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Chapter 7Resource Masters
CONSUMABLE WORKBOOKS Many of the worksheets contained in the ChapterResource Masters booklets are available as consumable workbooks in bothEnglish and Spanish.
Study Guide and Intervention Workbook 0-07-827753-1Study Guide and Intervention Workbook (Spanish) 0-07-827754-XSkills Practice Workbook 0-07-827747-7Skills Practice Workbook (Spanish) 0-07-827749-3Practice Workbook 0-07-827748-5Practice Workbook (Spanish) 0-07-827750-7Reading to Learn Mathematics Workbook 0-07-861060-5
ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
StudentWorks™ This CD-ROM includes the entire Student Edition text alongwith the English workbooks listed above.
TeacherWorks™ All of the materials found in this booklet are included for view-ing and printing in the Glencoe Algebra 1 TeacherWorks CD-ROM.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Algebra 1. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-827731-0 Glencoe Algebra 1Chapter 7 Resource Masters
3 4 5 6 7 8 9 10 024 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 1
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 7-1Study Guide and Intervention . . . . . . . . 403–404Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 405Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 406Reading to Learn Mathematics . . . . . . . . . . 407Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 408
Lesson 7-2Study Guide and Intervention . . . . . . . . 409–410Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 411Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 412Reading to Learn Mathematics . . . . . . . . . . 413Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 414
Lesson 7-3Study Guide and Intervention . . . . . . . . 415–416Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 417Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 418Reading to Learn Mathematics . . . . . . . . . . 419Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 420
Lesson 7-4Study Guide and Intervention . . . . . . . . 421–422Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 423Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 424Reading to Learn Mathematics . . . . . . . . . . 425Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 426
Lesson 7-5Study Guide and Intervention . . . . . . . . 427–428Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 429Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 430Reading to Learn Mathematics . . . . . . . . . . 431Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 432
Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 433–434Chapter 7 Test, Form 2A . . . . . . . . . . . 435–436Chapter 7 Test, Form 2B . . . . . . . . . . . 437–438Chapter 7 Test, Form 2C . . . . . . . . . . . 439–440Chapter 7 Test, Form 2D . . . . . . . . . . . 441–442Chapter 7 Test, Form 3 . . . . . . . . . . . . 443–444Chapter 7 Open-Ended Assessment . . . . . . 445Chapter 7 Vocabulary Test/Review . . . . . . . 446Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 447Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 448Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 449Chapter 7 Cumulative Review . . . . . . . . . . . 450Chapter 7 Standardized Test Practice . . 451–452Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 453–454
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A27
© Glencoe/McGraw-Hill iv Glencoe Algebra 1
Teacher’s Guide to Using theChapter 7 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 1 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirAlgebra Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 1 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 1
Assessment OptionsThe assessment masters in the Chapter 7Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 1. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 404–405. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
77
© Glencoe/McGraw-Hill vii Glencoe Algebra 1
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
consistent
kuhn·SIHS·tuhnt
dependent
elimination
ih·LIH·muh·NAY·shuhn
independent
inconsistent
(continued on the next page)
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
⎧ ⎪ ⎨ ⎪ ⎩
© Glencoe/McGraw-Hill viii Glencoe Algebra 1
Vocabulary Term Found on Page Definition/Description/Example
substitution
SUHB·stuh·TOO·shuhn
system of equations
system of inequalities
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
77
Study Guide and InterventionGraphing Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 403 Glencoe Algebra 1
Less
on 7
-1
Number of Solutions Two or more linear equations involving the same variables forma system of equations. A solution of the system of equations is an ordered pair of numbersthat satisfies both equations. The table below summarizes information about systems oflinear equations.
Graph of a System intersecting lines same line parallel lines
Number of Solutions exactly one solution infinitely many solutions no solution
Terminologyconsistent and consistent and
inconsistent independent dependent
Use the graph at the right to determine whether the system has no solution, one solution, orinfinitely many solutions.
a. y ! "x # 2y ! x # 1Since the graphs of y ! "x # 2 and y ! x # 1 intersect,there is one solution.
b. y ! "x # 23x # 3y ! "3Since the graphs of y ! "x # 2 and 3x # 3y ! "3 are parallel, there are no solutions.
c. 3x # 3y ! "3y ! "x " 1Since the graphs of 3x # 3y ! "3 and y ! "x " 1 coincide,there are infinitely many solutions.
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
1. y ! "x " 3 2. 2x # 2y ! "6y ! x " 1 y ! "x " 3
3. y ! "x " 3 4. 2x # 2y ! "62x # 2y ! 4 3x # y ! 3
x
y
O
y ! "x " 3
3x # y ! 3
2x # 2y ! "6
2x # 2y ! 4
y ! x " 1
x
y
O
y ! x # 1
y ! "x " 1
3x # 3y ! "3
y ! "x # 2
x
y
Ox
y
Ox
y
O
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 404 Glencoe Algebra 1
Solve by Graphing One method of solving a system of equations is to graph theequations on the same coordinate plane.
Graph each system of equations. Then determine whether thesystem has no solution, one solution, or infinitely many solutions. If the system hasone solution, name it.
a. x # y ! 2x " y ! 4The graphs intersect. Therefore, there is one solution. The point (3, "1) seems to lie on both lines. Check this estimate by replacing x with 3 and y with "1 in each equation.
x # y ! 23 # ("1) ! 2 ✓
x " y ! 43 " ("1) ! 3 # 1 or 4 ✓The solution is (3, "1).
b. y ! 2x # 12y ! 4x # 2The graphs coincide. Therefore there are infinitely many solutions.
Graph each system of equations. Then determine whether the system has nosolution, one solution, or infinitely many solutions. If the system has one solution,name it.
1. y ! "2 2. x ! 2 3. y ! x3x " y ! "1 2x # y ! 1 x # y ! 3
4. 2x # y ! 6 5. 3x # 2y ! 6 6. 2y ! "4x # 42x " y ! "2 3x # 2y ! "4 y ! "2x # 2
y ! "2x # 22y ! "4x # 4
x
y
O
3x # 2y ! "4
3x # 2y ! 6
x
y
O
2x # y ! 6
2x " y ! "2
(1, 4)
x
y
O
x # y ! 3
y ! 12x(2, 1)
x
y
O2x # y ! 1
x ! 2
(2, –3)
x
y
O
3x " y ! "1
y ! "2
(–1, –2)x
y
O
1$2
x
y
O
y ! 2x # 1 2y ! 4x # 2
x
y
O (3, –1)
x " y ! 4
x # y ! 2
Study Guide and Intervention (continued)
Graphing Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
ExampleExample
ExercisesExercises
Skills PracticeGraphing Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 405 Glencoe Algebra 1
Less
on 7
-1
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
1. y ! x " 1 2. x " y ! "4y ! "x # 1 y ! x # 4
3. y ! x # 4 4. y ! 2x " 32x " 2y ! 2 2x " 2y ! 2
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. 2x " y ! 1 6. x ! 1 7. 3x # y ! "3y ! "3 2x # y ! 4 3x # y ! 3
8. y ! x # 2 9. x # 3y ! "3 10. y " x ! "1x " y ! "2 x " 3y ! "3 x # y ! 3
11. x " y ! 3 12. x # 2y ! 4 13. y ! 2x # 3x " 2y ! 3 y ! " x # 2 3y ! 6x " 6
y ! 2x # 3
3y ! 6x " 6
x
y
O
y ! "1–2x # 2
x # 2y ! 4
x
y
O
x " 2y ! 3
(3, 0)
x " y ! 3
x
y
O
1$2
x # y ! 3
(2, 1)
y " x ! "1 x
y
O
x " 3y ! "3
(–3, 0)
x # 3y ! "3
x
y
O
y ! x # 2
x " y ! "2
x
y
O
3x # y ! 3
3x # y ! "3
x
y
O
(1, 2)
x ! 1
2x # y ! 4x
y
O
(–1, –3)
2x " y ! 1
y ! "3x
y
O
x
y
Oy ! "x # 1
x " y ! "4 2x " 2y ! 2
y ! 2x " 3
y ! x " 1
y ! x # 4
© Glencoe/McGraw-Hill 406 Glencoe Algebra 1
Use the graph at the right to determine whether each system has no solution, one solution, or infinitely many solutions.
1. x # y ! 3 2. 2x " y ! "3x # y ! "3 4x " 2y ! "6
3. x # 3y ! 3 4. x # 3y ! 3x # y ! "3 2x " y ! "3
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 6. y ! 2x " 3 7. x # 2y ! 33x " y ! 0 4x ! 2y # 6 3x " y ! "5
BUSINESS For Exercises 8 and 9, use the following information.Nick plans to start a home-based business producing and selling gourmet dog treats. He figures it will cost $20 inoperating costs per week plus $0.50 to produce each treat.He plans to sell each treat for $1.50.
8. Graph the system of equations y ! 0.5x # 20 and y ! 1.5x to represent the situation.
9. How many treats does Nick need to sell per week tobreak even?
SALES For Exercises 10–12, use the following information.A used book store also started selling used CDs and videos.In the first week, the store sold 40 used CDs and videos, at$4.00 per CD and $6.00 per video. The sales for both CDs and videos totaled $180.00
10. Write a system of equations to represent the situation.
11. Graph the system of equations.
12. How many CDs and videos did the store sell in the firstweek?
4c # 6v ! 180
c # v ! 40
(30, 10)
y ! 1.5x
y ! 0.5x # 20(20, 30)
Sales ($)
Dog Treats
Cost
($)
5 1510 20 25 30 35 40 450
40
35
30
25
20
15
10
5
3x " y ! "5
x # 2y ! 3(–1, 2)
4x ! 2y # 6
y ! 2x " 3x
y
O3x " y ! 0
3x " y ! "2
x
y
O
x
y
O
x # y ! "3
x # 3y ! 3 2x " y ! "3
4x " 2y ! "6
x # y ! 3
Practice Graphing Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Reading to Learn MathematicsGraphing Systems of Equations
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
© Glencoe/McGraw-Hill 407 Glencoe Algebra 1
Less
on 7
-1
Pre-Activity How can you use graphs to compare the sales of two products?
Read the introduction to Lesson 7-1 at the top of page 369 in your textbook.
• What is meant by the term linear function?
• What does it mean to say that two lines intersect?
Reading the Lesson
1. Each figure shows the graph of a system of two equations. Write the letter of the figuresthat illustrate each statement.
A. B.
C. D.
a. A system of two linear equations can have an infinite number of solutions.
b. A system of equations is consistent if there is at least one ordered pair that satisfiesboth equations.
c. If two graphs are parallel, there are no ordered pairs that satisfy both equations.
d. If a system of equations has exactly one solution, it is independent.
e. If a system of equations has an infinite number of solutions, it is dependent.
Helping You Remember
2. Describe how you can solve a system of equations by graphing.
x
y
Ox
y
O
x
y
Ox
y
O
© Glencoe/McGraw-Hill 408 Glencoe Algebra 1
Graphing a TripThe distance formula, d ! rt, is used to solve many types of problems. If you graph an equation such as d ! 50t, the graph is a model for a car going at 50 mi/h. The time the car travels is t;the distance in miles the car covers is d. The slope of the line is the speed.
Suppose you drive to a nearby town and return. You average 50 mi/h on the trip out but only 25 mi/h on the trip home. The round trip takes 5 hours. How far away is the town?
The graph at the right represents your trip. Notice that the return trip is shown with a negative slope because you are driving in the opposite direction.
Solve each problem.
1. Estimate the answer to the problem in the above example. About how far away is the town?
2. Graph this trip and solve the problem. An airplane has enough fuel for 3 hours of safe flying. On the trip out the pilotaverages 200 mi/h flying against a headwind. On the trip back,the pilot averages 250 mi/h. How long a trip out can the pilot make?
3. Graph this trip and solve the 4. Graph this trip and solve the problem. Youproblem. You drive to a town drive at an average speed of 50 mi/h to a100 miles away. On the trip out you discount shopping plaza, spend 2 hours average 25 mi/h. On the trip back you shopping, and then return at an average average 50 mi/h. How many hours do speed of 25 mi/h. The entire trip takes you spend driving? 8 hours. How far away is the shopping plaza?
t
d
O
50
2t
d
O 2
50
t
d
O 1
100
t
d
O
slope is 50
slope is –25
2
50
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-17-1
Study Guide and InterventionSubstitution
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 409 Glencoe Algebra 1
Less
on 7
-2
Substitution One method of solving systems of equations is substitution.
Use substitution tosolve the system of equations.y ! 2x4x " y ! "4
Substitute 2x for y in the secondequation.
4x " y ! "4 Second equation4x " 2x ! "4 y ! 2x
2x ! "4 Combine like terms.
! Divide each side by 2.
x ! "2 Simplify.
Use y ! 2x to find the value of y.y ! 2x First equationy ! 2("2) x ! "2y ! "4 Simplify.
The solution is ("2, "4).
"4$2
2x$2
Solve for one variable, thensubstitute.x # 3y ! 72x " 4y ! "6
Solve the first equation for x since the coefficientof x is 1.
x # 3y ! 7 First equationx # 3y " 3y ! 7 " 3y Subtract 3y from each side.
x ! 7 " 3y Simplify.
Find the value of y by substituting 7 " 3y for xin the second equation.
2x " 4y ! "6 Second equation2(7 " 3y) " 4y ! "6 x ! 7 " 3y
14 " 6y " 4y ! "6 Distributive Property14 " 10y ! "6 Combine like terms.
14 " 10y " 14 ! "6 " 14 Subtract 14 from each side."10y ! "20 Simplify.
! Divide each side by "10.
y ! 2 Simplify.
Use y ! 2 to find the value of x.x ! 7 " 3yx ! 7 " 3(2)x ! 1The solution is (1, 2).
"20$"10
"10y$"10
ExampleExample ExampleExample
ExercisesExercises
Use substitution to solve each system of equations. If the system does not haveexactly one solution, state whether it has no solution or infinitely many solutions.
1. y ! 4x 2. x ! 2y 3. x ! 2y " 33x " y ! 1 y ! x " 2 x ! 2y # 4
4. x " 2y ! "1 5. c " 4d ! 1 6. x # 2y ! 03y ! x # 4 2c " 8d ! 2 3x # 4y ! 4
7. 2b ! 6a " 14 8. x # y ! 16 9. y ! "x # 33a " b ! 7 2y ! "2x # 2 2y # 2x ! 4
10. x ! 2y 11. x " 2y ! "5 12. "0.2x # y ! 0.50.25x # 0.5y ! 10 x # 2y ! "1 0.4x # y ! 1.1
© Glencoe/McGraw-Hill 410 Glencoe Algebra 1
Real-World Problems Substitution can also be used to solve real-world problemsinvolving systems of equations. It may be helpful to use tables, charts, diagrams, or graphsto help you organize data.
CHEMISTRY How much of a 10% saline solution should be mixedwith a 20% saline solution to obtain 1000 milliliters of a 12% saline solution?
Let s ! the number of milliliters of 10% saline solution.Let t ! the number of milliliters of 20% saline solution.Use a table to organize the information.
10% saline 20% saline 12% saline
Total milliliters s t 1000
Milliliters of saline 0.10s 0.20t 0.12(1000)
Write a system of equations.s # t ! 10000.10s # 0.20t ! 0.12(1000)Use substitution to solve this system.
s # t ! 1000 First equations ! 1000 " t Solve for s.
0.10s # 0.20t ! 0.12(1000) Second equation0.10(1000 " t) # 0.20t ! 0.12(1000) s ! 1000 " t
100 " 0.10t # 0.20t ! 0.12(1000) Distributive Property100 # 0.10t ! 0.12(1000) Combine like terms.
0.10t ! 20 Simplify.
! Divide each side by 0.10.
t ! 200 Simplify.s # t ! 1000 First equation
s # 200 ! 1000 t ! 200s ! 800 Solve for s.
800 milliliters of 10% solution and 200 milliliters of 20% solution should be used.
1. SPORTS At the end of the 2000-2001 football season, 31 Super Bowl games had beenplayed with the current two football leagues, the American Football Conference (AFC) andthe National Football Conference (NFC). The NFC won five more games than the AFC.How many games did each conference win? Source: New York Times Almanac
2. CHEMISTRY A lab needs to make 100 gallons of an 18% acid solution by mixing a 12%acid solution with a 20% solution. How many gallons of each solution are needed?
3. GEOMETRY The perimeter of a triangle is 24 inches. The longest side is 4 inches longerthan the shortest side, and the shortest side is three-fourths the length of the middleside. Find the length of each side of the triangle.
20$0.10
0.10t$0.10
Study Guide and Intervention (continued)
Substitution
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
ExampleExample
ExercisesExercises
Skills PracticeSubstitution
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 411 Glencoe Algebra 1
Less
on 7
-2
Use substitution to solve each system of equations. If the system does not haveexactly one solution, state whether it has no solution or infinitely many solutions.
1. y ! 4x 2. y ! 2xx # y ! 5 x # 3y ! "14
3. y ! 3x 4. x ! "4y2x # y ! 15 3x # 2y ! 20
5. y ! x " 1 6. x ! y " 7x # y ! 3 x # 8y ! 2
7. y ! 4x " 1 8. y ! 3x # 8y ! 2x " 5 5x # 2y ! 5
9. 2x " 3y ! 21 10. y ! 5x " 8y ! 3 " x 4x # 3y ! 33
11. x # 2y ! 13 12. x # 5y ! 43x " 5y ! 6 3x # 15y ! "1
13. 3x " y ! 4 14. x # 4y ! 82x " 3y ! "9 2x " 5y ! 29
15. x " 5y ! 10 16. 5x " 2y ! 142x " 10y ! 20 2x " y ! 5
17. 2x # 5y ! 38 18. x " 4y ! 27x " 3y ! "3 3x # y ! "23
19. 2x # 2y ! 7 20. 2.5x # y ! "2x " 2y ! "1 3x # 2y ! 0
© Glencoe/McGraw-Hill 412 Glencoe Algebra 1
Use substitution to solve each system of equations. If the system does not haveexactly one solution, state whether it has no solution or infinitely many solutions.
1. y ! 6x 2. x ! 3y 3. x ! 2y # 72x # 3y ! "20 3x " 5y ! 12 x ! y # 4
4. y ! 2x " 2 5. y ! 2x # 6 6. 3x # y ! 12y ! x # 2 2x " y ! 2 y ! "x " 2
7. x # 2y ! 13 8. x " 2y ! 3 9. x " 5y ! 36"2x " 3y ! "18 4x " 8y ! 12 2x # y ! "16
10. 2x " 3y ! "24 11. x # 14y ! 84 12. 0.3x " 0.2y ! 0.5x # 6y ! 18 2x " 7y ! "7 x " 2y ! "5
13. 0.5x # 4y ! "1 14. 3x " 2y ! 11 15. x # 2y ! 121$2
Practice Substitution
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
x # 2.5y ! 3.5 x " y ! 4 x " 2y ! 6
16. x " y ! 3 17. 4x " 5y ! "7 18. x " 3y ! "4
2x # y ! 25 y ! 5x 2x # 6y ! 5
EMPLOYMENT For Exercises 19–21, use the following information.Kenisha sells athletic shoes part-time at a department store. She can earn either $500 permonth plus a 4% commission on her total sales, or $400 per month plus a 5% commission ontotal sales.
19. Write a system of equations to represent the situation.
20. What is the total price of the athletic shoes Kenisha needs to sell to earn the sameincome from each pay scale?
21. Which is the better offer?
MOVIE TICKETS For Exercises 22 and 23, use the following information.Tickets to a movie cost $7.25 for adults and $5.50 for students. A group of friends purchased8 tickets for $52.75.
22. Write a system of equations to represent the situation.
23. How many adult tickets and student tickets were purchased?
1$3
1$2
Reading to Learn MathematicsSubstitution
NAME ______________________________________________ DATE ____________ PERIOD _____
7-27-2
© Glencoe/McGraw-Hill 413 Glencoe Algebra 1
Less
on 7
-2
Pre-Activity How can a system of equations be used to predict media use?
Read the introduction to Lesson 7-2 at the top of page 376 in your textbook.
• What is the system of equations?
• Based on the graph, are there 0, 1, or infinitely many solutions of thesystem?
Reading the Lesson
1. Describe how you would use substitution to solve each system of equations.
a. y ! "2xx # 3y ! 15
b. 3x " 2y ! 12x ! 2y
c. x # 2y ! 72x " 8y ! 8
d. "3x # 5y ! 812x # y ! 24
2. Jess solved a system of equations and her result was "8 ! "8. All of her work wascorrect. Describe the graph of the system. Explain.
3. Miguel solved a system of equations and his result was 5 ! "2. All of his work wascorrect. Describe the graph of the system. Explain.
Helping You Remember
4. What is usually the first step in solving a system of equations by substitution?
© Glencoe/McGraw-Hill 414 Glencoe Algebra 1
Equations of Lines and Planes in Intercept FormOne form that a linear equation may take is interceptform. The constants a and b are the x- and y-intercepts of the graph.
! " 1
In three-dimensional space, the equation of a plane takes a similar form.
! ! " 1
Here, the constants a, b, and c are the points where theplane meets the x, y, and z-axes.
Solve each problem.
z#c
y#b
x#a
y#b
x#a
z
y
x
O
x–8 ! y–7 ! z–6 " 1
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
7-27-2
1. Graph the equation ! ! " 1.
5. Graph the equation ! ! " 1.
2. For the plane in Exercise 1, write anequation for the line where the planeintersects the xy-plane. Use interceptforms.
! " 1
3. Write an equation for the line wherethe plane intersects the xz-plane.
! " 1
4. Write an equation for the line wherethe plane intersects the yz-plane.
! " 1
6. Write an equation for the xy-plane.
z " 0
7. Write an equation for the yz-plane.
x " 0
8. Write an equation for a plane parallelto the xy-plane with a z-intercept of 2.
z " 2
9. Write an equation for a plane parallelto the yz-plane with an x-intercept of 23.
x " #3
z$1
y$2
z$1
x$3
y$2
x$3
z
y
x
O
z#2
y#4
x#1
z
y
x
O
z#1
y#2
x#3
Study Guide and InterventionElimination Using Addition and Subtraction
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 415 Glencoe Algebra 1
Less
on 7
-3
Elimination Using Addition In systems of equations in which the coefficients of thex or y terms are additive inverses, solve the system by adding the equations. Because one ofthe variables is eliminated, this method is called elimination.
Use addition to solve thesystem of equations.x " 3y ! 73x # 3y ! 9
Write the equations in column form and addto eliminate y.
x " 3y ! 7(#) 3x # 3y ! 9
4x ! 16Solve for x.
!
x ! 4Substitute 4 for x in either equation andsolve for y.
4 " 3y ! 74 " 3y " 4 ! 7 " 4
"3y ! 3
!
y ! "1The solution is (4, "1).
3$"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
ExampleExample ExampleExample
ExercisesExercises
Use elimination to solve each system of equations.
1. x # y ! "4 2. 2m " 3n ! 14 3. 3a " b ! "9x " y ! 2 m # 3n ! "11 "3a " 2b ! 0
4. "3x " 4y ! "1 5. 3c # d ! 4 6. "2x # 2y ! 93x " y ! "4 2c " d ! 6 2x " y ! "6
7. 2x # 2y ! "2 8. 4x " 2y ! "1 9. x " y ! 23x " 2y ! 12 "4x # 4y ! "2 x # y ! "3
10. 2x " 3y ! 12 11. "0.2x # y ! 0.5 12. 0.1x # 0.3y ! 0.94x # 3y ! 24 0.2x # 2y ! 1.6 0.1x " 0.3y ! 0.2
13. Rema is older than Ken. The difference of their ages is 12 and the sum of their ages is50. Find the age of each.
14. The sum of the digits of a two-digit number is 12. The difference of the digits is 2. Findthe number if the units digit is larger than the tens digit.
© Glencoe/McGraw-Hill 416 Glencoe Algebra 1
Elimination Using Subtraction In systems of equations where the coefficients of thex or y terms are the same, solve the system by subtracting the equations.
Use subtraction to solve the system of equations.2x " 3y ! 115x " 3y ! 14
2x " 3y ! 11 Write the equations in column form and subtract.(") 5x " 3y ! 14
"3x ! "3 Subtract the two equations. y is eliminated.
! Divide each side by "3.
x ! 1 Simplify.
2(1) " 3y ! 11 Substitute 1 for x in either equation.2 " 3y ! 11 Simplify.
2 " 3y " 2 ! 11 " 2 Subtract 2 from each side."3y ! 9 Simplify.
! Divide each side by "3.
y ! "3 Simplify.
The solution is (1, "3).
Use elimination to solve each system of equations.
1. 6x # 5y ! 4 2. 3m " 4n ! "14 3. 3a # b ! 16x " 7y ! "20 3m # 2n ! "2 a # b ! 3
4. "3x " 4y ! "23 5. c " 3d ! 11 6. x " 2y ! 6"3x # y ! 2 2c " 3d ! 16 x # y ! 3
7. 2a " 3b ! "13 8. 4x # 2y ! 6 9. 5s " t ! 62a # 2b ! 7 4x # 4y ! 10 5s # 2t ! 3
10. 6x " 3y ! 12 11. x # 2y ! 3.5 12. 0.2x # y ! 0.74x " 3y ! 24 x " 3y ! "9 0.2x # 2y ! 1.2
13. The sum of two numbers is 70. One number is ten more than twice the other number.Find the numbers.
14. GEOMETRY Two angles are supplementary. The measure of one angle is 10° more thanthree times the other. Find the measure of each angle.
9$"3
"3y$"3
"3$"3
"3x$"3
Study Guide and Intervention (continued)
Elimination Using Addition and Subtraction
NAME ______________________________________________ DATE ____________ PERIOD _____
Skills PracticeElimination Using Addition and Subtraction
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 417 Glencoe Algebra 1
Less
on 7
-3
Use elimination to solve each system of equations.
1. x ! y " 1 2. !x # y " 1x # y " 3 x # y " 11
3. x # 4y " 11 4. !x # 3y " 6x ! 6y " 11 x # 3y " 18
5. 3x # 4y " 19 6. x # 4y " !83x # 6y " 33 x ! 4y " !8
7. 3a # 4b " 2 8. 3c ! d " !14a ! 4b " 12 !3c ! d " 5
9. 2x ! 3y " 9 10. x ! y " 4!5x ! 3y " 30 2x # y " !4
11. 3m ! n " 26 12. 5x ! y " !6!2m ! n " !24 !x # y " 2
13. 6x ! 2y " 32 14. 3x # 2y " !194x ! 2y " 18 !3x ! 5y " 25
15. 7m # 4n " 2 16. 2x ! 5y " !287m # 2n " 8 4x # 5y " 4
17. The sum of two numbers is 28 and their difference is 4. What are the numbers?
18. Find the two numbers whose sum is 29 and whose difference is 15.
19. The sum of two numbers is 24 and their difference is 2. What are the numbers?
20. Find the two numbers whose sum is 54 and whose difference is 4.
21. Two times a number added to another number is 25. Three times the first number minusthe other number is 20. Find the numbers.
© Glencoe/McGraw-Hill 418 Glencoe Algebra 1
Use elimination to solve each system of equations.
1. x ! y " 1 2. p # q " !2 3. 4x # y " 23x # y " !9 p ! q " 8 3x ! y " 12
4. 2x # 5y " !3 5. 3x # 2y " !1 6. 5x # 3y " 222x # 2y " 6 4x # 2y " !6 5x ! 2y " 2
7. 5x # 2y " 7 8. 3x ! 9y " !12 9. !4c ! 2d " !2!2x # 2y " !14 3x ! 15y " !6 2c ! 2d " !14
10. 2x ! 6y " 6 11. 7x # 2y " 2 12. 4.25x ! 1.28y " !9.22x # 3y " 24 7x ! 2y " !30 x # 1.28y " 17.6
13. 2x # 4y " 10 14. 2.5x # y " 10.7 15. 6m ! 8n " 3x ! 4y " !2.5 2.5x # 2y " 12.9 2m ! 8n " !3
16. 4a # b " 2 17. ! x ! y " !2 18. x ! y " 84a # 3b " 10
x ! y " 4 x # y " 19
19. The sum of two numbers is 41 and their difference is 5. What are the numbers?
20. Four times one number added to another number is 36. Three times the first numberminus the other number is 20. Find the numbers.
21. One number added to three times another number is 24. Five times the first numberadded to three times the other number is 36. Find the numbers.
22. LANGUAGES English is spoken as the first or primary language in 78 more countriesthan Farsi is spoken as the first language. Together, English and Farsi are spoken as afirst language in 130 countries. In how many countries is English spoken as the firstlanguage? In how many countries is Farsi spoken as the first language?
23. DISCOUNTS At a sale on winter clothing, Cody bought two pairs of gloves and four hatsfor $43.00. Tori bought two pairs of gloves and two hats for $30.00. What were the pricesfor the gloves and hats?
1$2
3$2
2$3
1$3
1$2
3$4
4$3
1$3
Practice Elimination Using Addition and Subtraction
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Reading to Learn MathematicsElimination Using Addition and Subtraction
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
© Glencoe/McGraw-Hill 419 Glencoe Algebra 1
Less
on 7
-3
Pre-Activity How can you use a system of equations to solve problems aboutweather?
Read the introduction to Lesson 7-3 at the top of page 382 in your textbook.
What fact explains why the variable d gets eliminated from the system ofequations?
Reading the Lesson
1. Write addition or subtraction to tell which operation it would be easiest to use toeliminate a variable of the system. Explain your choice.
System of Equations Operation Explanation
a. 3x # 5y " 12!3x # 2y " 6
b. 3x # 5y " 73x ! 2y " 8
c. !x ! 4y " 94x ! 4y " 6
d. 5x ! 7y " 178x # 7y " 9
Helping You Remember
2. Tell how you can decide whether to use addition or subtraction to eliminate a variable ina system of equations.
© Glencoe/McGraw-Hill 420 Glencoe Algebra 1
Rózsa PéterRózsa Péter (1905–1977) was a Hungarian mathematician dedicated toteaching others about mathematics. As professor of mathematics at ateachers’ college in Budapest, she wrote several mathematics textbooksand championed reforms in the teaching of mathematics. In 1945 shewrote Playing with Infinity: Mathematical Explorations and Excursions,a popular work in which she attempted to convey the spirit ofmathematics to the general public.
By far Péter’s greatest contribution to mathematics was her pioneeringresearch in the field of recursive function theory. When you evaluate afunction recursively, you begin with one initial value of x. Working fromthis single number, you can use the function to generate an entiresequence of numbers. For instance, here is how you use an initialvalue of x " 1 to evaluate the function f(x) " 3x recursively.
f(1) " 3(1) " 3 ↓
f(3) " 3(3) " 9 ↓
f(9) " 3(9) " 27 ↓
f(27) " 3(27) " 81 ↓
f(81) " 3(81) " 243 ↓
The first five numbers of the sequence generated by this function are 3, 9, 27, 81, and 243.
Write the first five numbers of the sequence generated by each function, using the given number as the initial value of x.
1. f(x) " 3x; x " 2 2. g(x) " x ! 5; x " 1
3. f(x) " 2x # 1; x " !3 4. f(x) " x2; x " 2
5. h(x) " !x; x " 3 6. k(x) " ; x " 10 1$x
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-37-3
Study Guide and InterventionElimination Using Multiplication
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 421 Glencoe Algebra 1
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
ExampleExample ExampleExample
ExercisesExercises
Use elimination to solve each system of equations.
1. 2x # 3y " 6 2. 2m # 3n " 4 3. 3a ! b " 2x # 2y " 5 !m # 2n " 5 a # 2b " 3
4. 4x # 5y " 6 5. 4c ! 3d " 22 6. 3x ! 4y " !46x ! 7y " !20 2c ! d " 10 x # 3y " !10
7. 4s ! t " 9 8. 4a ! 3b " !8 9. 2x # 2y " 55s # 2t " 8 2a # 2b " 3 4x ! 4y " 10
10. 6x ! 4y " !8 11. 4x # 2y " !5 12. 2x # y " 3.54x # 2y " !3 !2x ! 4y " 1 !x # 2y " 2.5
13. GARDENING The length of Sally’s garden is 4 meters greater than 3 times the width.The perimeter of her garden is 72 meters. What are the dimensions of Sally’s garden?
14. Anita is 4 years older than Basilio. Three times Anita’s age added to six times Basilio’s
age is 36. How old are Anita and Basilio?
1$2
© Glencoe/McGraw-Hill 422 Glencoe Algebra 1
Determine the Best Method The methods to use for solving systems of linearequations are summarized in the table below.
Method The Best Time to Use
Graphing to estimate the solution, since graphing usually does not give an exact solution
Substitution if one of the variables in either equation has a coefficient of 1 or !1
Elimination Using Addition if one of the variables has opposite coefficients in the two equations
Elimination Using Subtraction if one of the variables has the same coefficient in the two equations
Elimination Using Multiplicationif none of the coefficients are 1 or !1 and neither of the variables can beeliminated by simply adding or subtracting the equations
Determine the best method to solve the system of equations. Thensolve the system.6x ! 2y " 20#2x ! 4y " #16
Since the coefficients of x will be additive inverses of each other if you multiply the secondequation by 3, use elimination.
Study Guide and Intervention (continued)
Elimination Using Multiplication
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
ExampleExample
ExercisesExercises
6x # 2y " 20(#) !6x # 12y " !48 Multiply the second equation by 3.
14y " !28 Add the two equations. x is eliminated.
" Divide each side by 14.
y " !2 Simplify.
!28$14
14y$14
6x # 2(!2) " 20 Substitute !2 for y ineither equation.
6x ! 4 " 20 Simplify.6x ! 4 # 4 " 20 # 4 Add 4 to each side.
6x " 24 Simplify.
" Divide each side by 6.
x " 4 Simplify.
24$6
6x$6
The solution is (4, !2).
Determine the best method to solve each system of equations. Then solve the system.
1. x # 2y " 3 2. m # 6n " !8 3. a ! b " 6x # y " 1 m " 2n # 8 a " 2b # 7
4. 4x # y " 15 5. 3c ! d " 14 6. x # 2y " !9!x ! 3y " !12 c ! d " 2 y " 4x
7. 4x " 2y ! 10 8. x " !2y 9. 2s ! 3t " 42x # 2y " 5 4x # 4y " !10 3s # 2t " 24
10. 4a ! 4b " !10 11. 4x # 10y " !6 12. 2x " y ! 32a # 4b " !2 !2x ! 10y " 2 !x # y " 0
Skills PracticeElimination Using Multiplication
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 423 Glencoe Algebra 1
Less
on 7
-4
Use elimination to solve each system of equations.
1. x # y " !9 2. 3x # 2y " !95x ! 2y " 32 x ! y " !13
3. 2x # 5y " 3 4. 2x # y " 3!x # 3y " !7 !4x ! 4y " !8
5. 4x ! 2y " !14 6. 2x # y " 03x ! y " !8 5x # 3y " 2
7. 5x # 3y " !10 8. 2x # 3y " 143x # 5y " !6 3x ! 4y " 4
9. 2x ! 3y " 21 10. 3x # 2y " !265x ! 2y " 25 4x ! 5y " !4
11. 3x ! 6y " !3 12. 5x # 2y " !32x # 4y " 30 3x # 3y " 9
13. Two times a number plus three times another number equals 13. The sum of the twonumbers is 7. What are the numbers?
14. Four times a number minus twice another number is !16. The sum of the two numbersis !1. Find the numbers.
Determine the best method to solve each system of equations. Then solve the system.
15. 2x # 3y " 10 16. 8x ! 7y " 185x # 2y " !8 3x # 7y " 26
17. y " 2x 18. 3x # y " 63x # 2y " 35 3x # y " 3
19. 3x ! 4y " 17 20. y " 3x # 14x # 5y " 2 3x ! y " !1
© Glencoe/McGraw-Hill 424 Glencoe Algebra 1
Use elimination to solve each system of equations.
1. 2x ! y " !1 2. 5x ! 2y " !10 3. 7x # 4y " !43x ! 2y " 1 3x # 6y " 66 5x # 8y " 28
4. 2x ! 4y " !22 5. 3x # 2y " !9 6. 4x ! 2y " 323x # 3y " 30 5x ! 3y " 4 !3x ! 5y " !11
7. 3x # 4y " 27 8. 0.5x # 0.5y " !2 9. 2x ! y " !75x ! 3y " 16 x ! 0.25y " 6
x # y " 0
10. Eight times a number plus five times another number is !13. The sum of the twonumbers is 1. What are the numbers?
11. Two times a number plus three times another number equals 4. Three times the firstnumber plus four times the other number is 7. Find the numbers.
Determine the best method to solve each system of equations. Then solve thesystem.
12. 5x # 7y " 3 13. 7x # 2y " 2 14. !6x ! 2y " 142x ! 7y " !38 2x ! 3y " !28 6x # 8y " !20
15. x " 2y # 6 16. 4x # 3y " !2 17. y " x1$2
1$2
3$4
Practice Elimination Using Multiplication
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
x ! y " 3 4x # 3y " 3x ! 2y " 9
18. FINANCE Gunther invested $10,000 in two mutual funds. One of the funds rose 6% inone year, and the other rose 9% in one year. If Gunther’s investment rose a total of $684in one year, how much did he invest in each mutual fund?
19. CANOEING Laura and Brent paddled a canoe 6 miles upstream in four hours. Thereturn trip took three hours. Find the rate at which Laura and Brent paddled the canoein still water.
20. NUMBER THEORY The sum of the digits of a two-digit number is 11. If the digits arereversed, the new number is 45 more than the original number. Find the number.
5$2
1$2
Reading to Learn MathematicsElimination Using Multiplication
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
© Glencoe/McGraw-Hill 425 Glencoe Algebra 1
Less
on 7
-4
Pre-Activity How can a manager use a system of equations to plan employee time?
Read the introduction to Lesson 7-4 at the top of page 387 in your textbook.
Can the system of equations be solved by elimination with addition orsubtraction? Explain.
Reading the Lesson
1. Could elimination by multiplication be used to solve the system shown below? Explain.3x ! 5y " 15!6x # 7y " 11
2. Tell whether it would be easiest to use substitution, elimination by addition, eliminationby subtraction, or elimination by multiplication to solve the system. Explain your choice.
System of Equations Solution Method Explanation
a. !3x # 4y " 23x # 2y " 10
b. x ! 2y " 05x ! 4y " 8
c. 6x ! 5y " !182x # 10y " 27
d. !2x # 3y " 93x # 3y " 12
Helping You Remember
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?
© Glencoe/McGraw-Hill 426 Glencoe Algebra 1
George Washington Carver and Percy Julian In 1990, George Washington Carver and Percy Julian became the first AfricanAmericans elected to the National Inventors Hall of Fame. Carver (1864–1943)was an agricultural scientist known worldwide for developing hundreds of usesfor the peanut and the sweet potato. His work revitalized the economy of thesouthern United States because it was no longer dependent solely upon cotton.Julian (1898–1975) was a research chemist who became famous for inventinga method of making a synthetic cortisone from soybeans. His discovery hashad many medical applications, particularly in the treatment of arthritis.
There are dozens of other African American inventors whose accomplishmentsare not as well known. Their inventions range from common household itemslike the ironing board to complex devices that have revolutionizedmanufacturing. The exercises that follow will help you identify just a few ofthese inventors and their inventions.
Match the inventors with their inventions by matching each systemwith its solution. (Not all the solutions will be used.)
1. Sara Boone x # y " 2 A. (1, 4) automatic traffic signal x ! y " 10
2. Sarah Goode x " 2 ! y B. (4, !2) eggbeater 2y # x " 9
3. Frederick M. y " 2x # 6 C. (!2, 3) fire extinguisher Jones y " !x ! 3
4. J. L. Love 2x # 3y " 8 D. (!5, 7) folding cabinet bed 2x ! y " !8
5. T. J. Marshall y ! 3x " 9 E. (6, !4) ironing board 2y # x " 4
6. Jan Matzeliger y # 4 " 2x F. (!2, 4) pencil sharpener 6x ! 3y " 12
7. Garrett A. 3x ! 2y " !5 G. (!3, 0) portable X-ray machine Morgan 3y ! 4x " 8
8. Norbert Rillieux 3x ! y " 12 H. (2, !3) player piano y ! 3x " 15
I. no solution evaporating pan for refining sugar
J. infinitely lasting (shaping) many machine for solutions manufacturing shoes
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-47-4
Study Guide and InterventionGraphing Systems of Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 427 Glencoe Algebra 1
Less
on 7
-5
Systems of Inequalities The solution of a system of inequalities is the set of allordered pairs that satisfy both inequalities. If you graph the inequalities in the samecoordinate plane, the solution is the region where the graphs overlap.
Solve the system of inequalities by graphing.y $ x ! 2y % #2x # 1
The solution includes the ordered pairs in the intersection of thegraphs. This region is shaded at the right. The graphs of y " x # 2and y " !2x ! 1 are boundaries of this region. The graph of y " x # 2 is dashed and is not included in the graph of y % x # 2.
Solve the system of inequalities by graphing.x ! y $ 4x ! y & #1
The graphs of x # y " 4 and x # y " !1 are parallel. Because the two regions have no points in common, the system of inequalities has no solution.
Solve each system of inequalities by graphing.
1. y % !1 2. y % !2x # 2 3. y & x # 1x & 0 y ' x # 1 3x # 4y ( 12
4. 2x # y ( 1 5. y ' 2x # 3 6. 5x ! 2y & 6x ! y ( !2 y ( !1 # 2x y % !x # 1
5x # 2y " 6
y " #x ! 1
x
y
O
y " 2x ! 3
y " #1 ! 2xx
y
O
2x ! y " 1
x # y " #2
x
y
O
y " x ! 1
3x ! 4y " 12
x
y
O
y " x ! 1
y " #2x ! 2
x
y
O
x " 0
y " #1
x
y
O
x
y
O
x ! y " #1
x ! y " 4
x
y
O
y " x ! 2
y " #2x # 1
ExampleExample
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 428 Glencoe Algebra 1
Real-World Problems In real-world problems, sometimes only whole numbers makesense for the solution, and often only positive values of x and y make sense.
BUSINESS AAA Gem Company produces necklaces and bracelets. In a 40-hour week, the company has 400 gems to use. A necklace requires 40 gems and a bracelet requires 10 gems. It takes 2 hours to produce anecklace and a bracelet requires one hour. How many of each type can be produced in a week?
Let n " the number of necklaces that will be produced and b " thenumber of bracelets that will be produced. Neither n or b can be anegative number, so the following system of inequalities represents the conditions of the problems.
n ( 0b ( 0b # 2n ' 4010b # 40n ' 400
The solution is the set ordered pairs in the intersection of the graphs. This region is shadedat the right. Only whole-number solutions, such as (5, 20) make sense in this problem.
For each exercise, graph the solution set. List three possible solutions to theproblem.
1. HEALTH Mr. Flowers is on a restricted 2. RECREATION Maria had $150 in gift diet that allows him to have between certificates to use at a record store. She 1600 and 2000 Calories per day. His bought fewer than 20 recordings. Each daily fat intake is restricted to between tape cost $5.95 and each CD cost $8.95.45 and 55 grams. What daily Calorie How many of each type of recording might and fat intakes are acceptable? she have bought?
t ! c " 20
5.95t ! 8.95c " 150
30
25
20
15
10
5
Tapes
Com
pact
Dis
cs
5 10 15 20 25 300
60
50
40
30
20
10
Calories
Fat
Gra
ms
1000 2000 30000
Necklaces
Brac
elet
s
10 20 30 40 500
50
40
30
20
10
10b ! 40n " 400
b ! 2n " 40
Study Guide and Intervention (continued)
Graphing Systems of Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
ExampleExample
ExercisesExercises
Skills PracticeGraphing Systems of Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 429 Glencoe Algebra 1
Less
on 7
-5
Solve each system of inequalities by graphing.
1. x % !1 2. y % 2 3. y % x # 3y ' !3 x & !2 y ' !1
4. x & 2 5. x # y ' !1 6. y ! x % 4y ! x ' 2 x # y ( 3 x # y % 2
7. y % x # 1 8. y ( !x # 2 9. y & 2x # 4y ( !x # 1 y & 2x ! 2 y ( x # 1
Write a system of inequalities for each graph.
10. 11. 12.
x
y
Ox
y
Ox
y
O
y " x ! 1
y " 2x ! 4
x
y
O
y " 2x # 2
y " #x ! 2
x
y
O
y " #x ! 1y " x ! 1
x
y
O
x ! y " 2
y # x " 4x
y
Ox
y
O
y # x " 2x " 2
x
y
O
y " x ! 3
y " #1
x
y
O
y " 2
x " #2x
y
O
x " #1
y " #3x
y
O
© Glencoe/McGraw-Hill 430 Glencoe Algebra 1
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.
SOUVENIRS For Exercises 9 and 10, use the following information.Emily wants to buy turquoise stones on her trip to New Mexicoto give to at least 4 of her friends. The gift shop sells stones foreither $4 or $6 per stone. Emily has no more than $30 to spend.
9. Make a graph showing the numbers of each price of stoneEmily can purchase.
10. List three possible solutions.
Gym (hours)
Diego’s Routine
Wak
ing
(mile
s)
1 32 4 5 6 7 80
16
14
12
10
8
6
4
2
2x # y " 2
x # 2y " 2
y " x # 4
2x ! y " 2y " 2x # 1
y " 2 # x
x ! y " 1
x ! 2y " 1 x
y
O
y " 2x ! 3
y " x ! 2
x
y
O
y " x # 2y " x
x
y
O
Practice Graphing Systems of Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
Reading to Learn MathematicsGraphing Systems of Inequalities
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill 431 Glencoe Algebra 1
Less
on 7
-5
Pre-Activity How can you use a system of inequalities to plan a sensible diet?
Read the introduction to Lesson 7-5 at the top of page 394 in your textbook.
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
Helping You Remember
5. Describe how you would explain the process of using a graph to solve a system ofinequalities to a friend who missed Lesson 7-5.
x
y
O
y " x ! 2
y " #2x # 1
x
y
O
y " x ! 2
y " #2x # 1
x
y
O
y " x ! 2
y " #2x # 1
x
y
O
y " x ! 2y " #2x # 1
© Glencoe/McGraw-Hill 432 Glencoe Algebra 1
Describing RegionsThe shaded region inside the triangle can be described with a system of three inequalities.
y & 2x # 1
y % x ! 3
y % 29x ! 31
Write systems of inequalities to describe each region. You may first need to divide a region into triangles or quadrilaterals.
1. 2.
3.
x
y
O
x
y
Ox
y
O
1$3 x
y
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
7-57-5
© Glencoe/McGraw-Hill A2 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Gra
phin
g S
yste
ms
of E
quat
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill40
3G
lenc
oe A
lgeb
ra 1
Lesson 7-1
Num
ber
of S
olut
ions
Tw
o or
mor
e lin
ear
equa
tion
s in
volv
ing
the
sam
e va
riab
les
form
a sy
stem
of
equ
atio
ns.
A s
olut
ion
of t
he s
yste
m o
f equ
atio
ns is
an
orde
red
pair
of n
umbe
rsth
at s
atis
fies
bot
h eq
uati
ons.
The
tab
le b
elow
sum
mar
izes
info
rmat
ion
abou
t sy
stem
s of
linea
r eq
uati
ons.
Gra
ph o
f a
Sys
tem
inte
rsec
ting
lines
sam
e lin
epa
ralle
l lin
es
Num
ber
of S
olut
ions
exac
tly o
ne s
olut
ion
infin
itely
man
y so
lutio
nsno
sol
utio
n
Term
inol
ogy
cons
iste
nt a
ndco
nsis
tent
and
inco
nsis
tent
in
depe
nden
tde
pend
ent
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er t
he
syst
em h
as n
oso
luti
on,o
ne
solu
tion
,or
infi
nit
ely
ma
ny
solu
tion
s.
a.y
!"
x#
2y
!x
#1
Sinc
e th
e gr
aphs
of y
!"
x#
2 an
d y
!x
#1
inte
rsec
t,th
ere
is o
ne s
olut
ion.
b.y
!"
x#
23x
#3y
!"
3Si
nce
the
grap
hs o
f y!
"x
#2
and
3x#
3y!
"3
are
para
llel,
ther
e ar
e no
sol
utio
ns.
c.3x
#3y
!"
3y
!"
x"
1Si
nce
the
grap
hs o
f 3x
#3y
!"
3 an
d y
!"
x"
1 co
inci
de,
ther
e ar
e in
fini
tely
man
y so
luti
ons.
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er e
ach
sy
stem
has
no
solu
tion
,on
eso
luti
on,o
r in
fin
itel
y m
an
yso
luti
ons.
1.y
!"
x"
32.
2x#
2y!
"6
y!
x"
1y
!"
x"
3on
ein
finite
ly m
any
3.y
!"
x"
34.
2x#
2y!
"6
2x#
2y!
43x
#y
!3
none
one
x
y
O
y !
"x
" 33x
# y
! 3
2x #
2y
! "
6
2x #
2y
! 4
y !
x "
1
x
y O
y !
x #
1
y !
"x
" 1
3x #
3y
! "
3
y !
"x
# 2
x
y
Ox
y
Ox
y
O
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill40
4G
lenc
oe A
lgeb
ra 1
Solv
e by
Gra
phin
gO
ne m
etho
d of
sol
ving
a s
yste
m o
f eq
uati
ons
is t
o gr
aph
the
equa
tion
s on
the
sam
e co
ordi
nate
pla
ne.
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s.T
hen
det
erm
ine
wh
eth
er t
he
syst
em h
as n
oso
luti
on,o
ne
solu
tion
,or
infi
nit
ely
ma
ny
solu
tion
s.If
th
e sy
stem
has
one
solu
tion
,nam
e it
.
a.x
#y
!2
x"
y!
4T
he g
raph
s in
ters
ect.
The
refo
re,t
here
is o
ne s
olut
ion.
The
po
int
(3,"
1) s
eem
s to
lie
on b
oth
lines
.Che
ck t
his
esti
mat
e by
rep
laci
ng x
wit
h 3
and
yw
ith
"1
in e
ach
equa
tion
.x
#y
!2
3 #
("1)
!2
✓x
"y
!4
3 "
("1)
!3
#1
or 4
✓T
he s
olut
ion
is (
3,"
1).
b.y
!2x
#1
2y!
4x#
2T
he g
raph
s co
inci
de.T
here
fore
the
re a
re in
fini
tely
m
any
solu
tion
s.
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s.T
hen
det
erm
ine
wh
eth
er t
he
syst
em h
as n
oso
luti
on,o
ne
solu
tion
,or
infi
nit
ely
ma
ny
solu
tion
s.If
th
e sy
stem
has
on
e so
luti
on,
nam
e it
.
1.y
!"
2on
e;("
1,"
2)2.
x!
2on
e;(2
,"3)
3.y
!x
one;
(2,1
)3x
"y
!"
12x
#y
!1
x#
y!
3
4.2x
#y
!6
one;
(1,4
)5.
3x#
2y!
6no
sol
utio
n6.
2y!
"4x
#4
infin
itely
2x
"y
!"
23x
#2y
!"
4y
!"
2x#
2m
any
y !
"2x
# 2
2y !
"4x
# 4 x
y
O
3x #
2y
! "
4
3x #
2y
! 6 x
y
O
2x #
y !
6
2x "
y !
"2
( 1, 4
)
x
y
O
x #
y !
3
y !
1 2x( 2
, 1)
x
y
O2x
# y
! 1
x !
2
( 2, –
3)
x
y
O
3x "
y !
"1
y !
"2
( –1,
–2)
x
y
O
1 $ 2
x
y O
y !
2x
# 1
2y !
4x
# 2x
y
O( 3
, –1) x "
y !
4
x #
y !
2
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Gra
phin
g S
yste
ms
of E
quat
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A3 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Gra
phin
g S
yste
ms
of E
quat
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill40
5G
lenc
oe A
lgeb
ra 1
Lesson 7-1
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er
each
sys
tem
has
no
solu
tion
,on
eso
luti
on,o
r in
fin
itel
y m
an
yso
luti
ons.
1.y
!x
"1
2.x
"y
!"
4in
finite
lyy
!"
x#
1on
ey
!x
#4
man
y
3.y
!x
#4
4.y
!2x
"3
2x"
2y!
22x
"2y
!2
no s
olut
ion
one
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s.T
hen
det
erm
ine
wh
eth
er t
he
syst
em h
as n
oso
luti
on,o
ne
solu
tion
,or
infi
nit
ely
ma
ny
solu
tion
s.If
th
e sy
stem
has
on
e so
luti
on,
nam
e it
.
5.2x
"y
!1
6.x
!1
7.3x
#y
!"
3no
y
!"
3on
e;("
1,"
3)2x
#y
!4
one;
(1,2
)3x
#y
!3
solu
tion
8.y
!x
#2
infin
itely
9.
x#
3y!
"3
one;
10.y
"x
!"
1x
"y
!"
2m
any
x"
3y!
"3
("3,
0)x
#y
!3
one;
(2,1
)
11.x
"y
!3
12.x
#2y
!4
infin
itely
13
.y!
2x#
3no
sol
utio
nx
"2y
!3
one;
(3,0
)y
!"
x#
2m
any
3y!
6x"
6
y !
2x
# 3
3y !
6x
" 6x
y
O
y !
"1 – 2x
# 2
x #
2y
! 4 x
y
O
x "
2y
! 3
( 3, 0
)
x "
y !
3
x
y
O
1 $ 2
x #
y !
3
( 2, 1
)
y "
x !
"1
x
y
O
x "
3y
! "
3
( –3,
0)
x #
3y
! "
3x
y
O
y !
x #
2
x "
y !
"2
x
y
O
3x #
y !
3
3x #
y !
"3
x
y
O
( 1, 2
)
x !
1 2x #
y !
4x
y
O
( –1,
–3)
2 x "
y !
1
y !
"3
x
y
O
x
y
Oy
! "
x #
1
x "
y !
"4
2x "
2y
! 2
y !
2x
" 3
y !
x "
1
y !
x #
4
©G
lenc
oe/M
cGra
w-H
ill40
6G
lenc
oe A
lgeb
ra 1
Use
th
e gr
aph
at
the
righ
t to
det
erm
ine
wh
eth
er
each
sys
tem
has
no
solu
tion
,on
eso
luti
on,o
r in
fin
itel
y m
an
yso
luti
ons.
1.x
#y
!3
2.2x
"y
!"
3x
#y
!"
34x
"2y
!"
6no
sol
utio
nin
finite
ly m
any
3.x
#3y
!3
4.x
#3y
!3
x#
y!
"3
one
2x"
y!
"3
one
Gra
ph
eac
h s
yste
m o
f eq
uat
ion
s.T
hen
det
erm
ine
wh
eth
er t
he
syst
em h
as n
oso
luti
on,o
ne
solu
tion
,or
infi
nit
ely
ma
ny
solu
tion
s.If
th
e sy
stem
has
on
e so
luti
on,
nam
e it
.
5.3x
"y
!"
2no
6.
y!
2x"
3in
finite
ly
7.x
#2y
!3
one;
3x"
y!
0so
lutio
n4x
!2y
#6
man
y3x
"y
!"
5("
1,2)
BU
SIN
ESS
For
Exe
rcis
es 8
an
d 9
,use
th
e fo
llow
ing
info
rmat
ion
.N
ick
plan
s to
sta
rt a
hom
e-ba
sed
busi
ness
pro
duci
ng a
nd
selli
ng g
ourm
et d
og t
reat
s.H
e fi
gure
s it
will
cos
t $2
0 in
oper
atin
g co
sts
per
wee
k pl
us $
0.50
to
prod
uce
each
tre
at.
He
plan
s to
sel
l eac
h tr
eat
for
$1.5
0.
8.G
raph
the
sys
tem
of
equa
tion
s y
!0.
5x#
20 a
nd
y!
1.5x
to r
epre
sent
the
sit
uati
on.
9.H
ow m
any
trea
ts d
oes
Nic
k ne
ed t
o se
ll pe
r w
eek
tobr
eak
even
?20
SALE
SF
or E
xerc
ises
10–
12,u
se t
he
foll
owin
g in
form
atio
n.
A u
sed
book
sto
re a
lso
star
ted
selli
ng u
sed
CD
s an
d vi
deos
.In
the
fir
st w
eek,
the
stor
e so
ld 4
0 us
ed C
Ds
and
vide
os,a
t$4
.00
per
CD
and
$6.
00 p
er v
ideo
.The
sal
es f
or b
oth
CD
s an
d vi
deos
tot
aled
$18
0.00
10.W
rite
a s
yste
m o
f eq
uati
ons
to
repr
esen
t th
e si
tuat
ion.
11.G
raph
the
sys
tem
of
equa
tion
s.
12.H
ow m
any
CD
s an
d vi
deos
did
the
sto
re s
ell i
n th
e fir
stw
eek?
30 C
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and
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ideo
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v !
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c #
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40
( 30,
10)
CD S
ales
($)
CD
an
d V
ideo
Sal
es
Video Sales ($)
515
1020
2530
3540
450
40 35 30 25 20 15 10 5
y !
1.5
x
y !
0.5
x #
20
( 20,
30) Sa
les
($)
Do
g T
reat
s
Cost ($)
515
1020
2530
3540
450
40 35 30 25 20 15 10 5
3x "
y !
"5
x #
2y
! 3 x
y O
( –1,
2)
4x !
2y
# 6
y !
2x
" 3
x
y
O3x
" y
! 0
3x "
y !
"2
x
y
O
x
y
O
x #
y !
"3
x #
3y
! 3
2x "
y !
"3
4x "
2y
! "
6
x #
y !
3
Pra
ctic
e (A
vera
ge)
Gra
phin
g S
yste
ms
of E
quat
ions
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A4 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csG
raph
ing
Sys
tem
s of
Equ
atio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
©G
lenc
oe/M
cGra
w-H
ill40
7G
lenc
oe A
lgeb
ra 1
Lesson 7-1
Pre-
Act
ivit
yH
ow c
an y
ou u
se g
rap
hs
to c
omp
are
the
sale
s of
tw
o p
rod
uct
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
1 at
the
top
of
page
369
in y
our
text
book
.
•W
hat
is m
eant
by
the
term
line
ar f
unct
ion?
Sam
ple
answ
er:
a fu
nctio
n w
hose
gra
ph is
a li
ne
•W
hat
does
it m
ean
to s
ay t
hat
two
lines
inte
rsec
t?S
ampl
e an
swer
:Th
e lin
es c
ross
.
Read
ing
the
Less
on
1.E
ach
figu
re s
how
s th
e gr
aph
of a
sys
tem
of
two
equa
tion
s.W
rite
the
lett
er o
f the
figu
res
that
illu
stra
te e
ach
stat
emen
t.
A.
B.
C.
D.
a.A
sys
tem
of
two
linea
r eq
uati
ons
can
have
an
infi
nite
num
ber
of s
olut
ions
.D
b.A
sys
tem
of
equa
tion
s is
con
sist
ent
if t
here
is a
t le
ast
one
orde
red
pair
tha
t sa
tisf
ies
both
equ
atio
ns.
B,C
,D
c.If
tw
o gr
aphs
are
par
alle
l,th
ere
are
no o
rder
ed p
airs
tha
t sa
tisf
y bo
th e
quat
ions
.A
d.If
a s
yste
m o
f eq
uati
ons
has
exac
tly
one
solu
tion
,it
is in
depe
nden
t.B
,C
e.If
a s
yste
m o
f eq
uati
ons
has
an in
fini
te n
umbe
r of
sol
utio
ns,i
t is
dep
ende
nt.
D
Hel
ping
You
Rem
embe
r
2.D
escr
ibe
how
you
can
sol
ve a
sys
tem
of
equa
tion
s by
gra
phin
g.S
ampl
e an
swer
:G
raph
the
equ
atio
ns o
n th
e sa
me
coor
dina
te p
lane
.Loc
ate
any
poin
ts o
fin
ters
ectio
n.
x
y
Ox
y
O
x
y
Ox
y O
©G
lenc
oe/M
cGra
w-H
ill40
8G
lenc
oe A
lgeb
ra 1
Gra
phin
g a
Trip
The
dis
tanc
e fo
rmul
a,d
!rt
,is
used
to
solv
e m
any
type
s of
pr
oble
ms.
If y
ou g
raph
an
equa
tion
suc
h as
d!
50t,
the
grap
h is
a
mod
el f
or a
car
goi
ng a
t 50
mi/h
.The
tim
e th
e ca
r tr
avel
s is
t;
the
dist
ance
in m
iles
the
car
cove
rs is
d.T
he s
lope
of t
he li
ne is
th
e sp
eed.
Supp
ose
you
driv
e to
a n
earb
y to
wn
and
retu
rn.Y
ou a
vera
ge
50 m
i/h o
n th
e tr
ip o
ut b
ut o
nly
25 m
i/h o
n th
e tr
ip h
ome.
The
ro
und
trip
tak
es 5
hou
rs.H
ow f
ar a
way
is t
he t
own?
The
gra
ph a
t th
e ri
ght
repr
esen
ts y
our
trip
.Not
ice
that
the
ret
urn
trip
is s
how
n w
ith
a ne
gati
ve s
lope
bec
ause
you
are
dri
ving
in t
he
oppo
site
dir
ecti
on.
Sol
ve e
ach
pro
blem
.
1.E
stim
ate
the
answ
er t
o th
e pr
oble
m in
the
abo
ve e
xam
ple.
Abo
ut
how
far
aw
ay is
the
tow
n?
abou
t 80
mile
s
2.G
raph
thi
s tr
ip a
nd s
olve
the
pro
blem
.An
airp
lane
has
en
ough
fue
l for
3 h
ours
of
safe
fly
ing.
On
the
trip
out
the
pilo
tav
erag
es 2
00 m
i/h f
lyin
g ag
ains
t a
head
win
d.O
n th
e tr
ip b
ack,
the
pilo
t av
erag
es 2
50 m
i/h.H
ow lo
ng a
tri
p ou
t ca
n th
e pi
lot
mak
e?
abou
t 1
hour
s an
d 33
0 m
iles
3.G
raph
thi
s tr
ip a
nd s
olve
the
4.
Gra
ph t
his
trip
and
sol
ve t
he p
robl
em.Y
oupr
oble
m.Y
ou d
rive
to
a to
wn
driv
e at
an
aver
age
spee
d of
50
mi/h
to
a10
0 m
iles
away
.On
the
trip
out
you
di
scou
nt s
hopp
ing
plaz
a,sp
end
2 ho
urs
av
erag
e 25
mi/h
.On
the
trip
bac
k yo
u
shop
ping
,and
the
n re
turn
at
an a
vera
ge
aver
age
50 m
i/h.H
ow m
any
hour
s do
sp
eed
of 2
5 m
i/h.T
he e
ntir
e tr
ip t
akes
yo
u sp
end
driv
ing?
8 ho
urs.
How
far
aw
ay is
the
sho
ppin
g pl
aza?
6 ho
urs
100
mile
s
t
d
O50
2t
d
O2
50
2 $ 3t
d
O1
100
t
d O
slop
e is
50
slop
e is
–25
2
50
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-1
7-1
Answers (Lesson 7-1)
© Glencoe/McGraw-Hill A5 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Sub
stitu
tion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill40
9G
lenc
oe A
lgeb
ra 1
Lesson 7-2
Subs
titu
tion
One
met
hod
of s
olvi
ng s
yste
ms
of e
quat
ions
is s
ubs
titu
tion
.
Use
su
bsti
tuti
on t
oso
lve
the
syst
em o
f eq
uat
ion
s.y
!2x
4x"
y!
"4
Subs
titu
te 2
xfo
r y
in t
he s
econ
deq
uati
on.
4x"
y!
"4
Sec
ond
equa
tion
4x"
2x!
"4
y!
2x2x
!"
4C
ombi
ne li
ke te
rms.
!D
ivid
e ea
ch s
ide
by 2
.
x!
"2
Sim
plify
.
Use
y!
2xto
fin
d th
e va
lue
of y
.y
!2x
Firs
t equ
atio
ny
!2(
"2)
x!
"2
y!
"4
Sim
plify
.
The
sol
utio
n is
("2,
"4)
.
"4
$2
2x $ 2
Sol
ve f
or o
ne
vari
able
,th
ensu
bsti
tute
.x
#3y
!7
2x"
4y!
"6
Solv
e th
e fi
rst
equa
tion
for
xsi
nce
the
coef
fici
ent
of x
is 1
.x
#3y
!7
Firs
t equ
atio
nx
#3y
"3y
!7
"3y
Sub
trac
t 3y
from
eac
h si
de.
x!
7 "
3yS
impl
ify.
Fin
d th
e va
lue
of y
by s
ubst
itut
ing
7 "
3yfo
r x
in t
he s
econ
d eq
uati
on.
2x"
4y!
"6
Sec
ond
equa
tion
2(7
"3y
) "4y
!"
6x
!7
"3y
14 "
6y"
4y!
"6
Dis
trib
utiv
e P
rope
rty
14 "
10y
!"
6C
ombi
ne li
ke te
rms.
14 "
10y
"14
!"
6 "
14S
ubtr
act 1
4 fr
om e
ach
side
."
10y
!"
20S
impl
ify.
!D
ivid
e ea
ch s
ide
by "
10.
y!
2S
impl
ify.
Use
y!
2 to
fin
d th
e va
lue
of x
.x
!7
"3y
x!
7 "
3(2)
x!
1T
he s
olut
ion
is (
1,2)
.
"20
$ "10
"10
y$ "
10
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Use
su
bsti
tuti
on t
o so
lve
each
sys
tem
of
equ
atio
ns.
If t
he
syst
em d
oes
not
hav
eex
actl
y on
e so
luti
on,s
tate
wh
eth
er i
t h
as n
oso
luti
on o
r in
fin
itel
y m
an
yso
luti
ons.
1.y
!4x
2.x
!2y
3.x
!2y
"3
3x"
y!
1("
1,"
4)y
!x
"2
(4,2
)x
!2y
#4
no s
olut
ion
4.x
"2y
!"
15.
c"
4d!
1in
finite
ly
6.x
#2y
!0
3y!
x#
4(5
,3)
2c"
8d!
2m
any
3x#
4y!
4(4
,"2)
7.2b
!6a
"14
infin
itely
8.
x#
y!
169.
y!
"x
#3
no3a
"b
!7
man
y2y
!"
2x#
2no
sol
utio
n2y
#2x
!4
solu
tion
10.x
!2y
(20,
10)
11.x
"2y
!"
512
."0.
2x#
y!
0.5
0.25
x#
0.5y
!10
x#
2y!
"1
("3,
1)0.
4x#
y!
1.1
(1,0
.7)
©G
lenc
oe/M
cGra
w-H
ill41
0G
lenc
oe A
lgeb
ra 1
Real
-Wor
ld P
robl
ems
Subs
titu
tion
can
als
o be
use
d to
sol
ve r
eal-
wor
ld p
robl
ems
invo
lvin
g sy
stem
s of
equ
atio
ns.I
t m
ay b
e he
lpfu
l to
use
tabl
es,c
hart
s,di
agra
ms,
or g
raph
sto
hel
p yo
u or
gani
ze d
ata.
CH
EMIS
TRY
How
mu
ch o
f a
10%
sal
ine
solu
tion
sh
ould
be
mix
edw
ith
a 2
0% s
alin
e so
luti
on t
o ob
tain
100
0 m
illi
lite
rs o
f a
12%
sal
ine
solu
tion
?
Let
s!
the
num
ber
of m
illili
ters
of
10%
sal
ine
solu
tion
.L
et t
!th
e nu
mbe
r of
mill
ilite
rs o
f 20
% s
alin
e so
luti
on.
Use
a t
able
to
orga
nize
the
info
rmat
ion.
10%
sal
ine
20%
sal
ine
12%
sal
ine
Tota
l mill
ilite
rss
t10
00
Mill
ilite
rs o
f sa
line
0.10
s0.
20t
0.12
(100
0)
Wri
te a
sys
tem
of
equa
tion
s.s
#t
!10
000.
10s
#0.
20t
!0.
12(1
000)
Use
sub
stit
utio
n to
sol
ve t
his
syst
em.
s#
t!
1000
Firs
t equ
atio
ns
!10
00 "
tS
olve
for
s.0.
10s
#0.
20t
!0.
12(1
000)
Sec
ond
equa
tion
0.10
(100
0 "
t) #
0.20
t!
0.12
(100
0)s
!10
00 "
t10
0 "
0.10
t#
0.20
t!
0.12
(100
0)D
istr
ibut
ive
Pro
pert
y10
0 #
0.10
t!
0.12
(100
0)C
ombi
ne li
ke te
rms.
0.10
t!
20S
impl
ify.
!D
ivid
e ea
ch s
ide
by 0
.10.
t!
200
Sim
plify
.s
#t
!10
00F
irst e
quat
ion
s#
200
!10
00t!
200
s!
800
Sol
ve fo
r s.
800
mill
ilite
rs o
f 10
% s
olut
ion
and
200
mill
ilite
rs o
f 20
% s
olut
ion
shou
ld b
e us
ed.
1.SP
ORT
SA
t th
e en
d of
the
200
0-20
01 fo
otba
ll se
ason
,31
Supe
r B
owl g
ames
had
bee
npl
ayed
wit
h th
e cu
rren
t tw
o fo
otba
ll le
ague
s,th
e A
mer
ican
Foo
tbal
l Con
fere
nce
(AF
C) a
ndth
e N
atio
nal F
ootb
all C
onfe
renc
e (N
FC
).T
he N
FC
won
fiv
e m
ore
gam
es t
han
the
AF
C.
How
man
y ga
mes
did
eac
h co
nfer
ence
win
?So
urce
:New
York
Tim
es A
lman
acA
FC 1
3;N
FC 1
8
2.C
HEM
ISTR
YA
lab
need
s to
mak
e 10
0 ga
llons
of
an 1
8% a
cid
solu
tion
by
mix
ing
a 12
%ac
id s
olut
ion
wit
h a
20%
sol
utio
n.H
ow m
any
gallo
ns o
f ea
ch s
olut
ion
are
need
ed?
25 g
al o
f 12
% s
olut
ion
and
75 g
al o
f 20
% s
olut
ion
3.G
EOM
ETRY
The
per
imet
er o
f a
tria
ngle
is 2
4 in
ches
.The
long
est
side
is 4
inch
es lo
nger
than
the
sho
rtes
t si
de,a
nd t
he s
hort
est
side
is t
hree
-fou
rths
the
leng
th o
f th
e m
iddl
esi
de.F
ind
the
leng
th o
f ea
ch s
ide
of t
he t
rian
gle.
6 in
.,8
in.,
10 in
.
20$ 0.
100.
10t
$ 0.10
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sub
stitu
tion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A6 Glencoe Algebra 1
Skil
ls P
ract
ice
Sub
stitu
tion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill41
1G
lenc
oe A
lgeb
ra 1
Lesson 7-2
Use
su
bsti
tuti
on t
o so
lve
each
sys
tem
of
equ
atio
ns.
If t
he
syst
em d
oes
not
hav
eex
actl
y on
e so
luti
on,s
tate
wh
eth
er i
t h
as n
oso
luti
on o
r in
fin
itel
y m
an
yso
luti
ons.
1.y
!4x
2.y
!2x
x#
y!
5(1
,4)
x#
3y!
"14
("2,
"4)
3.y
!3x
4.x
!"
4y2x
#y
!15
(3,9
)3x
#2y
!20
(8,"
2)
5.y
!x
"1
6.x
!y
"7
x#
y!
3(2
,1)
x#
8y!
2("
6,1)
7.y
!4x
"1
8.y
!3x
#8
y!
2x"
5("
2,"
9)5x
#2y
!5
("1,
5)
9.2x
"3y
!21
10.y
!5x
"8
y!
3 "
x(6
,"3)
4x#
3y!
33(3
,7)
11.x
#2y
!13
12.x
#5y
!4
3x"
5y!
6(7
,3)
3x#
15y
!"
1no
sol
utio
n
13.3
x"
y!
414
.x#
4y!
82x
"3y
!"
9(3
,5)
2x"
5y!
29(1
2,"
1)
15.x
"5y
!10
16.5
x"
2y!
142x
"10
y!
20in
finite
ly m
any
2x"
y!
5(4
,3)
17.2
x#
5y!
3818
.x"
4y!
27x
"3y
!"
3(9
,4)
3x#
y!
"23
("5,
"8)
19.2
x#
2y!
720
.2.5
x#
y!
"2
x"
2y!
"1
!2,"
3x#
2y!
0("
2,3)
3 $ 2
©G
lenc
oe/M
cGra
w-H
ill41
2G
lenc
oe A
lgeb
ra 1
Use
su
bsti
tuti
on t
o so
lve
each
sys
tem
of
equ
atio
ns.
If t
he
syst
em d
oes
not
hav
eex
actl
y on
e so
luti
on,s
tate
wh
eth
er i
t h
as n
oso
luti
on o
r in
fin
itel
y m
an
yso
luti
ons.
1.y
!6x
2.x
!3y
3.x
!2y
#7
2x#
3y!
"20
("1,
"6)
3x"
5y!
12(9
,3)
x!
y#
4(1
,"3)
4.y
!2x
"2
5.y
!2x
#6
6.3x
#y
!12
y!
x#
2(4
,6)
2x"
y!
2no
sol
utio
ny
!"
x"
2(7
,"9)
7.x
#2y
!13
("3,
8)8.
x"
2y!
3in
finite
ly9.
x"
5y!
36("
4,"
8)"
2x"
3y!
"18
4x"
8y!
12m
any
2x#
y!
"16
10.2
x"
3y!
"24
11.x
#14
y!
8412
.0.3
x"
0.2y
!0.
5x
#6y
!18
("6,
4)2x
"7y
!"
7(1
4,5)
x"
2y!
"5
(5,5
)
13.0
.5x
#4y
!"
114
.3x
"2y
!11
15.
x#
2y!
121 $ 2
Pra
ctic
e (A
vera
ge)
Sub
stitu
tion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
x#
2.5y
!3.
5(6
,"1)
x"
y!
4(5
,2)
x"
2y!
6(1
2,3)
16.
x"
y!
317
.4x
"5y
!"
718
.x"
3y!
"4
2x#
y!
25y
!5x
2x#
6y!
5
(12,
1)!
,1"
!",1
"E
MP
LO
YM
EN
TF
or E
xerc
ises
19–
21,u
se t
he
foll
owin
g in
form
atio
n.
Ken
isha
sel
ls a
thle
tic
shoe
s pa
rt-t
ime
at a
dep
artm
ent
stor
e.Sh
e ca
n ea
rn e
ithe
r $5
00 p
erm
onth
plu
s a
4% c
omm
issi
on o
n he
r to
tal s
ales
,or
$400
per
mon
th p
lus
a 5%
com
mis
sion
on
tota
l sal
es.
19.W
rite
a s
yste
m o
f eq
uati
ons
to r
epre
sent
the
sit
uati
on.
y!
0.04
x#
500
and
y!
0.05
x#
400
20.W
hat
is t
he t
otal
pri
ce o
f th
e at
hlet
ic s
hoes
Ken
isha
nee
ds t
o se
ll to
ear
n th
e sa
me
inco
me
from
eac
h pa
y sc
ale?
$10,
000
21.W
hich
is t
he b
ette
r of
fer?
the
first
off
er if
she
exp
ects
to
sell
less
tha
n$1
0,00
0 in
sho
es,a
nd t
he s
econ
d of
fer
if sh
e ex
pect
s to
sel
l mor
e th
an$1
0,00
0 in
sho
es
MO
VIE
TIC
KET
SF
or E
xerc
ises
22
and
23,
use
th
e fo
llow
ing
info
rmat
ion
.T
icke
ts t
o a
mov
ie c
ost
$7.2
5 fo
r ad
ults
and
$5.
50 f
or s
tude
nts.
A g
roup
of
frie
nds
purc
hase
d8
tick
ets
for
$52.
75.
22.W
rite
a s
yste
m o
f eq
uati
ons
to r
epre
sent
the
sit
uati
on.
x#
y!
8 an
d 7.
25x
#5.
5y!
52.7
5
23.H
ow m
any
adul
t ti
cket
s an
d st
uden
t ti
cket
s w
ere
purc
hase
d?5
adul
t an
d 3
stud
ent
1 $ 123 $ 4
2 $ 31 $ 3
1 $ 3
1 $ 2
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A7 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csS
ubst
itutio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
©G
lenc
oe/M
cGra
w-H
ill41
3G
lenc
oe A
lgeb
ra 1
Lesson 7-2
Pre-
Act
ivit
yH
ow c
an a
sys
tem
of
equ
atio
ns
be u
sed
to
pre
dic
t m
edia
use
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
2 at
the
top
of
page
376
in y
our
text
book
.
•W
hat
is t
he s
yste
m o
f eq
uati
ons?
y!
"2.
8x#
170,
y!
14.4
x#
2•
Bas
ed o
n th
e gr
aph,
are
ther
e 0,
1,or
infi
nite
ly m
any
solu
tion
s of
the
syst
em?
1 so
lutio
n
Read
ing
the
Less
on
1.D
escr
ibe
how
you
wou
ld u
se s
ubst
itut
ion
to s
olve
eac
h sy
stem
of
equa
tion
s.
a.y
!"
2xx
#3y
!15
b.3x
"2y
!12
x!
2y
c.x
#2y
!7
2x"
8y!
8
d."
3x#
5y!
812x
#y
!24
2.Je
ss s
olve
d a
syst
em o
f eq
uati
ons
and
her
resu
lt w
as "
8 !
"8.
All
of h
er w
ork
was
corr
ect.
Des
crib
e th
e gr
aph
of t
he s
yste
m.E
xpla
in.
It is
a li
ne.T
here
are
infin
itely
man
y so
lutio
ns,s
ince
"8
!"
8 is
alw
ays
true
.
3.M
igue
l sol
ved
a sy
stem
of
equa
tion
s an
d hi
s re
sult
was
5 !
"2.
All
of h
is w
ork
was
corr
ect.
Des
crib
e th
e gr
aph
of t
he s
yste
m.E
xpla
in.
The
grap
h ha
s tw
o pa
ralle
llin
es,s
ince
5 !
"2
is a
lway
s fa
lse.
Hel
ping
You
Rem
embe
r
4.W
hat
is u
sual
ly t
he f
irst
ste
p in
sol
ving
a s
yste
m o
f eq
uati
ons
by s
ubst
itut
ion?
Sam
ple
answ
er:S
olve
one
of
the
equa
tions
for
one
vari
able
in t
erm
s of
the
oth
er.
Sol
ve t
he s
econ
d eq
uatio
n fo
r y.
Sub
stitu
te t
heex
pres
sion
you
fin
d fo
r y
in t
he f
irst
equ
atio
n.S
impl
ifyan
d so
lve
for
x.Th
en u
se t
he v
alue
of
xto
fin
d th
eva
lue
of y
.
Sol
ve t
he f
irst
equ
atio
n fo
r x.
Sub
stitu
te t
he e
xpre
ssio
nyo
u fin
d fo
r x
in t
he s
econ
d eq
uatio
n.S
impl
ify a
nd s
olve
for
y.Th
en u
se t
he v
alue
of
yto
fin
d th
e va
lue
of x
.
Sub
stitu
te 2
yfo
r x
in t
he f
irst
equ
atio
n.S
impl
ify a
ndso
lve
for
y.Th
en u
se t
he v
alue
of
yto
fin
d th
e va
lue
of x
.
Sub
stitu
te "
2xfo
r y
in t
he s
econ
d eq
uatio
n.S
impl
ifyan
d so
lve
for
x.Th
en u
se t
he v
alue
of
xto
fin
d th
eva
lue
of y
.
©G
lenc
oe/M
cGra
w-H
ill41
4G
lenc
oe A
lgeb
ra 1
Equ
atio
ns o
f Li
nes
and
Pla
nes
in In
terc
ept
Form
One
for
m t
hat
a lin
ear
equa
tion
may
tak
e is
inte
rcep
tfo
rm.T
he c
onst
ants
aan
d b
are
the
x- a
nd y
-int
erce
pts
of t
he g
raph
.
#!
1
In t
hree
-dim
ensi
onal
spa
ce,t
he e
quat
ion
of a
pla
ne
take
s a
sim
ilar
form
.
##
!1
Her
e,th
e co
nsta
nts
a,b,
and
car
e th
e po
ints
whe
re t
hepl
ane
mee
ts t
he x
,y,a
nd z
-axe
s.
Sol
ve e
ach
pro
blem
.
z $ cy $ b
x $ a
y $ bx $ a
z
y
x
O
x – 8 # y – 7 #
z – 6 ! 1
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-2
7-2
1.G
raph
the
equ
atio
n #
#!
1.
5.G
raph
the
equ
atio
n #
#!
1.
2.Fo
r th
e pl
ane
in E
xerc
ise
1,w
rite
an
equa
tion
for
the
line
whe
re t
he p
lane
inte
rsec
ts t
he x
y-pl
ane.
Use
inte
rcep
tfo
rms.
#!
1
3.W
rite
an
equa
tion
for
the
line
whe
reth
e pl
ane
inte
rsec
ts t
he x
z-pl
ane.
#!
1
4.W
rite
an
equa
tion
for
the
line
whe
reth
e pl
ane
inte
rsec
ts t
he y
z-pl
ane.
#!
1
6.W
rite
an
equa
tion
for
the
xy-
plan
e.
z!
0
7.W
rite
an
equa
tion
for
the
yz-
plan
e.
x!
0
8.W
rite
an
equa
tion
for
a p
lane
par
alle
lto
the
xy-
plan
e w
ith
a z-
inte
rcep
t of
2.
z!
2
9.W
rite
an
equa
tion
for
a p
lane
par
alle
lto
the
yz-
plan
e w
ith
an x
-inte
rcep
t of
23.
x!
"3z $
y $
z $x $
y $x $
z
y
x
O
z $ 2y $ 4
x $ 1
z
y
x
O
z $ 1y $ 2
x $ 3
Answers (Lesson 7-2)
© Glencoe/McGraw-Hill A8 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Elim
inat
ion
Usi
ng A
dditi
on a
nd S
ubtr
actio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill41
5G
lenc
oe A
lgeb
ra 1
Lesson 7-3
Elim
inat
ion
Usi
ng A
ddit
ion
In s
yste
ms
of e
quat
ions
in w
hich
the
coe
ffic
ient
s of
the
xor
yte
rms
are
addi
tive
inve
rses
,sol
ve t
he s
yste
m b
y ad
ding
the
equ
atio
ns.B
ecau
se o
ne o
fth
e va
riab
les
is e
limin
ated
,thi
s m
etho
d is
cal
led
elim
inat
ion
.
Use
ad
dit
ion
to
solv
e th
esy
stem
of
equ
atio
ns.
x"
3y!
73x
#3y
!9
Wri
te t
he e
quat
ions
in c
olum
n fo
rm a
nd a
ddto
elim
inat
e y.
x"
3y!
7(#
) 3x
#3y
!9
4x!
16So
lve
for
x.
!
x!
4Su
bsti
tute
4 f
or x
in e
ithe
r eq
uati
on a
ndso
lve
for
y.4
"3y
!7
4 "
3y"
4 !
7 "
4"
3y!
3
!
y!
"1
The
sol
utio
n is
(4,
"1)
.
3$ "
3"
3y$ "
3
16 $ 44x $ 4
Th
e su
m o
f tw
o n
um
bers
is 7
0 an
d t
hei
r d
iffe
ren
ce i
s 24
.Fin
dth
e n
um
bers
.
Let
xre
pres
ent
one
num
ber
and
yre
pres
ent
the
othe
r nu
mbe
r.x
#y
!70
(#) x
"y
!24
2x!
94
!
x!
47Su
bsti
tute
47
for
xin
eit
her
equa
tion
.47
#y
!70
47 #
y"
47 !
70 "
47y
!23
The
num
bers
are
47
and
23.
94 $ 22x $ 2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.x
#y
!"
42.
2m "
3n!
143.
3a"
b!
"9
x"
y!
2("
1,"
3)m
#3n
!"
11(1
,"4)
"3a
"2b
!0
("2,
3)
4."
3x"
4y!
"1
5.3c
#d
!4
6."
2x#
2y!
93x
"y
!"
4("
1,1)
2c"
d!
6(2
,"2)
2x"
y!
"6
!",3
"7.
2x#
2y!
"2
8.4x
"2y
!"
19.
x"
y!
23x
"2y
!12
(2,"
3)"
4x#
4y!
"2
!"1,
""
x#
y!
"3
!","
"10
.2x
"3y
!12
11."
0.2x
#y
!0.
512
.0.1
x#
0.3y
!0.
94x
#3y
!24
(6,0
)0.
2x#
2y!
1.6
(1,0
.7)
0.1x
"0.
3y!
0.2
!,
"13
.Rem
a is
old
er t
han
Ken
.The
dif
fere
nce
of t
heir
age
s is
12
and
the
sum
of
thei
r ag
es is
50.F
ind
the
age
of e
ach.
Rem
a is
31
and
Ken
is 1
9.
14.T
he s
um o
f th
e di
gits
of
a tw
o-di
git
num
ber
is 1
2.T
he d
iffe
renc
e of
the
dig
its
is 2
.Fin
dth
e nu
mbe
r if
the
uni
ts d
igit
is la
rger
tha
n th
e te
ns d
igit
.57
7 $ 611 $ 25 $ 2
1 $ 23 $ 2
3 $ 2
©G
lenc
oe/M
cGra
w-H
ill41
6G
lenc
oe A
lgeb
ra 1
Elim
inat
ion
Usi
ng S
ubtr
acti
onIn
sys
tem
s of
equ
atio
ns w
here
the
coe
ffic
ient
s of
the
xor
yte
rms
are
the
sam
e,so
lve
the
syst
em b
y su
btra
ctin
g th
e eq
uati
ons.
Use
su
btra
ctio
n t
o so
lve
the
syst
em o
f eq
uat
ion
s.2x
"3y
!11
5x"
3y!
14
2x"
3y!
11W
rite
the
equa
tions
in c
olum
n fo
rm a
nd s
ubtr
act.
(") 5
x"
3y!
14"
3x!
"3
Sub
trac
t the
two
equa
tions
. yis
elim
inat
ed.
!D
ivid
e ea
ch s
ide
by "
3.
x!
1S
impl
ify.
2(1)
"3y
!11
Sub
stitu
te 1
for
xin
eith
er e
quat
ion.
2 "
3y!
11S
impl
ify.
2 "
3y"
2 !
11 "
2S
ubtr
act 2
from
eac
h si
de.
"3y
!9
Sim
plify
.
!D
ivid
e ea
ch s
ide
by "
3.
y!
"3
Sim
plify
.
The
sol
utio
n is
(1,
"3)
.
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.6x
#5y
!4
2.3m
"4n
!"
143.
3a#
b!
16x
"7y
!"
203m
#2n
!"
2a
#b
!3
("1,
2)("
2,2)
("1,
4)
4."
3x"
4y!
"23
5.c
"3d
!11
6.x
"2y
!6
"3x
#y
!2
2c"
3d!
16x
#y
!3
(1,5
)(5
,"2)
(4,"
1)
7.2a
"3b
!"
138.
4x#
2y!
69.
5s"
t!
62a
#2b
!7
4x#
4y!
105s
#2t
!3
!",4
"!
,2"
(1,"
1)
10.6
x"
3y!
1211
.x#
2y!
3.5
12.0
.2x
#y
!0.
74x
"3y
!24
x"
3y!
"9
0.2x
#2y
!1.
2("
6,"
16)
("1.
5,2.
5)(1
,0.5
)
13.T
he s
um o
f tw
o nu
mbe
rs is
70.
One
num
ber
is t
en m
ore
than
tw
ice
the
othe
r nu
mbe
r.F
ind
the
num
bers
.50
;20
14.G
EOM
ETRY
Tw
o an
gles
are
sup
plem
enta
ry.T
he m
easu
re o
f on
e an
gle
is 1
0°m
ore
than
thre
e ti
mes
the
oth
er.F
ind
the
mea
sure
of
each
ang
le.
42.5
%;13
7.5%
1 $ 21 $ 2
9$ "
3"
3y$ "
3
"3
$ "3
"3x
$ "3
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Elim
inat
ion
Usi
ng A
dditi
on a
nd S
ubtr
actio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A9 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Elim
inat
ion
Usi
ng A
dditi
on a
nd S
ubtr
actio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill41
7G
lenc
oe A
lgeb
ra 1
Lesson 7-3
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.x
"y
!1
2."
x#
y!
1x
#y
!3
(2,1
)x
#y
!11
(5,6
)
3.x
#4y
!11
4."
x#
3y!
6x
"6y
!11
(11,
0)x
#3y
!18
(6,4
)
5.3x
#4y
!19
6.x
#4y
!"
83x
#6y
!33
("3,
7)x
"4y
!"
8("
8,0)
7.3a
#4b
!2
8.3c
"d
!"
14a
"4b
!12
(2,"
1)"
3c"
d!
5("
1,"
2)
9.2x
"3y
!9
10.x
"y
!4
"5x
"3y
!30
("3,
"5)
2x#
y!
"4
(0,"
4)
11.3
m"
n!
2612
.5x
"y
!"
6"
2m"
n!
"24
(10,
4)"
x#
y!
2("
1,1)
13.6
x"
2y!
3214
.3x
#2y
!"
194x
"2y
!18
(7,5
)"
3x"
5y!
25("
5,"
2)
15.7
m#
4n!
216
.2x
"5y
!"
287m
#2n
!8
(2,"
3)4x
#5y
!4
("4,
4)
17.T
he s
um o
f tw
o nu
mbe
rs is
28
and
thei
r di
ffer
ence
is 4
.Wha
t ar
e th
e nu
mbe
rs?
12,1
6
18.F
ind
the
two
num
bers
who
se s
um is
29
and
who
se d
iffe
renc
e is
15.
7,22
19.T
he s
um o
f tw
o nu
mbe
rs is
24
and
thei
r di
ffer
ence
is 2
.Wha
t ar
e th
e nu
mbe
rs?
13,1
1
20.F
ind
the
two
num
bers
who
se s
um is
54
and
who
se d
iffe
renc
e is
4.
25,2
9
21.T
wo
tim
es a
num
ber
adde
d to
ano
ther
num
ber
is 2
5.T
hree
tim
es t
he f
irst
num
ber
min
usth
e ot
her
num
ber
is 2
0.F
ind
the
num
bers
.9,
7
©G
lenc
oe/M
cGra
w-H
ill41
8G
lenc
oe A
lgeb
ra 1
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.x
"y
!1
2.p
#q
!"
23.
4x#
y!
23x
#y
!"
9p
"q
!8
3x"
y!
12
("4,
"5)
(3,"
5)(5
,3)
4.2x
#5y
!"
35.
3x#
2y!
"1
6.5x
#3y
!22
2x#
2y!
64x
#2y
!"
65x
"2y
!2
(6,"
3)("
5,7)
(2,4
)
7.5x
#2y
!7
8.3x
"9y
!"
129.
"4c
"2d
!"
2"
2x#
2y!
"14
3x"
15y
!"
62c
"2d
!"
14
(3,"
4)("
7,"
1)("
2,5)
10.2
x"
6y!
611
.7x
#2y
!2
12.4
.25x
"1.
28y
!"
9.2
2x#
3y!
247x
"2y
!"
30x
#1.
28y
!17
.6
(9,2
)("
2,8)
(1.6
,12.
5)
13.2
x#
4y!
1014
.2.5
x#
y!
10.7
15.6
m"
8n!
3x
"4y
!"
2.5
2.5x
#2y
!12
.92m
"8n
!"
3
(2.5
,1.2
5)(3
.4,2
.2)
!1,
"16
.4a
#b
!2
17."
x"
y!
"2
18.
x"
y!
84a
#3b
!10
x"
y!
4x
#y
!19
(10,
"1)
(12,
2)
19.T
he s
um o
f tw
o nu
mbe
rs is
41
and
thei
r di
ffer
ence
is 5
.Wha
t ar
e th
e nu
mbe
rs?
18,2
3
20.F
our
tim
es o
ne n
umbe
r ad
ded
to a
noth
er n
umbe
r is
36.
Thr
ee t
imes
the
fir
st n
umbe
rm
inus
the
oth
er n
umbe
r is
20.
Fin
d th
e nu
mbe
rs.
8,4
21.O
ne n
umbe
r ad
ded
to t
hree
tim
es a
noth
er n
umbe
r is
24.
Fiv
e ti
mes
the
fir
st n
umbe
rad
ded
to t
hree
tim
es t
he o
ther
num
ber
is 3
6.F
ind
the
num
bers
.3,
7
22.L
AN
GU
AG
ESE
nglis
h is
spo
ken
as t
he f
irst
or
prim
ary
lang
uage
in 7
8 m
ore
coun
trie
sth
an F
arsi
is s
poke
n as
the
fir
st la
ngua
ge.T
oget
her,
Eng
lish
and
Fars
i are
spo
ken
as a
firs
t la
ngua
ge in
130
cou
ntri
es.I
n ho
w m
any
coun
trie
s is
Eng
lish
spok
en a
s th
e fi
rst
lang
uage
? In
how
man
y co
untr
ies
is F
arsi
spo
ken
as t
he f
irst
lang
uage
?E
nglis
h:10
4 co
untr
ies,
Fars
i:26
cou
ntri
es
23.D
ISC
OU
NTS
At
a sa
le o
n w
inte
r cl
othi
ng,C
ody
boug
ht t
wo
pair
s of
glo
ves
and
four
hat
sfo
r $4
3.00
.Tor
i bou
ght
two
pair
s of
glo
ves
and
two
hats
for
$30
.00.
Wha
t w
ere
the
pric
esfo
r th
e gl
oves
and
hat
s?gl
oves
:$8.
50,h
ats:
$6.5
0.
1 $ 23 $ 2
2 $ 31 $ 3
1 $ 23 $ 4
4 $ 31 $ 3
3 $ 41 $ 2
Pra
ctic
e (A
vera
ge)
Elim
inat
ion
Usi
ng A
dditi
on a
nd S
ubtr
actio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3 !"
,4"
1 $ 2
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A10 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csE
limin
atio
n U
sing
Add
ition
and
Sub
trac
tion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3
©G
lenc
oe/M
cGra
w-H
ill41
9G
lenc
oe A
lgeb
ra 1
Lesson 7-3
Pre-
Act
ivit
yH
ow c
an y
ou u
se a
sys
tem
of
equ
atio
ns
to s
olve
pro
blem
s ab
out
wea
ther
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
3 at
the
top
of
page
382
in y
our
text
book
.
Wha
t fa
ct e
xpla
ins
why
the
var
iabl
e d
gets
elim
inat
ed f
rom
the
sys
tem
of
equa
tion
s?d
"d
!0
Read
ing
the
Less
on
1.W
rite
add
itio
nor
sub
trac
tion
to t
ell w
hich
ope
rati
on it
wou
ld b
e ea
sies
t to
use
to
elim
inat
e a
vari
able
of
the
syst
em.E
xpla
in y
our
choi
ce.
Sys
tem
of
Equ
atio
nsO
pera
tion
Exp
lana
tion
a.3x
#5y
!12
addi
tion
The
coef
ficie
nts
of t
he x
term
s ar
e "
3x#
2y!
6ad
ditiv
e in
vers
es.
b.3x
#5y
!7
subt
ract
ion
The
coef
ficie
nts
of t
he x
term
s ar
e 3x
"2y
!8
the
sam
e.
c."
x"
4y!
9su
btra
ctio
nTh
e co
effic
ient
s of
the
yte
rms
are
4x"
4y!
6th
e sa
me.
d.5x
"7y
!17
addi
tion
The
coef
ficie
nts
of t
he y
term
s ar
e 8x
#7y
!9
addi
tive
inve
rses
.
Hel
ping
You
Rem
embe
r
2.Te
ll ho
w y
ou c
an d
ecid
e w
heth
er t
o us
e ad
diti
on o
r su
btra
ctio
n to
elim
inat
e a
vari
able
ina
syst
em o
f eq
uati
ons.
Sam
ple
answ
er:L
ook
at t
he c
oeff
icie
nts
of e
ach
vari
able
.If
the
coef
ficie
nts
of o
ne o
f th
e va
riab
les
are
addi
tive
inve
rses
,yo
u ca
n us
e ad
ditio
n to
elim
inat
e th
e va
riab
le.I
f th
e co
effic
ient
s of
one
vari
able
are
the
sam
e,yo
u ca
n us
e su
btra
ctio
n to
elim
inat
e th
e va
riab
le.
©G
lenc
oe/M
cGra
w-H
ill42
0G
lenc
oe A
lgeb
ra 1
Róz
sa P
éter
Róz
sa P
éter
(190
5–19
77) w
as a
Hun
gari
an m
athe
mat
icia
n de
dica
ted
tote
achi
ng o
ther
s ab
out
mat
hem
atic
s.A
s pr
ofes
sor
of m
athe
mat
ics
at a
teac
hers
’ col
lege
in B
udap
est,
she
wro
te s
ever
al m
athe
mat
ics
text
book
san
d ch
ampi
oned
ref
orm
s in
the
tea
chin
g of
mat
hem
atic
s.In
194
5 sh
ew
rote
Pla
ying
wit
h In
fini
ty:M
athe
mat
ical
Exp
lora
tion
s an
d E
xcur
sion
s,a
popu
lar
wor
k in
whi
ch s
he a
ttem
pted
to
conv
ey t
he s
piri
t of
mat
hem
atic
s to
the
gen
eral
pub
lic.
By
far
Pét
er’s
grea
test
con
trib
utio
n to
mat
hem
atic
s w
as h
er p
ione
erin
gre
sear
ch in
the
fie
ld o
f re
curs
ive
func
tion
the
ory.
Whe
n yo
u ev
alua
te a
func
tion
rec
ursi
vely
,you
beg
in w
ith
one
init
ial v
alue
of x
.Wor
king
from
this
sin
gle
num
ber,
you
can
use
the
func
tion
to
gene
rate
an
enti
rese
quen
ce o
f nu
mbe
rs.F
or in
stan
ce,h
ere
is h
ow y
ou u
se a
n in
itia
lva
lue
of x
!1
to e
valu
ate
the
func
tion
f(x
) !3x
recu
rsiv
ely.
f(1)
!3(
1) !
3 ↓f(
3) !
3(3)
!9 ↓
f(9)
!3(
9) !
27
↓f(
27)
!3(
27)
!81
↓
f(81
) !
3(81
) !
243 ↓
The
fir
st f
ive
num
bers
of
the
sequ
ence
gen
erat
ed b
y th
is f
unct
ion
are
3,9,
27,8
1,an
d 24
3.
Wri
te t
he
firs
t fi
ve n
um
bers
of
the
sequ
ence
gen
erat
ed b
y ea
ch
fun
ctio
n,u
sin
g th
e gi
ven
nu
mbe
r as
th
e in
itia
l va
lue
of x
.
1.f(
x) !
3x;x
!2
2.g(
x) !
x"
5;x
!1
6,18
,54,
162,
486
"4,
"9,
"14
,"19
,"24
3.f(
x) !
2x#
1;x
!"
34.
f(x)
!x2
;x!
2
"5,
"9,
"17
,"33
,"65
4;16
;256
;65,
536;
4,29
4,96
7,29
6
5.h(
x) !
"x;
x!
36.
k(x)
!;x
!10
"3,
3,"
3,3,
"3
,10,
,10,
1 $ 101 $ 10
1 $ 10
1 $ x
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-3
7-3
Answers (Lesson 7-3)
© Glencoe/McGraw-Hill A11 Glencoe Algebra 1
Ans
wer
s
Stu
dy
Gu
ide
and I
nte
rven
tion
Elim
inat
ion
Usi
ng M
ultip
licat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill42
1G
lenc
oe A
lgeb
ra 1
Lesson 7-4
Elim
inat
ion
Usi
ng M
ulti
plic
atio
nSo
me
syst
ems
of e
quat
ions
can
not
be s
olve
dsi
mpl
y by
add
ing
or s
ubtr
acti
ng t
he e
quat
ions
.In
such
cas
es,o
ne o
r bo
th e
quat
ions
mus
tfi
rst
be m
ulti
plie
d by
a n
umbe
r be
fore
the
sys
tem
can
be
solv
ed b
y el
imin
atio
n.
Use
eli
min
atio
n t
o so
lve
the
syst
em o
f eq
uat
ion
s.x
#10
y!
34x
#5y
!5
If y
ou m
ulti
ply
the
seco
nd e
quat
ion
by "
2,yo
u ca
n el
imin
ate
the
yte
rms.
x#
10y
!3
(#) "
8x"
10y
!"
10"
7x!
"7
!
x!
1Su
bsti
tute
1 f
or x
in e
ithe
r eq
uati
on.
1 #
10y
!3
1 #
10y
"1
!3
"1
10y
!2
!
y!
The
sol
utio
n is
!1,
".1 $ 5
1 $ 52 $ 1010
y$ 10
"7
$ "7
"7x
$ "7
Use
eli
min
atio
n t
o so
lve
the
syst
em o
f eq
uat
ion
s.3x
"2y
!"
72x
"5y
!10
If y
ou m
ulti
ply
the
firs
t eq
uati
on b
y 2
and
the
seco
nd e
quat
ion
by "
3,yo
u ca
nel
imin
ate
the
xte
rms.
6x"
4y!
"14
(#) "
6x#
15y
!"
3011
y!
"44
!
y!
"4
Subs
titu
te "
4 fo
r y
in e
ithe
r eq
uati
on.
3x"
2("
4) !
"7
3x#
8 !
"7
3x#
8 "
8 !
"7
"8
3x!
"15
!
x!
"5
The
sol
utio
n is
("5,
"4)
.
"15
$3
3x $ 3
"44
$11
11y
$ 11
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.2x
#3y
!6
2.2m
#3n
!4
3.3a
"b
!2
x#
2y!
5("
3,4)
"m
#2n
!5
("1,
2)a
#2b
!3
(1,1
)
4.4x
#5y
!6
5.4c
"3d
!22
6.3x
"4y
!"
46x
"7y
!"
20("
1,2)
2c"
d!
10(4
,"2)
x#
3y!
"10
("4,
"2)
7.4s
"t
!9
8.4a
"3b
!"
89.
2x#
2y!
55s
#2t
!8
(2,"
1)2a
#2b
!3
!",2
"4x
"4y
!10
!,0
"10
.6x
"4y
!"
811
.4x
#2y
!"
512
.2x
#y
!3.
5(0
.9,1
.7)
4x#
2y!
"3
!"1,
""
2x"
4y!
1!"
,"
"x
#2y
!2.
5
13.G
AR
DEN
ING
The
leng
th o
f Sa
lly’s
gar
den
is 4
met
ers
grea
ter
than
3 t
imes
the
wid
th.
The
per
imet
er o
f he
r ga
rden
is 7
2 m
eter
s.W
hat
are
the
dim
ensi
ons
of S
ally
’s g
arde
n?28
m b
y 8
m
14.A
nita
is 4
year
s ol
der
than
Bas
ilio.
Thr
ee t
imes
Ani
ta’s
age
add
ed t
o si
x ti
mes
Bas
ilio’
s
age
is 3
6.H
ow o
ld a
re A
nita
and
Bas
ilio?
1 $ 2
1 $ 23 $ 2
1 $ 2
5 $ 21 $ 2
Ani
ta:7
yr;
Bas
ilio:
2yr
1 $ 2
©G
lenc
oe/M
cGra
w-H
ill42
2G
lenc
oe A
lgeb
ra 1
Det
erm
ine
the
Best
Met
hod
The
met
hods
to
use
for
solv
ing
syst
ems
of li
near
equa
tion
s ar
e su
mm
ariz
ed in
the
tab
le b
elow
.
Met
hod
The
Bes
t Tim
e to
Use
Gra
phin
gto
est
imat
e th
e so
lutio
n, s
ince
gra
phin
g us
ually
doe
s no
t giv
e an
exa
ct s
olut
ion
Sub
stitu
tion
if on
e of
the
varia
bles
in e
ither
equ
atio
n ha
s a
coef
ficie
nt o
f 1 o
r "
1
Elim
inat
ion
Usi
ng A
dditi
onif
one
of th
e va
riabl
es h
as o
ppos
ite c
oeffi
cien
ts in
the
two
equa
tions
Elim
inat
ion
Usi
ng S
ubtr
actio
nif
one
of th
e va
riabl
es h
as th
e sa
me
coef
ficie
nt in
the
two
equa
tions
Elim
inat
ion
Usi
ng M
ultip
licat
ion
if no
ne o
f the
coe
ffici
ents
are
1 o
r "
1 an
d ne
ither
of t
he v
aria
bles
can
be
elim
inat
ed b
y si
mpl
y ad
ding
or
subt
ract
ing
the
equa
tions
Det
erm
ine
the
best
met
hod
to
solv
e th
e sy
stem
of
equ
atio
ns.
Th
enso
lve
the
syst
em.
6x#
2y!
20"
2x#
4y!
"16
Sinc
e th
e co
effi
cien
ts o
f xw
ill b
e ad
diti
ve in
vers
es o
f ea
ch o
ther
if y
ou m
ulti
ply
the
seco
ndeq
uati
on b
y 3,
use
elim
inat
ion.
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Elim
inat
ion
Usi
ng M
ultip
licat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
6x#
2y!
20(#
) "6x
#12
y!
"48
Mul
tiply
the
seco
nd e
quat
ion
by 3
.
14y
!"
28A
dd th
e tw
o eq
uatio
ns. x
is e
limin
ated
.
!D
ivid
e ea
ch s
ide
by 1
4.
y!
"2
Sim
plify
.
"28
$14
14y
$ 14
6x#
2("
2) !
20S
ubst
itute
"2
for
yin
eith
er e
quat
ion.
6x"
4 !
20S
impl
ify.
6x"
4 #
4 !
20 #
4A
dd 4
to e
ach
side
.6x
!24
Sim
plify
.
!D
ivid
e ea
ch s
ide
by 6
.
x!
4S
impl
ify.
24 $ 66x $ 6
The
sol
utio
n is
(4,
"2)
.
Det
erm
ine
the
best
met
hod
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.T
hen
sol
ve t
he
syst
em.
1.x
#2y
!3
2.m
#6n
!"
83.
a"
b!
6x
#y
!1
m!
2n#
8a
!2b
#7
elim
inat
ion
(");
("1,
2)su
bstit
utio
n;(4
,"2)
subs
titut
ion;
(5,"
1)
4.4x
#y
!15
5.3c
"d
!14
6.x
#2y
!"
9"
x"
3y!
"12
c"
d!
2y
!4x
subs
titut
ion;
(3,3
)el
imin
atio
n ("
);(6
,4)
subs
titut
ion;
("1,
"4)
7.4x
!2y
"10
8.x
!"
2y9.
2s"
3t!
42x
#2y
!5
4x#
4y!
"10
3s#
2t!
24su
bstit
utio
n;("
1,3)
subs
titut
ion;
!"5,
"el
imin
atio
n (&
);(1
2,"
6)
10.4
a"
4b!
"10
11.4
x#
10y
!"
612
.2x
!y
"3
2a#
4b!
"2
"2x
"10
y!
2"
x#
y!
0el
imin
atio
n (#
);!"
2,"
elim
inat
ion
(#);
!"2,
"su
bstit
utio
n;("
3,"
3)1 $ 5
1 $ 2
5 $ 2
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A12 Glencoe Algebra 1
Skil
ls P
ract
ice
Elim
inat
ion
Usi
ng M
ultip
licat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill42
3G
lenc
oe A
lgeb
ra 1
Lesson 7-4
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.x
#y
!"
92.
3x#
2y!
"9
5x"
2y!
32(2
,"11
)x
"y
!"
13("
7,6)
3.2x
#5y
!3
4.2x
#y
!3
"x
#3y
!"
7(4
,"1)
"4x
"4y
!"
8(1
,1)
5.4x
"2y
!"
146.
2x#
y!
03x
"y
!"
8("
1,5)
5x#
3y!
2("
2,4)
7.5x
#3y
!"
108.
2x#
3y!
143x
#5y
!"
6("
2,0)
3x"
4y!
4(4
,2)
9.2x
"3y
!21
10.3
x#
2y!
"26
5x"
2y!
25(3
,"5)
4x"
5y!
"4
("6,
"4)
11.3
x"
6y!
"3
12.5
x#
2y!
"3
2x#
4y!
30(7
,4)
3x#
3y!
9("
3,6)
13.T
wo
tim
es a
num
ber
plus
thr
ee t
imes
ano
ther
num
ber
equa
ls 1
3.T
he s
um o
f th
e tw
onu
mbe
rs is
7.W
hat
are
the
num
bers
?8,
"1
14.F
our
tim
es a
num
ber
min
us t
wic
e an
othe
r nu
mbe
r is
"16
.The
sum
of
the
two
num
bers
is "
1.F
ind
the
num
bers
."
3,2
Det
erm
ine
the
best
met
hod
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.T
hen
sol
ve t
he
syst
em.
15.2
x#
3y!
10el
imin
atio
n (&
);16
.8x
"7y
!18
elim
inat
ion
(#);
5x#
2y!
"8
("4,
6)3x
#7y
!26
(4,2
)
17.y
!2x
subs
titut
ion;
18.3
x#
y!
6el
imin
atio
n ("
);3x
#2y
!35
(5,1
0)3x
#y
!3
no s
olut
ion
19.3
x"
4y!
17el
imin
atio
n (&
);20
.y!
3x#
1su
bstit
utio
n;4x
#5y
!2
(3,"
2)3x
"y
!"
1in
finite
ly m
any
solu
tions
©G
lenc
oe/M
cGra
w-H
ill42
4G
lenc
oe A
lgeb
ra 1
Use
eli
min
atio
n t
o so
lve
each
sys
tem
of
equ
atio
ns.
1.2x
"y
!"
12.
5x"
2y!
"10
3.7x
#4y
!"
43x
"2y
!1
3x#
6y!
665x
#8y
!28
("3,
"5)
(2,1
0)("
4,6)
4.2x
"4y
!"
225.
3x#
2y!
"9
6.4x
"2y
!32
3x#
3y!
305x
"3y
!4
"3x
"5y
!"
11(3
,7)
("1,
"3)
(7,"
2)
7.3x
#4y
!27
8.0.
5x#
0.5y
!"
29.
2x"
y!
"7
5x"
3y!
16x
"0.
25y
!6
x#
y!
0(5
,3)
(4,"
8)("
2,4)
10.E
ight
tim
es a
num
ber
plus
fiv
e ti
mes
ano
ther
num
ber
is "
13.T
he s
um o
f th
e tw
onu
mbe
rs is
1.W
hat
are
the
num
bers
?"
6,7
11.T
wo
tim
es a
num
ber
plus
thr
ee t
imes
ano
ther
num
ber
equa
ls 4
.Thr
ee t
imes
the
fir
stnu
mbe
r pl
us f
our
tim
es t
he o
ther
num
ber
is 7
.Fin
d th
e nu
mbe
rs.
5,"
2
Det
erm
ine
the
best
met
hod
to
solv
e ea
ch s
yste
m o
f eq
uat
ion
s.T
hen
sol
ve t
he
syst
em.
12.5
x#
7y!
313
.7x
#2y
!2
14."
6x"
2y!
142x
"7y
!"
382x
"3y
!"
286x
#8y
!"
20el
imin
atio
n (#
);el
imin
atio
n (&
);el
imin
atio
n (#
);("
5,4)
("2,
8)("
2,"
1)
15.x
!2y
#6
16.4
x#
3y!
"2
17.y
!x
1 $ 21 $ 23 $ 4
Pra
ctic
e (A
vera
ge)
Elim
inat
ion
Usi
ng M
ultip
licat
ion
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4 x
"y
!3
4x#
3y!
3x
"2y
!9
subs
titut
ion;
infin
itely
el
imin
atio
n ("
);su
bstit
utio
n;(6
,3)
man
y so
lutio
nsno
sol
utio
n
18.F
INA
NC
EG
unth
er in
vest
ed $
10,0
00 in
tw
o m
utua
l fun
ds.O
ne o
f th
e fu
nds
rose
6%
inon
e ye
ar,a
nd t
he o
ther
ros
e 9%
in o
ne y
ear.
If G
unth
er’s
inve
stm
ent
rose
a t
otal
of
$684
in o
ne y
ear,
how
muc
h di
d he
inve
st in
eac
h m
utua
l fun
d?$7
200
in t
he 6
% f
und
and
$280
0 in
the
9%
fun
d
19.C
AN
OEI
NG
Lau
ra a
nd B
rent
pad
dled
a c
anoe
6 m
iles
upst
ream
in f
our
hour
s.T
here
turn
tri
p to
ok t
hree
hou
rs.F
ind
the
rate
at
whi
ch L
aura
and
Bre
nt p
addl
ed t
he c
anoe
in s
till
wat
er.
1.75
mi/h
20.N
UM
BER
TH
EORY
The
sum
of
the
digi
ts o
f a
two-
digi
t nu
mbe
r is
11.
If t
he d
igit
s ar
ere
vers
ed,t
he n
ew n
umbe
r is
45
mor
e th
an t
he o
rigi
nal n
umbe
r.F
ind
the
num
ber.
38
5 $ 2
1 $ 2
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A13 Glencoe Algebra 1
Ans
wer
s
Rea
din
g t
o L
earn
Math
emati
csE
limin
atio
n U
sing
Mul
tiplic
atio
n
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4
©G
lenc
oe/M
cGra
w-H
ill42
5G
lenc
oe A
lgeb
ra 1
Lesson 7-4
Pre-
Act
ivit
yH
ow c
an a
man
ager
use
a s
yste
m o
f eq
uat
ion
s to
pla
n e
mp
loye
e ti
me?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
4 at
the
top
of
page
387
in y
our
text
book
.
Can
the
sys
tem
of
equa
tion
s be
sol
ved
by e
limin
atio
n w
ith
addi
tion
or
subt
ract
ion?
Exp
lain
.N
o;ne
ither
var
iabl
e ha
s co
effic
ient
s th
at a
read
ditiv
e in
vers
es o
r eq
ual.
Read
ing
the
Less
on
1.C
ould
elim
inat
ion
by m
ulti
plic
atio
n be
use
d to
sol
ve t
he s
yste
m s
how
n be
low
? E
xpla
in.
3x"
5y!
15"
6x#
7y!
11Ye
s;yo
u ca
n m
ultip
ly t
he f
irst
equ
atio
n by
2 t
o m
ake
the
coef
ficie
nts
ofth
e x
term
s ad
ditiv
e in
vers
es.
2.Te
ll w
heth
er it
wou
ld b
e ea
sies
t to
use
sub
stit
utio
n,el
imin
atio
n by
add
itio
n,el
imin
atio
nby
sub
trac
tion
,or
elim
inat
ion
by m
ulti
plic
atio
n to
sol
ve t
he s
yste
m.E
xpla
in y
our
choi
ce.
Sys
tem
of
Equ
atio
nsS
olut
ion
Met
hod
Exp
lana
tion
a."
3x#
4y!
2el
imin
atio
n by
Th
e co
effic
ient
s of
the
xte
rms
are
3x#
2y!
10ad
ditio
nad
ditiv
e in
vers
es.
b.x
"2y
!0
subs
titut
ion
It is
eas
y to
sol
ve t
he f
irst
equ
atio
n 5x
"4y
!8
for
x.
c.6x
"5y
!"
18el
imin
atio
n by
S
ampl
e an
swer
:You
can
mul
tiply
the
2x
#10
y!
27m
ultip
licat
ion
first
equ
atio
n by
2 t
o el
imin
ate
yby
addi
tion.
d."
2x#
3y!
9el
imin
atio
n by
Th
e co
effic
ient
s of
the
yte
rms
are
3x#
3y!
12su
btra
ctio
nth
e sa
me.
Hel
ping
You
Rem
embe
r
3.If
you
are
goi
ng t
o so
lve
a sy
stem
by
elim
inat
ion,
how
do
you
deci
de w
heth
er y
ou w
illne
ed t
o m
ulti
ply
one
or b
oth
equa
tion
s by
a n
umbe
r?S
ampl
e an
swer
:If
both
addi
ng t
he e
quat
ions
and
sub
trac
ting
the
equa
tions
res
ult
in e
quat
ions
that
stil
l hav
e tw
o va
riab
les,
you
will
nee
d to
use
mul
tiplic
atio
n.
©G
lenc
oe/M
cGra
w-H
ill42
6G
lenc
oe A
lgeb
ra 1
Geo
rge
Was
hing
ton
Car
ver
and
Perc
y Ju
lian
In 1
990,
Geo
rge
Was
hing
ton
Car
ver
and
Perc
y Ju
lian
beca
me
the
firs
t A
fric
anA
mer
ican
s el
ecte
d to
the
Nat
iona
l Inv
ento
rs H
all o
f Fam
e.C
arve
r (1
864–
1943
)w
as a
n ag
ricu
ltur
al s
cien
tist
kno
wn
wor
ldw
ide
for
deve
lopi
ng h
undr
eds
of u
ses
for
the
pean
ut a
nd t
he s
wee
t po
tato
.His
wor
k re
vita
lized
the
eco
nom
y of
the
sout
hern
Uni
ted
Stat
es b
ecau
se it
was
no
long
er d
epen
dent
sol
ely
upon
cot
ton.
Julia
n (1
898–
1975
) w
as a
res
earc
h ch
emis
t w
ho b
ecam
e fa
mou
s fo
r in
vent
ing
a m
etho
d of
mak
ing
a sy
nthe
tic
cort
ison
e fr
om s
oybe
ans.
His
dis
cove
ry h
asha
d m
any
med
ical
app
licat
ions
,par
ticu
larl
y in
the
tre
atm
ent
of a
rthr
itis
.
The
re a
re d
ozen
s of
oth
er A
fric
an A
mer
ican
inve
ntor
s w
hose
acc
ompl
ishm
ents
are
not
as w
ell k
now
n.T
heir
inve
ntio
ns r
ange
fro
m c
omm
on h
ouse
hold
item
slik
e th
e ir
onin
g bo
ard
to c
ompl
ex d
evic
es t
hat
have
rev
olut
ioni
zed
man
ufac
turi
ng.T
he e
xerc
ises
tha
t fo
llow
will
hel
p yo
u id
enti
fy ju
st a
few
of
thes
e in
vent
ors
and
thei
r in
vent
ions
.
Mat
ch t
he
inve
nto
rs w
ith
th
eir
inve
nti
ons
by m
atch
ing
each
sys
tem
wit
h i
ts s
olu
tion
.(N
ot a
ll t
he
solu
tion
s w
ill
be u
sed
.)
1.Sa
ra B
oone
x#
y!
2E
A.(
1,4)
auto
mat
ic t
raff
ic s
igna
l x
"y
!10
2.Sa
rah
Goo
dex
!2
"y
DB
.(4,
"2)
eggb
eate
r 2y
#x
!9
3.F
rede
rick
M.
y!
2x#
6G
C.(
"2,
3)fi
re e
xtin
guis
her
Jone
sy
!"
x"
3
4.J.
L.L
ove
2x#
3y !
8F
D.(
"5,
7)fo
ldin
g ca
bine
t be
d 2x
"y
!"
8
5.T.
J.M
arsh
all
y"
3x!
9C
E.(
6,"
4)ir
onin
g bo
ard
2y#
x!
4
6.Ja
n M
atze
liger
y#
4 !
2xJ
F.("
2,4)
penc
il sh
arpe
ner
6x"
3y!
12
7.G
arre
tt A
.3x
"2y
!"
5A
G.(
"3,
0)po
rtab
le X
-ray
mac
hine
M
orga
n3y
"4x
!8
8.N
orbe
rt R
illie
ux3x
"y
!12
IH
.(2,
"3)
play
er p
iano
y
"3x
!15
I.no
sol
utio
nev
apor
atin
g pa
n fo
r re
fini
ng s
ugar
J.in
fini
tely
last
ing
(sha
ping
) m
any
mac
hine
for
so
luti
ons
man
ufac
turi
ng s
hoes
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-4
7-4
Answers (Lesson 7-4)
© Glencoe/McGraw-Hill A14 Glencoe Algebra 1
Stu
dy
Gu
ide
and I
nte
rven
tion
Gra
phin
g S
yste
ms
of In
equa
litie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill42
7G
lenc
oe A
lgeb
ra 1
Lesson 7-5
Syst
ems
of In
equa
litie
sT
he s
olut
ion
of a
sys
tem
of
ineq
ual
itie
sis
the
set
of
all
orde
red
pair
s th
at s
atis
fy b
oth
ineq
ualit
ies.
If y
ou g
raph
the
ineq
ualit
ies
in t
he s
ame
coor
dina
te p
lane
,the
sol
utio
n is
the
reg
ion
whe
re t
he g
raph
s ov
erla
p.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by g
rap
hin
g.y
'x
#2
y(
"2x
"1
The
sol
utio
n in
clud
es t
he o
rder
ed p
airs
in t
he in
ters
ecti
on o
f th
egr
aphs
.Thi
s re
gion
is s
hade
d at
the
rig
ht.T
he g
raph
s of
y!
x#
2an
d y
!"
2x"
1 ar
e bo
unda
ries
of
this
reg
ion.
The
gra
ph o
f y
!x
#2
is d
ashe
d an
d is
not
incl
uded
in t
he g
raph
of y
%x
#2.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by g
rap
hin
g.x
#y
'4
x#
y)
"1
The
gra
phs
of x
#y
!4
and
x#
y!
"1
are
para
llel.
Bec
ause
the
tw
o re
gion
s ha
ve n
o po
ints
in c
omm
on,t
he s
yste
m o
f in
equa
litie
s ha
s no
sol
utio
n.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.y
%"
12.
y%
"2x
#2
3.y
&x
#1
x&
0y
'x
#1
3x#
4y(
12
4.2x
#y
(1
5.y
'2x
#3
6.5x
"2y
&6
x"
y(
"2
y(
"1
#2x
y%
"x
#1
5x "
2y
! 6
y !
"x
# 1
x
y
O
y !
2x
# 3
y !
"1
# 2
xx
y
O
2x #
y !
1
x "
y !
"2
x
y
O
y !
x #
1
3x #
4y
! 1
2
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 !
4x
y O
y !
x #
2
y !
"2x
" 1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill42
8G
lenc
oe A
lgeb
ra 1
Real
-Wor
ld P
robl
ems
In r
eal-
wor
ld p
robl
ems,
som
etim
es o
nly
who
le n
umbe
rs m
ake
sens
e fo
r th
e so
luti
on,a
nd o
ften
onl
y po
siti
ve v
alue
s of
xan
d y
mak
e se
nse.
BU
SIN
ESS
AA
A G
em C
omp
any
pro
du
ces
nec
kla
ces
and
bra
cele
ts.I
n a
40-
hou
r w
eek
,th
e co
mp
any
has
400
gem
s to
use
.A n
eck
lace
req
uir
es 4
0 ge
ms
and
a
brac
elet
req
uir
es 1
0 ge
ms.
It t
akes
2 h
ours
to
pro
du
ce a
nec
kla
ce a
nd
a b
race
let
requ
ires
on
e h
our.
How
man
y of
ea
ch t
ype
can
be
pro
du
ced
in
a w
eek
?
Let
n!
the
num
ber
of n
eckl
aces
tha
t w
ill b
e pr
oduc
ed a
nd b
!th
enu
mbe
r of
bra
cele
ts t
hat
will
be
prod
uced
.Nei
ther
nor
bca
n be
ane
gati
ve n
umbe
r,so
the
fol
low
ing
syst
em o
f in
equa
litie
s re
pres
ents
th
e co
ndit
ions
of
the
prob
lem
s.
n(
0b
(0
b#
2n'
4010
b#
40n
'40
0
The
sol
utio
n is
the
set
ord
ered
pai
rs in
the
inte
rsec
tion
of
the
grap
hs.T
his
regi
on is
sha
ded
at t
he r
ight
.Onl
y w
hole
-num
ber
solu
tion
s,su
ch a
s (5
,20)
mak
e se
nse
in t
his
prob
lem
.
For
eac
h e
xerc
ise,
grap
h t
he
solu
tion
set
.Lis
t th
ree
pos
sibl
e so
luti
ons
to t
he
pro
blem
.
1.H
EALT
HM
r.F
low
ers
is o
n a
rest
rict
ed
2.R
ECR
EATI
ON
Mar
ia h
ad $
150
in g
ift
diet
tha
t al
low
s hi
m t
o ha
ve b
etw
een
cert
ific
ates
to
use
at a
rec
ord
stor
e.Sh
e 16
00 a
nd 2
000
Cal
orie
s pe
r da
y.H
is
boug
ht f
ewer
tha
n 20
rec
ordi
ngs.
Eac
h da
ily f
at in
take
is r
estr
icte
d to
bet
wee
nta
pe c
ost
$5.9
5 an
d ea
ch C
D c
ost
$8.9
5.45
and
55
gram
s.W
hat
daily
Cal
orie
H
ow m
any
of e
ach
type
of r
ecor
ding
mig
ht
and
fat
inta
kes
are
acce
ptab
le?
she
have
bou
ght?
Sam
ple
answ
ers:
1600
Cal
orie
s,S
ampl
e an
swer
s:10
tap
es,9
CD
s;45
fat
gra
ms;
1800
Cal
orie
s,50
fat
0 ta
pes,
16 C
Ds;
14 t
apes
,5 C
Ds
gram
s;20
00 C
alor
ies,
55 f
at g
ram
s
t # c
! 2
0 5.95
t # 8
.95c
! 1
50
30 25 20 15 10 5
Tape
s
Compact Discs
510
1520
2530
0
60 50 40 30 20 10
Calo
ries
Fat Grams
1000
2000
3000
0
Nec
klac
es
Bracelets
1020
3040
500
50 40 30 20 10
10b
# 4
0n !
400
b #
2n
! 4
0
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Gra
phin
g S
yste
ms
of In
equa
litie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A15 Glencoe Algebra 1
Ans
wer
s
Skil
ls P
ract
ice
Gra
phin
g S
yste
ms
of In
equa
litie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill42
9G
lenc
oe A
lgeb
ra 1
Lesson 7-5
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.x
%"
12.
y%
23.
y%
x#
3y
'"
3x
&"
2y
'"
1
4.x
&2
5.x
#y
'"
16.
y"
x%
4y
"x
'2
x#
y(
3*
x#
y%
2
7.y
%x
#1
8.y
("
x#
29.
y&
2x#
4y
("
x#
1y
&2x
"2
y(
x#
1
Wri
te a
sys
tem
of
ineq
ual
itie
s fo
r ea
ch g
rap
h.
10.
11.
12.
y(
x#
2,y
+x
"3
y'
"x,
y'
xy
+x
#1,
y)
1x
y
Ox
y
Ox
y
O
y !
x #
1
y !
2x
# 4
x
y
O
y !
2x
" 2
y !
"x
# 2
x
y
O
y !
"x
# 1
y !
x #
1
x
y
O
x #
y !
2
y "
x !
4x
y
Ox
y
O
y "
x !
2x
! 2 x
y
O
y !
x #
3
y !
"1
x
y
O
y !
2
x !
"2
x
y
O
x !
"1
y !
"3
x
y
O
©G
lenc
oe/M
cGra
w-H
ill43
0G
lenc
oe A
lgeb
ra 1
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.y
%x
"2
2.y
(x
#2
3.x
#y
(1
y'
xy
%2x
#3
x#
2y%
1
4.y
&2x
"1
5.y
%x
"4
6.2x
"y
(2
y%
2 "
x2x
#y
'2
x"
2y(
2
FITN
ESS
For
Exe
rcis
es 7
an
d 8
,use
th
e fo
llow
ing
info
rmat
ion
.D
iego
sta
rted
an
exer
cise
pro
gram
in w
hich
eac
h w
eek
he
wor
ks o
ut a
t th
e gy
m b
etw
een
4.5
and
6 ho
urs
and
wal
ksbe
twee
n 9
and
12 m
iles.
7.M
ake
a gr
aph
to s
how
the
num
ber
of h
ours
Die
go w
orks
ou
t at
the
gym
and
the
num
ber
of m
iles
he w
alks
per
wee
k.
8.L
ist
thre
e po
ssib
le c
ombi
nati
ons
of w
orki
ng o
ut a
nd
wal
king
tha
t m
eet
Die
go’s
goa
ls.
Sam
ple
answ
ers:
gym
5 h
,wal
k 9
mi;
gym
6 h
,wal
k 10
mi,
gym
5.5
h,w
alk
11 m
i
SOU
VEN
IRS
For
Exe
rcis
es 9
an
d 1
0,u
se t
he
foll
owin
g in
form
atio
n.
Em
ily w
ants
to
buy
turq
uois
e st
ones
on
her
trip
to
New
Mex
ico
to g
ive
to a
t le
ast
4 of
her
fri
ends
.The
gif
t sh
op s
ells
sto
nes
for
eith
er $
4 or
$6
per
ston
e.E
mily
has
no
mor
e th
an $
30 t
o sp
end.
9.M
ake
a gr
aph
show
ing
the
num
bers
of
each
pri
ce o
f st
one
Em
ily c
an p
urch
ase.
10.L
ist
thre
e po
ssib
le s
olut
ions
.S
ampl
e an
swer
:one
$4
ston
e an
d fo
ur $
6 st
ones
;thr
ee $
4 st
ones
and
th
ree
$6 s
tone
s;fiv
e $4
sto
nes
and
one
$6 s
tone
$4 S
tone
s
Turq
uo
ise
Sto
nes
$6 Stones
13
24
56
78
06 5 4 3 2 1
Gym
(hou
rs)
Die
go
’s R
ou
tin
e
Waking (miles)
13
24
56
78
0
16 14 12 10 8 6 4 2
2x "
y !
2
x "
2y
! 2
x
y
O
y !
x "
4
2x #
y !
2
x
y
O
y !
2x
" 1
y !
2 "
x
x
y
O
x #
y !
1
x #
2y
! 1
x
y
O
y !
2x
# 3
y !
x #
2 x
y
O
y !
x "
2y
! x
x
y
OPra
ctic
e (A
vera
ge)
Gra
phin
g S
yste
ms
of In
equa
litie
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
Answers (Lesson 7-5)
© Glencoe/McGraw-Hill A16 Glencoe Algebra 1
Rea
din
g t
o L
earn
Math
emati
csG
raph
ing
Sys
tem
s of
Ineq
ualit
ies
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
©G
lenc
oe/M
cGra
w-H
ill43
1G
lenc
oe A
lgeb
ra 1
Lesson 7-5
Pre-
Act
ivit
yH
ow c
an y
ou u
se a
sys
tem
of
ineq
ual
itie
s to
pla
n a
sen
sibl
e d
iet?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 7-
5 at
the
top
of
page
394
in y
our
text
book
.
The
gre
en s
ecti
on o
n th
e gr
aph
repr
esen
ts a
ran
ge o
f
Cal
orie
s a
day
and
gram
s of
fat
per
day
.
Read
ing
the
Less
onW
rite
th
e in
equ
alit
y sy
mbo
ls t
hat
you
nee
d t
o ge
t a
syst
em w
hos
e gr
aph
loo
ks
lik
eth
e on
e sh
own
.Use
),(
,',o
r +
.
1.2.
yx
#2
yx
#2
y"
2x"
1y
"2x
"1
3.4.
yx
#2
yx
#2
y"
2x"
1y
"2x
"1
Hel
ping
You
Rem
embe
r
5.D
escr
ibe
how
you
wou
ld e
xpla
in t
he p
roce
ss o
f us
ing
a gr
aph
to s
olve
a s
yste
m o
fin
equa
litie
s to
a f
rien
d w
ho m
isse
d L
esso
n 7-
5.G
raph
eac
h in
equa
lity
on t
hesa
me
coor
dina
te p
lane
.The
sol
utio
ns a
re t
he o
rder
ed p
airs
for
the
poin
tsin
bot
h gr
aphs
.
'(
('
x
y O
y !
x #
2
y !
"2x
" 1
x
y O
y !
x #
2
y !
"2x
" 1
)+
)+
x
y O
y !
x #
2
y !
"2x
" 1
x
y O
y !
x #
2y
! "
2x "
1
60 t
o 75
2000
to
2400
©G
lenc
oe/M
cGra
w-H
ill43
2G
lenc
oe A
lgeb
ra 1
Des
crib
ing
Reg
ions
The
sha
ded
regi
on in
side
the
tri
angl
e ca
n be
des
crib
ed
wit
h a
syst
em o
f th
ree
ineq
ualit
ies.
y&
2x#
1
y%
x"
3
y%
29x
"31
Wri
te s
yste
ms
of i
neq
ual
itie
s to
des
crib
e ea
ch r
egio
n.Y
ou m
ay
firs
t n
eed
to
div
ide
a re
gion
in
to t
rian
gles
or
quad
rila
tera
ls.
1.2.
y)
x#
12y
'"
x"
2y
)x
#2
y)
"x
#y
'x
"2
y)
"x
#2
y)
2y
'"
3
y)
5y
'"
5
3.to
p:y
)x
#7,
y)
"x
#7,
y'
4
mid
dle:
y)
4,y
'0,
y'
"x
"4,
y'
x"
4
bott
om le
ft:y
)4x
#12
,y'
x"
2,y
)0,
x)
0
bott
om r
ight
:y)
"4x
#12
,y
'"
x"
2,y
)0,
x,
01 $ 2
1 $ 2
4 $ 3
4 $ 3
3 $ 23 $ 2
x
y
O19 $ 25 $ 2
5 $ 2
x
y
Ox
y
O
1 $ 3x
y
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
7-5
7-5
Answers (Lesson 7-5)
Recommended