution, one solution, or infinitely many solutions. If the system hasone solution, name it. 4–6. See margin for graphs.4. y � x � 2 5. x � 2y � 11 6. 3x � y � 5

y � 2x � 7 one; (�5, �3) x � 14 � 2y no solution 2y � 10 � �6x infinitely many

Use substitution or elimination to solve each system of equations.

7. 2x � 5y � 16 8. y � 2x � �1 9. 2x � y � �45x � 2y � 11 (3, 2) y � 4 � �2x no solution 5x � 3y � �6 (�6, 8)

10. y � 7 � x 11. x � 2y � 7 12. x � y � 10x � y � �3 (2, 5) y � 3x � �9 (5, 6) x � y � 2 (6, 4)

13. 3x � y � 11 14. 3x � y � 10 15. 5x � 3y � 12x � 2y � �36 (�2, �17) 3x � 2y � 16 (4, �2) �2x � 3y � �3 (3, 1)

16. 2x � 5y � 12 17. x � y � 6 18. 3x � �13

�y � 10x � 6y � �11 (1, 2) 3x � 3y � 13 �5�

16

�, �56

��2x � �

53

�y � 35 (5, �15)

19. NUMBER THEORY The units digit of a two-digit number exceeds twice thetens digit by 1. Find the number if the sum of its digits is 10. 37

20. GEOMETRY The difference between the length and width of a rectangle is 7 centimeters. Find the dimensions of the rectangle if its perimeter is50 centimeters. 16 cm by 9 cm

Solve each system of inequalities by graphing. 21–23. See pp. 405A–405D.21. y �4 22. y � 3 23. x � 2y

y � �1 y �x � 2 2x � 3y � 7

24. FINANCE Last year, Jodi invested $10,000, part at 6% annual interest and therest at 8% annual interest. If she received $760 in interest at the end of the year,how much did she invest at each rate? $2000 at 6%, $8000 at 8%

25. STANDARDIZED TEST PRACTICE Which graph represents the system ofinequalities y 2x � 1 and y � �x � 2? D

y

xO

Dy

xO

Cy

xO

By

xO

A

www.algebra1.com/chapter_test

a. consistent

b. elimination

c. inconsistent

Chapter 7 Practice Test 403

Introduction Have you ever found that your preference for completing a taskmay be different from those of others?Ask Students Select one of the systems of equations from this chapter thatcould be solved by various methods. Demonstrate how to solve it using some ofthe methods you learned in this chapter. Write some pros and cons for usingeach method. Which method do you prefer, and why?

Portfolio Suggestion

Assessment Options

Vocabulary Test A vocabularytest/review for Chapter 7 can befound on p. 446 of the Chapter 7Resource Masters.

Chapter Tests There are sixChapter 7 Tests and an Open-Ended Assessment task availablein the Chapter 7 Resource Masters.

Open-Ended AssessmentPerformance tasks for Chapter 7can be found on p. 445 of theChapter 7 Resource Masters. Asample scoring rubric for thesetasks appears on p. A22.

Unit 2 Test A unit test/reviewcan be found on pp. 453–454 ofthe Chapter 7 Resource Masters.

TestCheck andWorksheet Builder

This networkable software hasthree modules for assessment.

• Worksheet Builder to makeworksheets and tests.

• Student Module to take testson-screen.

• Management System to keepstudent records.

Chapter 7 TestsForm Type Level Pages

1 MC basic 433–434

2A MC average 435–436

2B MC average 437–438

2C FR average 439–440

2D FR average 441–442

3 FR advanced 443–444

MC = multiple-choice questionsFR = free-response questions



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

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

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

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

11 (grid in) 11 12 13

12 (grid in)

13 (grid in)

14

15

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

16

17

18

19

Record your answers for Question 20 on the back of this paper.

DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

NAME DATE PERIOD

77

An

swer

s

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

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison

Part 4 Open-EndedPart 4 Open-Ended

Part 1 Multiple ChoicePart 1 Multiple Choice

Standardized Test PracticeStudent Recording Sheet, p. A1

Additional Practice

See pp. 451–452 in the Chapter 7Resource Masters for additionalstandardized test practice.

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

1. What is the value of x in 4x � 2(x � 2) � 8 � 0?(Lesson 3-4) B

�2 2

5 6

2. Noah paid $17.11 for a CD, including tax. If the tax rate is 7%, then what was the price of the CD before tax? (Lesson 3-5) C

$10.06 $11.98

$15.99 $17.04

3. What is the range of f(x) � 2x � 3 when thedomain is {3, 4, 5}? (Lesson 4-3) B

{0, 1, 2} {3, 5, 7}

{6, 8, 10} {9, 11, 13}

4. Jolene kept a log of the numbers of birds thatvisited a birdfeeder over periods of severalhours. On the table below, she recorded thenumber of hours she watched and thecumulative number of birds that she saw eachsession. Which equation best represents thisdata set shown in the table? (Lesson 4-8) D

y � x � 5 y � 3x � 3

y � 3x � 5 y � 4x � 2

5. Which equation describes the graph?(Lesson 5-3) C

3y � 4x � �12

4y � 3x � �16

3y � 4x � �12

3y � 4x � �9

6. Which equation represents a line parallel tothe line given by y � 3x � 6? (Lesson 5-6) B

y � �3x � 4 y � 3x � 2

y � �13

�x � 6 y � ��13

�x � 4

7. Tamika has $185 in her bank account. Sheneeds to deposit enough money so that shecan withdraw $230 for her car payment andstill have at least $200 left in the account.Which inequality describes d, the amount she needs to deposit? (Lesson 6-1) D

d(185 � 230) � 200

185 � 230d � 200

185 � 230 � d � 200

185 � d � 230 � 200

8. The perimeter of a rectangular garden is 68 feet. The length of the garden is 4 morethan twice the width. Which system ofequations will determine the length � and the width w of the garden? (Lesson 7-2) B

2� � 2w � 68 2� � 2w � 68� � 4 � 2w � � 2w � 4

2 � 2w � 68 2� � 2w � 682� � w � 4 w � 2� � 4

9. Ernesto spent a total of $64 for a pair of jeansand a shirt. The jeans cost $6 more than theshirt. What was the cost of the jeans?(Lesson 7-2) C

$26 $29

$35 $58

10. What is the value of y in the following systemof equations? (Lesson 7-3) C3x � 4y � 83x � 2y � �2

�2 4

5 6DC

BA

DC

BA

DC

BA

D

C

B

A

DC

BA

D

C

B

y

xO

A

DC

BA

DC

BA

DC

BA

DC

BA

Part 1 Multiple Choice


1 3 4 6

6 14 18 26

Number of hours, x

Number of birds, y


These two pages contain practicequestions in the various formatsthat can be found on the mostfrequently given standardizedtests.

A practice answer sheet for thesetwo pages can be found on p. A1of the Chapter 7 Resource Masters.

Log On for Test Practice The Princeton Review offersadditional test-taking tips and

practice problems at their web site. Visitwww.princetonreview.com orwww.review.com

TestCheck andWorksheet Builder

Special banks of standardized testquestions similar to those on the SAT,ACT, TIMSS 8, NAEP 8, and Algebra 1End-of-Course tests can be found onthis CD-ROM.

http://www.princetonreview.com

http://www.review.com

Evaluating Open-EndedAssessment Questions

Open-Ended Assessmentquestions are graded by using amultilevel rubric that guides youin assessing a student’s knowl-edge of a particular concept.

Goal: Use a system of equationsto find ticket sales and moneyearned from sales.

Sample Scoring Rubric: The fol-lowing rubric is a sample scoringdevice. You may wish to addmore detail to this sample to meetyour individual scoring needs.

Answers

20a. A � C � 650, 7.5A � 4.5C � 3675

20b. A � C � 650A � C � C � 650 � C

A � 650 � C

7.5A � 4.5C � 36757.5 (650 � C) � 4.5C � 36754875 � 7.5C � 4.5C � 3675

4875 � 3C � 36754875 � 3C � 4875 � 3675 � 4875

�3C � �1200

�

C � 400

A � 650 � CA � 650 � 400 or A � 250

250 adult tickets and 400 child tickets

�1200�

�3�3C��3

Aligned and verified by

Part 2 Short Response/Grid In Part 3 Quantitative Comparison

Chapter 7 Standardized Test Practice 405www.algebra1.com/standardized_test

Record your answers on the answer sheetprovided by your teacher or on a sheet ofpaper.

11. The diagram shows the dimensions of thecargo area of a delivery truck.

What is the maximum volume of cargo, in cubic feet, that can fit in the truck?(Prerequisite Skill) 1644

12. The perimeter of the square below is 204 feet.What is the value of x? (Lesson 3-4) 9

13. What is the x-intercept of the graph of 4x � 3y � 12? (Lesson 4-5) 3

14. What are the slope and the y-intercept of the graph of the equation 4x � 2y � 5?(Lesson 5-4) m � 2, b � ��5

2�

15. Solve the following system of equations.(Lesson 7-2) (3, 5)

5x � y � 107x � 2y � 11

Compare the quantity in Column A and thequantity in Column B. Then determinewhether:

the quantity in Column A is greater,

the quantity in Column B is greater,

the two quantities are equal, or

the relationship cannot be determinedfrom the information given.

16.

C (Lesson 1-1)

17.

A (Lesson 5-1)

x � 3y � 113x � y � 13

18.

B (Lesson 7-4)

3x � 2y � 195x � 4y � 17

19.

A (Lesson 7-4)

Record your answers on a sheet of paper.Show your work.

20. The manager of a movie theater found thatSaturday’s sales were $3675. He knew that a total of 650 tickets were sold Saturday.Adult tickets cost $7.50, and children’s ticketscost $4.50. (Lesson 7-2) a–b. See margin.

a. Write equations to represent the numberof tickets sold and the amount of moneycollected.

b. How many of each kind of ticket weresold? Show your work. Include all steps.

D

C

B

A

5x � 6 feet

3 ft

2 ft

6 ft

8 ft12

22 ft

Test-Taking TipQuestions 11 and 12 To prepare for astandardized test, make flash cards of keymathematical terms, such as “perimeter” and“volume.” Use the glossary of your textbook todetermine the important terms and their correctdefinitions.

Column A Column B

34 92

the slope of the line the slope of the linethat contains A(2, 4) that contains C(�2, 1)

and B(�1, 3) and D (5, 3)

y 0

x y

Part 4 Open Ended

Chapter 7 Standardized Test Practice 405

Score Criteria4 A correct solution that is supported

by well-developed, accurateexplanations

3 A generally correct solution, butmay contain minor flaws inreasoning or computation

2 A partially correct interpretationand/or solution to the problem

1 A correct solution with nosupporting evidence or explanation

0 An incorrect solution indicating nomathematical understanding of the concept or task, or no solution is given


Page 367, Chapter 7 Getting Started1. 2.

3. 4.

5. 6.

Page 371–374, Lesson 7-11. Sample answer: 2. Always; if the system of

linear equations has 2 solu-tions, their graphs are thesame line and there areinfinitely many solutions.

3. Sample answer: Thegraphs of the equations x � y � 3 and 2x � 2y � 6have a slope of �1. Sincethe graphs of the equa-tions coincide, there areinfinitely many solutions.

8. 9.

10. 11.

12. 13.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32. y

xO

2x � 3y � 4

�4x � 6y � �8

y

xO

3x � 2y � 6

3x � 2y � 12

y

xO

x � y � 2

2y � x � 10

(�2, 4)

y

xO

3x � y � 6

(2, 0 )

x � 2y � 2

y

xO

y � 2x � 6

y � �x � 3

(�3, 0)

y

xO

y � �3x � 4

y � 3x � 4

(0, �4)

y

xO

y � �x

y � 2x � 6

(2, �2)

y

xO

2x � y � 10

(4, 2) y � x 1

2

y

xO

x � 2

3x � y � 8

(2, �2)

y

xO

y � �6

4x � y � 2

(2, �6)

y

xO

x � y � 2

y � 4x � 7

(�1, 3)

y

xO

(3, 1)

x � y � 2

3y � 2x � 9

y

xO

x � y � 4

x � y � 1

y

xO

2x � 4y � 2

3x � 6y � 3

y

xO

x � y � 8

x � y � 2 (5, 3)

y

xO

y � 2x

y � �x

(0, 0)

y

xO

(�2, 3)

y

xO

y � x � 2 12

y

xO

y � 5 � 2x

y

xO

y � 2x � 3

y

xO

y � 4 � x

y

xO

y � �2x

y

xO

y � 1

405A Chapter 7 Additional Answers

Addit

ion

al

An

sw

ers

for

Ch

apte

r 7

33. 34.

35. 36.

37. 38.

39. 40.

53.

Pages 379–381, Lesson 7-2

43. 44.

45. 46.

47. 48.


41. Elimination can be used to solve problems aboutmeteorology if the coefficients of one variable are thesame or are additive inverses. Answers should includethe following.

• The two equations in the system of equations areadded or subtracted so that one of the variables iseliminated. You then solve for the remaining variable.This number is substituted into one of the originalequations, and that equation is solved for the othervariable.

• n � d � 24 Write the equations in (�) n � d � 12 column form and add.

2n � 36 Notice the d variable iseliminated.


n � 18 Simplify. n � d � 24 First equation

18 � d � 24 n � 1818 � d � 18 � 24 � 18 Subtract 18 from each side.

d � 6 Simplify.On the winter solstice, Seward, Alaska, has 18 hoursof nighttime and 6 hours of daylight.

47. 48. y

xO

3y � 7 � 2x

2x � 3y � 7

y

xO

3x � y � 1

x � y � 3 (1, �2)

36�2

2n�2

y

xO

2x � y � 6

y

xO

x � 4

y

xO

y � �5

y

xO

2x � y � 3

4x � 2y � 6

y

xO (3, �1)

2x � y � 5x � 2y � 1

y

xO

x � y � 4

x � y � 3

p

tO 2 4 6 8 10 12

20

40

60

80

Pop

uat

ion

(mill

ion

s o

f p

eop

le)

Years Since 1990

y

xO2468

1012

�2

(8, 6)

y � x � 2 12

x � y � 6 13

12

�2�4

2 4 6 8 10 12 14

y

xO2 4 6 8 10 12 14

468

10121416

�2

2

6 � y � x 38

x � y � 4 14

23

y

xO

3y � 2x

y � x � 523

yx

O

(�1, �5)

2x � 3y � �17

y � x � 4

y

xO

(�1, 2)

3y � x � 5

y � x � 3

y

xO

2y � �6x � 6

3x � y � 3

y

xO

(3, 4)

5x � 8y � �17

4x � 3y � 24

y

xO2x � y � �4

5x � 3y � �6

�16 �12 �8 �4

4

8

12

16

(�6, 8)

Chapter 7 Additional Answers 405B

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49.

Pages 390–392, Lesson 7-43. Sample answer: (1) You could solve the first equation for

a and substitute the resulting expression for a in thesecond equation. Then find the value of b. Use thisvalue for b and one of the equations to find the value ofa. (2) You could multiply the first equation by 3 and addthis new equation to the second equation. This will elim-inate the b-term. Find the value of a. Use this value fora and one of the equations to find the value of b. Seestudent’s work for their preference and explanation.

44. By having two equations that represent the timerestraints, a manager can determine the best use ofemployee time. Answers should include the following.

• 20c � 10b � 800 → 20c � 10b � 80010c � 30b � 900 → �20c � 60b � �1800

�50b � �1000

�

b � 2020c � 10b � 800

20c � 10(20) � 80020c � 200 � 800

20c � 200 � 200 � 800 � 20020c � 600

�

c � 30

• In order to make the most of the employee and oventime, the manager should make assignments to bake30 batches of cookies and 20 loaves of bread.

Page 395, Graphing Calculator Investigation

2. 4.

[�10, 10] scl: 1 by [�10, 10] scl: 1 by[�10, 10] scl: 1 [�10, 10] scl: 1


4. 5.

6. 7.

8. 9.

10.

12. 13.

14. 15. y

xO

x � 2

y � x � 5

y

xO

y � x � 1

y � �2

y

xO

y � �1

x � �4

y

xO

x � 0

y � 0

O

x � y � 30

Min

ute

s Jo

gg

ing

20

10

30

40

Minutes Walking2010 30 40

Natasha’s Daily Exercise

x � y � 3860

460

y

x

y

xO

3x � 4y � 12

x � 0

x � 2y � 2

y

xO

3x � y � 4

2y � x � 6

y

xOy � �2x � 1

2x � y � 4�

y

xO

y � �x � 3

y � x � 3

y

xO

y � 3

y � �x � 4

y

xO

x � 5y � 4

600�20

20c�20

�1000�

�50�50b��50

y

xO

12

8

4

4 8

4x � y � 12

x � 3 � y14

405C Chapter 7 Additional Answers

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Ch

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16. 17.

18. 19.

20. 21.

22. 23.

24. 25.

26.

29.

31.

35. By graphing a system of equations, you can see theappropriate range of Calories and fat intake. Answersshould include the following.

• Two sample appropriate Calorie and fat intakes are2200 Calories and 60 g of fat and 2300 Calories and65 g of fat.

• The graph represents 2000 � c � 2400 and 60 � f � 75.

Page 403, Practice Test

21. 22.

23.

xO

y

2x � 3y � 7

x � 2y

O x

yy � 3

y � �x � 2xO

y

y � �1

x � �4

y

xO

20

40

60

60 804020 120100

HD

L

LDL

AppropriateCholesterol Levels

y

xO 10

10

20

30

40 503020

Dar

k G

reen

Light Green

Green Paint

4x � y � 32

x � 6y � 54

xO

y

2x � y � �94y � x

x � 2

x � 3y � 9

y

xO

2x � 7y � 4

y � x � 3 3x � 2y � 6

y

xO2x � y � �4

�5x � 2y � 1

y

xO

x � y � 4

�2x � 3y � �12

y

xO

3x � 4y � 1

x � 2y � 7

y

xO

3x � y � 6

2x � y � 4

y

xO

y � x � 3y � x � 2

y

xO

y � x � 3

y � x � 1

�y

xO

y � �x � 3

y � 2x � 1

y

xO

y � 2x � 1

y � �x � 1

y

xO

x � 3x � y � 2

Chapter 7 Additional Answers 405D

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Documents

Chapter 7: Solving Systems of Linear Equations and ...manualmath.info/downloads/chap07_bk.pdf · 366C Chapter 7 Solving Systems of Linear Equations and ... Chapter 7 Solving Systems