chapter 3 resource masters - math class · graph systems of equations a system of equations is a...
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Chapter 3Resource Masters
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9
ANSWERS FOR WORKBOOKS The answers for Chapter 3 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-828006-0 Algebra 2Chapter 3 Resource Masters
2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03
Glencoe/McGraw-Hill
© Glencoe/McGraw-Hill iii Glencoe Algebra 2
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Lesson 3-1Study Guide and Intervention . . . . . . . . 119–120Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 121Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Reading to Learn Mathematics . . . . . . . . . . 123Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 124
Lesson 3-2Study Guide and Intervention . . . . . . . . 125–126Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 127Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 128Reading to Learn Mathematics . . . . . . . . . . 129Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 130
Lesson 3-3Study Guide and Intervention . . . . . . . . 131–132Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 133Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 134Reading to Learn Mathematics . . . . . . . . . . 135Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 136
Lesson 3-4Study Guide and Intervention . . . . . . . . 137–138Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 139Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Reading to Learn Mathematics . . . . . . . . . . 141Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 142
Lesson 3-5Study Guide and Intervention . . . . . . . . 143–144Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 145Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Reading to Learn Mathematics . . . . . . . . . . 147Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 148
Chapter 3 AssessmentChapter 3 Test, Form 1 . . . . . . . . . . . . 149–150Chapter 3 Test, Form 2A . . . . . . . . . . . 151–152Chapter 3 Test, Form 2B . . . . . . . . . . . 153–154Chapter 3 Test, Form 2C . . . . . . . . . . . 155–156Chapter 3 Test, Form 2D . . . . . . . . . . . 157–158Chapter 3 Test, Form 3 . . . . . . . . . . . . 159–160Chapter 3 Open-Ended Assessment . . . . . . 161Chapter 3 Vocabulary Test/Review . . . . . . . 162Chapter 3 Quizzes 1 & 2 . . . . . . . . . . . . . . . 163Chapter 3 Quizzes 3 & 4 . . . . . . . . . . . . . . . 164Chapter 3 Mid-Chapter Test . . . . . . . . . . . . 165Chapter 3 Cumulative Review . . . . . . . . . . . 166Chapter 3 Standardized Test Practice . . 167–168
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A26
© Glencoe/McGraw-Hill iv Glencoe Algebra 2
Teacher’s Guide to Using theChapter 3 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 3 Resource Masters includes the core materials neededfor Chapter 3. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theAlgebra 2 TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 3-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.
Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
© Glencoe/McGraw-Hill v Glencoe Algebra 2
Assessment OptionsThe assessment masters in the Chapter 3Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 150–151. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 3.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
bounded region
consistent system
constraints
kuhn·STRAYNTS
dependent system
elimination method
feasible region
FEE·zuh·buhl
inconsistent system
ihn·kuhn·SIHS·tuhnt
independent system
(continued on the next page)
© Glencoe/McGraw-Hill vii Glencoe Algebra 2
© Glencoe/McGraw-Hill viii Glencoe Algebra 2
Vocabulary Term Found on Page Definition/Description/Example
linear programming
ordered triple
substitution method
system of equations
system of inequalities
unbounded region
vertices
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
33
Study Guide and InterventionSolving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 119 Glencoe Algebra 2
Less
on
3-1
Graph Systems of Equations A system of equations is a set of two or moreequations containing the same variables. You can solve a system of linear equations bygraphing the equations on the same coordinate plane. If the lines intersect, the solution isthat intersection point.
Solve the system of equations by graphing. x ! 2y " 4x # y " !2
Write each equation in slope-intercept form.x ! 2y " 4 → y " ! 2
x # y " !2 → y " !x ! 2
The graphs appear to intersect at (0, !2).
CHECK Substitute the coordinates into each equation.x ! 2y " 4 x # y " !2
0 ! 2(!2) ! 4 0 # (!2) ! !24 " 4 ✓ !2 " !2 ✓
The solution of the system is (0, !2).
Solve each system of equations by graphing.
1. y " ! # 1 2. y " 2x ! 2 3. y " ! # 3
y " ! 4 (6, !1) y " !x # 4 (2, 2) y " (4, 1)
4. 3x ! y " 0 5. 2x # " !7 6. ! y " 2
x ! y " !2 (1, 3) # y " 1 (!4, 3) 2x ! y " !1 (!2, !3)
x
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ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 120 Glencoe Algebra 2
Classify Systems of Equations The following chart summarizes the possibilities forgraphs of two linear equations in two variables.
Graphs of Equations Slopes of Lines Classification of System Number of Solutions
Lines intersect Different slopes Consistent and independent One
Lines coincide (same line)Same slope, same
Consistent and dependent Infinitely many y-intercept
Lines are parallelSame slope, different
Inconsistent None y-intercepts
Graph the system of equations x ! 3y " 6and describe it as consistent and independent, 2x ! y " !3consistent and dependent, or inconsistent.
Write each equation in slope-intercept form.
x ! 3y " 6 → y " x ! 2
2x ! y " !3 → y " 2x # 3
The graphs intersect at (!3, !3). Since there is one solution, the system is consistent and independent.
Graph the system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
1. 3x # y " !2 2. x # 2y " 5 3. 2x ! 3y " 06x # 2y " 10 3x ! 15 " !6y consistent 4x ! 6y " 3inconsistent and dependent inconsistent
4. 2x ! y " 3 5. 4x # y " !2 6. 3x ! y " 2x # 2y " 4 consistent 2x # " !1 consistent x # y " 6 consistent and independent and dependent and independent
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Study Guide and Intervention (continued)
Solving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
ExercisesExercises
ExampleExample
Skills PracticeSolving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 121 Glencoe Algebra 2
Less
on
3-1
Solve each system of equations by graphing.
1. x " 2 2. y " !3x # 6 3. y " 4 ! 3x
y " 0 (2, 0) y " 2x ! 4 (2, 0) y " ! x ! 1 (2, !2)
4. y " 4 ! x 5. y " !2x # 2 6. y " x
y " x ! 2 (3, 1) y " x ! 5 (3, !4) y " !3x # 4 (1, 1)
7. x # y " 3 8. x ! y " 4 9. 3x ! 2y " 4x ! y " 1 (2, 1) 2x ! 5y " 8 (4, 0) 2x ! y " 1 (!2, !5)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
10. y " !3x 11. y " x ! 5 12. 2x ! 5y " 10y " !3x # 2 !2x # 2y " !10 3x # y " 15
inconsistent consistent and consistent and dependent independent
2
2 x
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x
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1$2
© Glencoe/McGraw-Hill 122 Glencoe Algebra 2
Solve each system of equations by graphing.
1. x ! 2y " 0 2. x # 2y " 4 3. 2x # y " 3
y " 2x ! 3 (2, 1) 2x ! 3y " 1 (2, 1) y " x ! (3, !3)
4. y ! x " 3 5. 2x ! y " 6 6. 5x ! y " 4y " 1 (!2, 1) x # 2y " !2 (2, !2) !2x # 6y " 4 (1, 1)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
7. 2x ! y " 4 8. y " !x ! 2 9. 2y ! 8 " x
x ! y " 2 x # y " !4 y " x # 4
consistent and indep. inconsistent consistent and dep.SOFTWARE For Exercises 10–12, use the following information.Location Mapping needs new software. Software A costs $13,000 plus $500 per additionalsite license. Software B costs $2500 plus $1200 per additional site license.
10. Write two equations that represent the cost of each software. A: y " 13,000 # 500x, B: y " 2500 # 1200x
11. Graph the equations. Estimate the break-even point of the software costs.15 additional licenses
12. If Location Mapping plans to buy 10 additional site licenses, which software will cost less? B
Software Costs
Additional Licenses
Tota
l Co
st (
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40 8 12 14 16 18 202 6 10
24,000
20,000
16,000
12,000
8,000
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Practice (Average)
Solving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
Reading to Learn MathematicsSolving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 123 Glencoe Algebra 2
Less
on
3-1
Pre-Activity How can a system of equations be used to predict sales?
Read the introduction to Lesson 3-1 at the top of page 110 in your textbook.
• Which are growing faster, in-store sales or online sales? online sales
• In what year will the in-store and online sales be the same? 2005
Reading the Lesson
1. The Study Tip on page 110 of your textbook says that when you solve a system ofequations by graphing and find a point of intersection of the two lines, you must alwayscheck the ordered pair in both of the original equations. Why is it not good enough tocheck the ordered pair in just one of the equations?Sample answer: To be a solution of the system, the ordered pair mustmake both of the equations true.
2. Under each system graphed below, write all of the following words that apply: consistent,inconsistent, dependent, and independent.
inconsistent consistent; consistent; dependent independent
Helping You Remember
3. Look up the words consistent and inconsistent in a dictionary. How can the meaning ofthese words help you distinguish between consistent and inconsistent systems ofequations?Sample answer: One meaning of consistent is “being in agreement,” or“compatible,” while one meaning of inconsistent is “not being inagreement” or “incompatible.” When a system is consistent, theequations are compatible because both can be true at the same time (forthe same values of x and y). When a system is inconsistent, theequations are incompatible because they can never be true at the same time.
x
y
Ox
y
Ox
y
O
© Glencoe/McGraw-Hill 124 Glencoe Algebra 2
InvestmentsThe following graph shows the value of two different investments over time.Line A represents an initial investment of $30,000 with a bank paying passbook savings interest. Line B represents an initial investment of $5,000 in a profitable mutual fund with dividends reinvested and capital gains accepted in shares. By deriving the linear equation y ! mx " b for A and B, you can predict the value of these investments for years to come.
1. The y-intercept, b, is the initial investment. Find b for each of the following.a. line A 30,000 b. line B 5000
2. The slope of the line, m, is the rate of return. Find m for each of the following.a. line A 0.588 b. line B 2.73
3. What are the equations of each of the following lines?a. line A y ! 0.588x " 30 b. line B y ! 2.73x " 5
4. What will be the value of the mutual fund after 11 years of investment? $35,030
5. What will be the value of the bank account after 11 years of investment? $36,468
6. When will the mutual fund and the bank account have equal value?after 11.67 years of investment
7. Which investment has the greatest payoff after 11 years of investment? the mutual fund
A
B25
20
30
35
10
5
0
15
1 2 3 4 5 6 7 8 9
Amou
nt In
vest
ed (t
hous
ands
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Years Invested
x
y
Enrichment
NAME ______________________________________________ DATE______________ PERIOD _____
3-13-1
Study Guide and InterventionSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 125 Glencoe Algebra 2
Less
on
3-2
Substitution To solve a system of linear equations by substitution, first solve for onevariable in terms of the other in one of the equations. Then substitute this expression intothe other equation and simplify.
Use substitution to solve the system of equations. 2x ! y " 9x # 3y " !6
Solve the first equation for y in terms of x.2x ! y " 9 First equation
!y " !2x # 9 Subtract 2x from both sides.y " 2x ! 9 Multiply both sides by !1.
Substitute the expression 2x ! 9 for y into the second equation and solve for x.x # 3y " !6 Second equation
x # 3(2x ! 9) " !6 Substitute 2x ! 9 for y.x # 6x ! 27 " !6 Distributive Property
7x ! 27 " !6 Simplify.7x " 21 Add 27 to each side.x " 3 Divide each side by 7.
Now, substitute the value 3 for x in either original equation and solve for y.2x ! y " 9 First equation
2(3) ! y " 9 Replace x with 3.6 ! y " 9 Simplify.
!y " 3 Subtract 6 from each side.y " !3 Multiply each side by !1.
The solution of the system is (3, !3).
Solve each system of linear equations by using substitution.
1. 3x # y " 7 2. 2x # y " 5 3. 2x # 3y " !34x # 2y " 16 3x ! 3y " 3 x # 2y " 2
(!1, 10) (2, 1) (!12, 7)
4. 2x ! y " 7 5. 4x ! 3y " 4 6. 5x # y " 66x ! 3y " 14 2x # y " !8 3 ! x " 0
no solution (!2, !4) (3, !9)
7. x # 8y " !2 8. 2x ! y " !4 9. x ! y " !2x ! 3y " 20 4x # y " 1 2x ! 3y " 2
(14, !2) !! , 3" (!8, !6)
10. x ! 4y " 4 11. x # 3y " 2 12. 2x # 2y " 42x # 12y " 13 4x # 12 y " 8 x ! 2y " 0
!5, " infinitely many ! , "2$
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ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 126 Glencoe Algebra 2
Elimination To solve a system of linear equations by elimination, add or subtract theequations to eliminate one of the variables. You may first need to multiply one or both of theequations by a constant so that one of the variables has the same (or opposite) coefficient inone equation as it has in the other.
Use the elimination method to solve the system of equations.2x ! 4y " !263x ! y " !24
Study Guide and Intervention (continued)
Solving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Example 1Example 1
Example 2Example 2
Multiply the second equation by 4. Then subtractthe equations to eliminate the y variable.2x ! 4y " !26 2x ! 4y " !263x ! y " !24 Multiply by 4. 12x ! 4y " !96
!10x " 70x " !7
Replace x with !7 and solve for y.2x ! 4y " !26
2(!7) !4y " !26!14 ! 4y " !26
!4y " !12y " 3
The solution is (!7, 3).
Multiply the first equation by 3 and the secondequation by 2. Then add the equations toeliminate the y variable.3x ! 2y " 4 Multiply by 3. 9x ! 6y " 125x # 3y " !25 Multiply by 2. 10x # 6y " !50
19x " !38x " !2
Replace x with !2 and solve for y.3x ! 2y " 4
3(!2) ! 2y " 4!6 ! 2y " 4
!2y " 10y " !5
The solution is (!2, !5).
Use the elimination method to solve the system of equations.3x ! 2y " 45x # 3y " !25
ExercisesExercises
Solve each system of equations by using elimination.
1. 2x ! y " 7 2. x ! 2y " 4 3. 3x # 4y " !10 4. 3x ! y " 123x # y " 8 !x # 6y " 12 x ! 4y " 2 5x # 2y " 20
(3, !1) (12, 4) (!2, !1) (4, 0)
5. 4x ! y " 6 6. 5x # 2y " 12 7. 2x # y " 8 8. 7x # 2y " !1
2x ! " 4 !6x ! 2y " !14 3x # y " 12 4x ! 3y " !13
no solution (2, 1) infinitely many (!1, 3)
9. 3x # 8y " !6 10. 5x # 4y " 12 11. !4x # y " !12 12. 5m # 2n " !8x ! y " 9 7x ! 6y " 40 4x # 2y " 6 4m # 3n " 2
(6, !3) (4, !2) ! , !2" (!4, 6)5$
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Skills PracticeSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 127 Glencoe Algebra 2
Less
on
3-2
Solve each system of equations by using substitution.
1. m # n " 20 2. x # 3y " !3 3. w ! z " 1m ! n " !4 (8, 12) 4x # 3y " 6 (3, !2) 2w # 3z " 12 (3, 2)
4. 3r # s " 5 5. 2b # 3c " !4 6. x ! y " 12r ! s " 5 (2, !1) b # c " 3 (13, !10) 2x # 3y " 12 (3, 2)
Solve each system of equations by using elimination.
7. 2p ! q " 5 8. 2j ! k " 3 9. 3c ! 2d " 23p # q " 5 (2, !1) 3j # k " 2 (1, !1) 3c # 4d " 50 (6, 8)
10. 2f # 3g " 9 11. !2x # y " !1 12. 2x ! y " 12f ! g " 2 (3, 1) x # 2y " 3 (1, 1) 2x ! y " 6 no solution
Solve each system of equations by using either substitution or elimination.
13. !r # t " 5 14. 2x ! y " !5 15. x ! 3y " !12!2r # t " 4 (1, 6) 4x # y " 2 !! , 4" 2x # y " 11 (3, 5)
16. 2p ! 3q " 6 17. 6w ! 8z " 16 18. c # d " 6!2p # 3q " !6 (3, 0) 3w ! 4z " 8 c ! d " 0 (3, 3)
infinitely many
19. 2u # 4v " !6 20. 3a # b " !1 21. 2x # y " 6u # 2v " 3 no solution !3a # b " 5 (!1, 2) 3x ! 2y " 16 (4, !2)
22. 3y ! z " !6 23. c # 2d " !2 24. 3r ! 2s " 1!3y ! z " 6 (!2, 0) !2c ! 5d " 3 (!4, 1) 2r ! 3s " 9 (!3, !5)
25. The sum of two numbers is 12. The difference of the same two numbers is !4.Find the numbers. 4, 8
26. Twice a number minus a second number is !1. Twice the second number added to three times the first number is 9. Find the two numbers. 1, 3
1$
© Glencoe/McGraw-Hill 128 Glencoe Algebra 2
Solve each system of equations by using substitution.
1. 2x # y " 4 2. x ! 3y " 9 3. g # 3h " 8 no3x # 2y " 1 (7, !10) x # 2y " !1 (3, !2) g # h " 9 solution
4. 2a ! 4b " 6 infinitely 5. 2m # n " 6 6. 4x ! 3y " !6!a # 2b " !3 many 5m # 6n " 1 (5, !4) !x ! 2y " 7 (!3, !2)
7. u ! 2v " 8. x ! 3y " 16 9. w # 3z " 1
!u # 2v " 5 no solution 4x ! y " 9 (1, !5) 3w ! 5z " !4 !! , "Solve each system of equations by using elimination.
10. 2r # s " 5 11. 2m ! n " !1 12. 6x # 3y " 63r ! s " 20 (5, !5) 3m # 2n " 30 (4, 9) 8x # 5y " 12 (!1, 4)
13. 3j ! k " 10 14. 2x ! y " !4 no 15. 2g # h " 64j ! k " 16 (6, 8) !4x # 2y " 6 solution 3g ! 2h " 16 (4, !2)
16. 2t # 4v " 6 infinitely 17. 3x ! 2y " 12 18. x # 3y " 11!t ! 2v " !3 many
2x # y " 14 (6, 3) 8x ! 5y " 17 (4, 3)
Solve each system of equations by using either substitution or elimination.
19. 8x # 3y " !5 20. 8q ! 15r " !40 21. 3x ! 4y " 12 infinitely10x # 6y " !13 ! , !3" 4q # 2r " 56 (10, 8) x ! y " many
22. 4b ! 2d " 5 no 23. s # 3y " 4 24. 4m ! 2p " 0!2b # d " 1 solution s " 1 (1, 1) !3m # 9p " 5 ! , "
25. 5g # 4k " 10 26. 0.5x # 2y " 5 27. h ! z " 3 no!3g ! 5k " 7 (6, !5) x ! 2y " !8 (!2, 3) !3h # 3z " 6 solution
SPORTS For Exercises 28 and 29, use the following information.Last year the volleyball team paid $5 per pair for socks and $17 per pair for shorts on atotal purchase of $315. This year they spent $342 to buy the same number of pairs of socksand shorts because the socks now cost $6 a pair and the shorts cost $18.
28. Write a system of two equations that represents the number of pairs of socks and shortsbought each year. 5x # 17y " 315, 6x # 18y " 342
29. How many pairs of socks and shorts did the team buy each year?socks: 12, shorts: 15
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Practice (Average)
Solving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Reading to Learn MathematicsSolving Systems of Equations Algebraically
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
© Glencoe/McGraw-Hill 129 Glencoe Algebra 2
Less
on
3-2
Pre-Activity How can systems of equations be used to make consumerdecisions?
Read the introduction to Lesson 3-2 at the top of page 116 in your textbook.
• How many more minutes of long distance time did Yolanda use inFebruary than in January? 13 minutes
• How much more were the February charges than the January charges?$1.04
• Using your answers for the questions above, how can you find the rateper minute? Find $1.04 % 13.
Reading the Lesson
1. Suppose that you are asked to solve the system of equations 4x ! 5y " 7at the right by the substitution method. 3x # y " !9
The first step is to solve one of the equations for one variable in terms of the other. To make your work as easy as possible,which equation would you solve for which variable? Explain.Sample answer: Solve the second equation for y because in thatequation the variable y has a coefficient of 1.
2. Suppose that you are asked to solve the system of equations 2x # 3y " !2at the right by the elimination method. 7x ! y " 39
To make your work as easy as possible, which variable would you eliminate? Describe how you would do this.Sample answer: Eliminate the variable y; multiply the second equationby 3 and then add the result to the first equation.
Helping You Remember
3. The substitution method and elimination method for solving systems both have severalsteps, and it may be difficult to remember them. You may be able to remember themmore easily if you notice what the methods have in common. What step is the same inboth methods?Sample answer: After finding the value of one of the variables, you findthe value of the other variable by substituting the value you have foundin one of the original equations.
© Glencoe/McGraw-Hill 130 Glencoe Algebra 2
Using CoordinatesFrom one observation point, the line of sight to a downed plane is given by y " x ! 1. This equation describes the distance from the observation point to the plane in a straight line. From another observation point, the line of sight is given by x # 3y " 21. What are the coordinates of the point at which the crash occurred?
Solve the system of equations ! y " x ! 1 .x # 3y " 21
x # 3y " 21x # 3(x ! 1) " 21 Substitute x ! 1 for y.
x # 3x ! 3 " 214x " 24x " 6
x # 3y " 216 # 3y " 21 Substitute 6 for x.
3y " 15y " 5
The coordinates of the crash are (6, 5).
Solve the following.
1. The lines of sight to a forest fire are as follows.
From Ranger Station A: 3x # y " 9From Ranger Station B: 2x # 3y " 13Find the coordinates of the fire.
2. An airplane is traveling along the line x ! y " !1 when it sees another airplane traveling along the line 5x # 3y " 19. If they continue along the same lines, at what point will their flight paths cross?
3. Two mine shafts are dug along the paths of the following equations.
x ! y " 14002x # y " 1300
If the shafts meet at a depth of 200 feet, what are the coordinates of the point at which they meet?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-23-2
Study Guide and InterventionSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 131 Glencoe Algebra 2
Less
on
3-3
Graph Systems of Inequalities To solve a system of inequalities, graph the inequalitiesin the same coordinate plane. The solution set is represented by the intersection of the graphs.
Solve the system of inequalities by graphing.y & 2x ! 1 and y ' # 2
The solution of y % 2x ! 1 is Regions 1 and 2.
The solution of y & # 2 is Regions 1 and 3.
The intersection of these regions is Region 1, which is the solution set of the system of inequalities.
Solve each system of inequalities by graphing.
1. x ! y % 2 2. 3x ! 2y % !1 3. y % 1x # 2y ' 1 x # 4y ' !12 x & 2
4. y ' ! 3 5. y ( # 2 6. y ' ! # 1
y ( 2x y ( !2x # 1 y ( 3x ! 1
7. x # y ' 4 8. x # 3y ( 3 9. x ! 2y & 62x ! y & 2 x ! 2y ' 4 x # 4y ( !4
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Region 3Region 1
Region 2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 132 Glencoe Algebra 2
Find Vertices of a Polygonal Region Sometimes the graph of a system ofinequalities forms a bounded region. You can find the vertices of the region by a combinationof the methods used earlier in this chapter: graphing, substitution, and/or elimination.
Find the coordinates of the vertices of the figure formed by 5x # 4y ( 20, y ( 2x # 3, and x ! 3y ( 4.
Graph the boundary of each inequality. The intersections of the boundary lines are the vertices of a triangle.The vertex (4, 0) can be determined from the graph. To find thecoordinates of the second and third vertices, solve the two systems of equations
y " 2x # 3 and y " 2x # 35x # 4y " 20 x ! 3y " 4
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Study Guide and Intervention (continued)
Solving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
For the first system of equations, rewrite the firstequation in standard form as 2x ! y " !3. Thenmultiply that equation by 4 and add to the secondequation.2x ! y " !3 Multiply by 4. 8x ! 4y " !125x # 4y " 20 (#) 5x # 4y " 20
13x " 8
x "
Then substitute x " in one of the original equations and solve for y.
2" # ! y " !3
! y " !3
y "
The coordinates of the second vertex are " , 4 #.3$13
8$13
55$13
16$13
8$13
8$13
8$13
For the second system ofequations, use substitution.Substitute 2x # 3 for y in thesecond equation to getx ! 3(2x # 3) " 4
x ! 6x ! 9 " 4!5x " 13
x " !
Then substitute x " ! in the
first equation to solve for y.
y " 2"! # # 3
y " ! # 3
y " !
The coordinates of the third
vertex are "!2 , !2 #.1$5
3$5
11$5
26$5
13$5
13$5
13$5
ExercisesExercises
Thus, the coordinates of the three vertices are (4, 0), " , 4 #, and "!2 , !2 #.
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
1. y % !3x # 7 2. x & !3 3. y ( ! x # 3
y ( x (2, 1), (!4, !2), y ( ! x # 3 (!3, 4), y & x # 1 (!2, 4), (2, 2),
y & !2(3, !2)
y & x ! 1(!3, !4), (3, 2)
y ( 3x # 10 !!3 , ! "4$
3$
1$2
1$3
1$2
1$2
1$5
3$5
3$13
8$13
ExampleExample
Skills PracticeSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 133 Glencoe Algebra 2
Less
on
3-3
Solve each system of inequalities by graphing.
1. x ( 1 2. x ' !3 3. x % 2y ' !1 y ' !3 x & 4 no solution
4. y ' x 5. y ( !4x 6. x ! y ' !1y ' !x y ' 3x ! 2 3x ! y % 4
7. y ( 3 8. y ( !2x # 3 9. x ! y % 4x # 2y ( 12 y % x ! 2 2x # y ( 4
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
10. y ( 0 11. y ( 3 ! x 12. x ' !2x ( 0 y ' 3 y & x ! 2y ' !x ! 1 x & !5 x # y % 2 (!2, 4), (0, 0), (0, !1), (!1, 0) (0, 3), (!5, 3), (!5, 8) (!2, !4), (2, 0)
x
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© Glencoe/McGraw-Hill 134 Glencoe Algebra 2
Solve each system of inequalities by graphing.
1. y # 1 ( !x 2. x & !2 3. y % 2x ! 3
y ' 1 2y ' 3x # 6 y % ! x # 2
4. x # y & !2 5. y % 1 6. 3y & 4x3x ! y ' !2 y ( x ! 1 2x ! 3y & !6
Find the coordinates of the vertices of the figure formed by each system ofinequalities.
7. y ' 1 ! x 8. x ! y % 2 9. y ' 2x ! 2y % x ! 1 x # y % 2 2x # 3y ' 6x % 3 x ' !2 y ( 4
(1, 0), (3, 2), (3, !2) (!2, 4), (!2, !4), (2, 0) (!3, 4), ! , 1", (3, 4)
DRAMA For Exercises 10 and 11, use the following information.The drama club is selling tickets to its play. An adult ticket costs $15 and a student ticket costs $11. The auditorium willseat 300 ticket-holders. The drama club wants to collect atleast $3630 from ticket sales.
10. Write and graph a system of four inequalities that describe how many of each type of ticket the club must sell to meets its goal.x ) 0, y ) 0, x # y & 300, 15x # 11y ) 3630
11. List three different combinations of tickets sold thatsatisfy the inequalities. Sample answer: 250 adult and 50 student, 200 adult and 100 student, 145 adult and 148 student
Play Tickets
Adult Tickets
Stu
den
t Ti
cket
s
1000 200 300 400
400
350
300
250
200
150
100
50
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Practice (Average)
Solving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Reading to Learn MathematicsSolving Systems of Inequalities by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
© Glencoe/McGraw-Hill 135 Glencoe Algebra 2
Less
on
3-3
Pre-Activity How can you determine whether your blood pressure is in a normal range?
Read the introduction to Lesson 3-3 at the top of page 123 in your textbook.
Satish is 37 years old. He has a blood pressure reading of 135/99. Is hisblood pressure within the normal range? Explain.Sample answer: No; his systolic pressure is normal, but hisdiastolic pressure is too high. It should be between 60 and 90.
Reading the Lesson
1. Without actually drawing the graph, describe the boundary lines x ( 3for the system of inequalities shown at the right. y % 5Two dashed vertical lines (x " 3 and x " !3) and two solid horizontallines (y " !5 and y " 5)
2. Think about how the graph would look for the system given above. What will be theshape of the shaded region? (It is not necessary to draw the graph. See if you canimagine it without drawing anything. If this is difficult to do, make a rough sketch tohelp you answer the question.)a rectangle
3. Which system of inequalities matches the graph shown at the right? BA. x ! y % !2 B. x ! y ' !2
x ! y & 2 x ! y ( 2
C. x # y % !2 D. x ! y & !2x # y & 2 x ! y % 2
Helping You Remember
4. To graph a system of inequalities, you must graph two or more boundary lines. Whenyou graph each of these lines, how can the inequality symbols help you rememberwhether to use a dashed or solid line?Use a dashed line if the inequality symbol is ' or (, because thesesymbols do not include equality and the dashed line reminds you thatthe line itself is not included in the graph. Use a solid line if the symbolis ) or &, because these symbols include equality and tell you that theline itself is included in the graph.
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© Glencoe/McGraw-Hill 136 Glencoe Algebra 2
Tracing StrategyTry to trace over each of the figures below without tracing the same segment twice.
The figure at the left cannot be traced, but the one at the right can. The rule is that a figure is traceable if it has no more than two points where an odd number of segments meet. The figure at the left has three segments meeting at each of the four corners. However, the figure at the right has only two points, L and Q, where an odd number of segments meet.
Determine if each figure can be traced without tracing the same segment twice. If it can, then name the starting point and name the segments in the order they should be traced.
1. 2.
yes; X; X#Y#, Y#F#, F#X#, X#G#, G#F#, yes; E; E#D#, D#A#, A#E#, E#B#, B#F#, F#C#,C#K#, F#E#, E#H#, H#X#, X#W#, W#H#, H#G# K#F#, F#J#, J#H#, H#F#, F#E#, E#H#, H#G#, G#E#
3. T
S R
U
P V
A
E FD K
B C
G JH
E F
H G
XW
Y
K
M
J LPQ
A B
D C
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-33-3
Study Guide and InterventionLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 137 Glencoe Algebra 2
Less
on
3-4
Maximum and Minimum Values When a system of linear inequalities produces abounded polygonal region, the maximum or minimum value of a related function will occurat a vertex of the region.
Graph the system of inequalities. Name the coordinates of thevertices of the feasible region. Find the maximum and minimum values of thefunction f(x, y) " 3x # 2y for this polygonal region.
y & 4y & !x # 6
y ) x !
y & 6x # 4First find the vertices of the bounded region. Graph the inequalities.The polygon formed is a quadrilateral with vertices at (0, 4), (2, 4), (5, 1), and (!1, !2). Use the table to find themaximum and minimum values of f(x, y) " 3x # 2y.
The maximum value is 17 at (5, 1). The minimum value is !7 at (!1, !2).
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. y ' 2 2. y ' !2 3. x # y ' 21 % x % 5 y ' 2x ! 4 4y % x # 8y % x # 3 x ! 2y ' !1 y ' 2x ! 5f(x, y) " 3x ! 2y f(x, y) " 4x ! y f(x, y) " 4x # 3y
vertices: (1, 2), (1, 4), vertices: (!5, !2), vertices (0, 2), (4, 3),(5, 8), (5, 2); max: 11; (3, 2), (1, !2); ! , ! "; max: 25; min: 6 min; !5 max: 10; min: !18
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(x, y ) 3x # 2y f (x, y )
(0, 4) 3(0) # 2(4) 8
(2, 4) 3(2) # 2(4) 14
(5, 1) 3(5) # 2(1) 17
(!1, !2) 3(!1) # 2(!2) !7
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3$2
1$2
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 138 Glencoe Algebra 2
Real-World Problems When solving linear programming problems, use thefollowing procedure.
1. Define variables.2. Write a system of inequalities.3. Graph the system of inequalities.4. Find the coordinates of the vertices of the feasible region.5. Write an expression to be maximized or minimized.6. Substitute the coordinates of the vertices in the expression.7. Select the greatest or least result to answer the problem.
A painter has exactly 32 units of yellow dye and 54 units of greendye. He plans to mix as many gallons as possible of color A and color B. Eachgallon of color A requires 4 units of yellow dye and 1 unit of green dye. Eachgallon of color B requires 1 unit of yellow dye and 6 units of green dye. Find themaximum number of gallons he can mix.
Step 1 Define the variables.x ! the number of gallons of color A madey ! the number of gallons of color B made
Step 2 Write a system of inequalities.Since the number of gallons made cannot benegative, x " 0 and y " 0.There are 32 units of yellow dye; each gallon of color A requires 4 units, and each gallon of color B requires 1 unit.So 4x # y $ 32.Similarly for the green dye, x # 6y $ 54.Steps 3 and 4 Graph the system of inequalities and find the coordinates of the vertices of the feasible region.The vertices of the feasible region are (0, 0), (0, 9), (6, 8), and (8, 0).Steps 5–7 Find the maximum number of gallons,x # y, that he can make.The maximum number of gallons the painter can make is 14,6 gallons of color A and 8 gallons of color B.
1. FOOD A delicatessen has 8 pounds of plain sausage and 10 pounds of garlic-flavoredsausage. The deli wants to make as much bratwurst as possible. Each pound of
bratwurst requires pound of plain sausage and pound of garlic-flavored sausage.
Find the maximum number of pounds of bratwurst that can be made. 10 lb2. MANUFACTURING Machine A can produce 30 steering wheels per hour at a cost of $16
per hour. Machine B can produce 40 steering wheels per hour at a cost of $22 per hour.At least 360 steering wheels must be made in each 8-hour shift. What is the least costinvolved in making 360 steering wheels in one shift? $194
2!3
1%4
3%4
(x, y) x " y f (x, y)
(0, 0) 0 # 0 0
(0, 9) 0 # 9 9
(6, 8) 6 # 8 14
(8, 0) 8 # 0 8
Color A (gallons)
Co
lor
B (
gal
lon
s)
5 10 15 20 25 30 35 40 45 50 550
40
35
30
25
20
15
10
5
(6, 8)
(8, 0)(0, 9)
Study Guide and Intervention (continued)
Linear Programming
NAME ______________________________________________ DATE______________ PERIOD _____
3-43-4
ExampleExample
ExercisesExercises
Skills PracticeLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 139 Glencoe Algebra 2
Less
on
3-4
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. x ' 2 2. x ' 1 3. x ' 0x % 5 y % 6 y ' 0y ' 1 y ' x ! 2 y % 7 ! xy % 4 f(x, y) " x ! y f(x, y) " 3x # yf(x, y) " x # y
max.: 9, min.: 3 max.: 2, min.: !5 max.: 21, min.: 0
4. x ' !1 5. y % 2x 6. y ' !x ! 2x # y % 6 y ' 6 ! x y ' 3x # 2f(x, y) " x # 2y y % 6 y % x # 4
f(x, y) " 4x # 3y f(x, y) " !3x # 5y
max.: 13, no min. no max., min.: 20 max.: 22, min.: !2
7. MANUFACTURING A backpack manufacturer produces an internal frame pack and anexternal frame pack. Let x represent the number of internal frame packs produced inone hour and let y represent the number of external frame packs produced in one hour.Then the inequalities x # 3y % 18, 2x # y % 16, x ' 0, and y ' 0 describe the constraintsfor manufacturing both packs. Use the profit function f(x) " 50x # 80y and theconstraints given to determine the maximum profit for manufacturing both backpacksfor the given constraints. $620
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© Glencoe/McGraw-Hill 140 Glencoe Algebra 2
Graph each system of inequalities. Name the coordinates of the vertices of thefeasible region. Find the maximum and minimum values of the given function forthis region.
1. 2x ! 4 % y 2. 3x ! y % 7 3. x ' 0!2x ! 4 % y 2x ! y ' 3 y ' 0y % 2 y ' x ! 3 y % 6f(x, y) " !2x # y f(x, y) " x ! 4y y % !3x # 15
f(x, y) " 3x # y
max.: 8, min.: !4 max.: 12, min.: !16 max.: 15, min.: 0
4. x % 0 5. y % 3x # 6 6. 2x # 3y ' 6y % 0 4y # 3x % 3 2x ! y % 24x # y ' !7 x ' !2 x ' 0f(x, y) " !x ! 4y f(x, y) " !x # 3y y ' 0
f(x, y) " x # 4y # 3
max.: 28, min.: 0 max.: , no min. no max., min.:
PRODUCTION For Exercises 7–9, use the following information.A glass blower can form 8 simple vases or 2 elaborate vases in an hour. In a work shift of nomore than 8 hours, the worker must form at least 40 vases.
7. Let s represent the hours forming simple vases and e the hours forming elaborate vases.Write a system of inequalities involving the time spent on each type of vase.s ) 0, e ) 0, s # e & 8, 8s # 2e ) 40
8. If the glass blower makes a profit of $30 per hour worked on the simple vases and $35per hour worked on the elaborate vases, write a function for the total profit on the vases.f(s, e) " 30s # 35e
9. Find the number of hours the worker should spend on each type of vase to maximizeprofit. What is that profit? 4 h on each; $260
17$
34$
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Practice (Average)
Linear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
Reading to Learn MathematicsLinear Programming
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
© Glencoe/McGraw-Hill 141 Glencoe Algebra 2
Less
on
3-4
Pre-Activity How is linear programming used in scheduling work?
Read the introduction to Lesson 3-4 at the top of page 129 in your textbook.
Name two or more facts that indicate that you will need to use inequalitiesto model this situation.Sample answer: The buoy tender can carry up to 8 new buoys.There seems to be a limit of 24 hours on the time the crew hasat sea. The crew will want to repair or replace the maximumnumber of buoys possible.
Reading the Lesson
1. Complete each sentence.
a. When you find the feasible region for a linear programming problem, you are solving
a system of linear called . The points
in the feasible region are of the system.
b. The corner points of a polygonal region are the of the feasible region.
2. A polygonal region always takes up only a limited part of the coordinate plane. One wayto think of this is to imagine a circle or rectangle that the region would fit inside. In thecase of a polygonal region, you can always find a circle or rectangle that is large enoughto contain all the points of the polygonal region. What word is used to describe a regionthat can be enclosed in this way? What word is used to describe a region that is too largeto be enclosed in this way? bounded; unbounded
3. How do you find the corner points of the polygonal region in a linear programmingproblem? You solve a system of two linear equations.
4. What are some everyday meanings of the word feasible that remind you of themathematical meaning of the term feasible region?Sample answer: possible or achievable
Helping You Remember
5. Look up the word constraint in a dictionary. If more than one definition is given, choosethe one that seems closest to the idea of a constraint in a linear programming problem.How can this definition help you to remember the meaning of constraint as it is used inthis lesson? Sample answer: A constraint is a restriction or limitation. Theconstraints in a linear programming problem are restrictions on thevariables that translate into inequality statements.
vertices
solutions
constraintsinequalities
© Glencoe/McGraw-Hill 142 Glencoe Algebra 2
Computer Circuits and LogicComputers operate according to the laws of logic. The circuits of a computer can be described using logic.
1. With switch A open, no current flows. The value 0 is assigned to an open switch.
2. With switch A closed, current flows. The value 1 is assigned to a closed switch.
3. With switches A and B open, no current flows. This circuit can be described by the conjunction, A ) B.
4. In this circuit, current flows if either A or B is closed. This circuit can be described by the disjunction, A # B.
Truth tables are used to describe the flow of current in a circuit.The table at the left describes the circuit in diagram 4. According to the table, the only time current does not flow through the circuit is when both switches A and B are open.
Draw a circuit diagram for each of the following.
1. (A ) B) # C 2. (A # B) ) C
3. (A # B) ) (C # D) 4. (A ) B) # (C ) D)
5. Construct a truth table for the following circuit.
B
C
A
A
A
A
B
A B
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-43-4
A B A # B
0 0 00 1 11 0 11 1 1
Study Guide and InterventionSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 143 Glencoe Algebra 2
Less
on
3-5
Systems in Three Variables Use the methods used for solving systems of linearequations in two variables to solve systems of equations in three variables. A system ofthree equations in three variables can have a unique solution, infinitely many solutions, orno solution. A solution is an ordered triple.
Solve this system of equations. 3x # y ! z " !62x ! y # 2z " 84x # y ! 3z " !21
Step 1 Use elimination to make a system of two equations in two variables.3x # y ! z " !6 First equation 2x ! y # 2z " 8 Second equation
(#) 2x ! y # 2z " 8 Second equation (#) 4x # y ! 3z " !21 Third equation
5x # z " 2 Add to eliminate y. 6x ! z " !13 Add to eliminate y.
Step 2 Solve the system of two equations.5x # z " 2
(#) 6x ! z " !1311x " !11 Add to eliminate z.
x " !1 Divide both sides by 11.
Substitute !1 for x in one of the equations with two variables and solve for z.5x # z " 2 Equation with two variables
5(!1) # z " 2 Replace x with !1.!5 # z " 2 Multiply.
z " 7 Add 5 to both sides.
The result so far is x " !1 and z " 7.
Step 3 Substitute !1 for x and 7 for z in one of the original equations with three variables.3x # y ! z " !6 Original equation with three variables
3(!1) # y ! 7 " !6 Replace x with !1 and z with 7.!3 # y ! 7 " !6 Multiply.
y " 4 Simplify.
The solution is (!1, 4, 7).
Solve each system of equations.
1. 2x # 3y ! z " 0 2. 2x ! y # 4z " 11 3. x ! 2y # z " 8x ! 2y ! 4z " 14 x # 2y ! 6z " !11 2x # y ! z " 03x # y ! 8z " 17 3x ! 2y !10z " 11 3x ! 6y # 3z " 24
(4, !3, !1) !2, !5, " infinitely manysolutions
4. 3x ! y ! z " 5 5. 2x ! 4y ! z " 10 6. x ! 6y # 4z " 23x # 2y ! z " 11 4x ! 8y ! 2z " 16 2x # 4y ! 8z " 166x ! 3y # 2z " !12 3x # y # z " 12 x ! 2y " 5
! , 2, !5" no solution !6, , ! "1$
1$
2$
1$
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 144 Glencoe Algebra 2
Real-World Problems
The Laredo Sports Shop sold 10 balls, 3 bats, and 2 bases for $99 onMonday. On Tuesday they sold 4 balls, 8 bats, and 2 bases for $78. On Wednesdaythey sold 2 balls, 3 bats, and 1 base for $33.60. What are the prices of 1 ball, 1 bat,and 1 base?
First define the variables.x " price of 1 bally " price of 1 batz " price of 1 base
Translate the information in the problem into three equations.
10x # 3y # 2z " 994x # 8y # 2z " 782x # 3y #z " 33.60
Study Guide and Intervention (continued)
Solving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
ExampleExample
Subtract the second equation from the firstequation to eliminate z.
10x # 3y # 2z " 99(!) 4x # 8y # 2z " 78
6x ! 5y " 21
Multiply the third equation by 2 andsubtract from the second equation.
4x # 8y # 2z " 78(!) 4x # 6y # 2z " 67.20
2y " 10.80y " 5.40
Substitute 5.40 for y in the equation 6x ! 5y " 21.
6x !5(5.40) " 216x " 48x " 8
Substitute 8 for x and 5.40 for y in one ofthe original equations to solve for z.
10x # 3y # 2z " 9910(8) # 3(5.40) # 2z " 99
80 # 16.20 # 2z " 992z " 2.80z " 1.40
So a ball costs $8, a bat $5.40, and a base $1.40.
1. FITNESS TRAINING Carly is training for a triathlon. In her training routine each week,she runs 7 times as far as she swims, and she bikes 3 times as far as she runs. One weekshe trained a total of 232 miles. How far did she run that week? 56 miles
2. ENTERTAINMENT At the arcade, Ryan, Sara, and Tim played video racing games, pinball,and air hockey. Ryan spent $6 for 6 racing games, 2 pinball games, and 1 game of airhockey. Sara spent $12 for 3 racing games, 4 pinball games, and 5 games of air hockey. Timspent $12.25 for 2 racing games, 7 pinball games, and 4 games of air hockey. How much dideach of the games cost? Racing game: $0.50; pinball: $0.75; air hockey: $1.50
3. FOOD A natural food store makes its own brand of trail mix out of dried apples, raisins,and peanuts. One pound of the mixture costs $3.18. It contains twice as much peanutsby weight as apples. One pound of dried apples costs $4.48, a pound of raisins $2.40, anda pound of peanuts $3.44. How many ounces of each ingredient are contained in 1 poundof the trail mix? 3 oz of apples, 7 oz of raisins, 6 oz of peanuts
ExercisesExercises
Skills PracticeSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 145 Glencoe Algebra 2
Less
on
3-5
Solve each system of equations.
1. 2a # c " !10 (5, !5, !20) 2. x # y # z " 3 (0, 2, 1)b ! c " 15 13x # 2z " 2a ! 2b # c " !5 !x ! 5z " !5
3. 2x # 5y # 2z " 6 (!3, 2, 1) 4. x # 4y ! z " 1 no solution5x ! 7y " !29 3x ! y # 8z " 0z " 1 x # 4y ! z " 10
5. !2z " !6 (2, !1, 3) 6. 3x ! 2y # 2z " !2 (!2, 1, 3)2x # 3y ! z " !2 x # 6y ! 2z " !2x # 2y # 3z " 9 x # 2y " 0
7. !x ! 5z " !5 (0, 0, 1) 8. !3r # 2t " 1 (1, !6, 2)y ! 3x " 0 4r # s ! 2t " !613x # 2z " 2 r # s # 4t " 3
9. x ! y # 3z " 3 no solution 10. 5m # 3n # p " 4 (!2, 3, 5)!2x # 2y ! 6z " 6 3m # 2n " 0y ! 5z " !3 2m ! n # 3p " 8
11. 2x # 2y # 2z " !2 infinitely many 12. x # 2y ! z " 4 (1, 2, 1)2x # 3y # 2z " 4 3x ! y # 2z " 3x # y # z " !1 !x # 3y # z " 6
13. 3x ! 2y # z " 1 (5, 7, 0) 14. 3x ! 5y # 2z " !12 infinitely many!x # y ! z " 2 x # 4y ! 2z " 85x # 2y # 10z " 39 !3x # 5y ! 2z " 12
15. 2x # y # 3z " !2 (!1, 3, !1) 16. 2x ! 4y # 3z " 0 (3, 0, !2)x ! y ! z " !3 x ! 2y ! 5z " 133x ! 2y # 3z " !12 5x # 3y ! 2z " 19
17. !2x # y # 2z " 2 (1, !2, 3) 18. x ! 2y # 2z " !1 infinitely many3x # 3y # z " 0 x # 2y ! z " 6x # y # z " 2 !3x # 6y ! 6z " 3
19. The sum of three numbers is 18. The sum of the first and second numbers is 15,and the first number is 3 times the third number. Find the numbers. 9, 6, 3
© Glencoe/McGraw-Hill 146 Glencoe Algebra 2
Solve each system of equations.
1. 2x ! y # 2z " 15 2. x ! 4y # 3z " !27 3. a # b " 3!x # y # z " 3 2x # 2y ! 3z " 22 !b # c " 33x ! y # 2z " 18 4z " !16 a # 2c " 10(3, 1, 5) (1, 4, !4) (2, 1, 4)
4. 3m ! 2n # 4p " 15 5. 2g # 3h ! 8j " 10 6. 2x # y ! z " !8m ! n # p " 3 g ! 4h " 1 4x ! y # 2z " !3m # 4n ! 5p " 0 !2g ! 3h # 8j " 5 !3x # y # 2z " 5(3, 3, 3) no solution (!2, !3, 1)
7. 2x ! 5y # z " 5 8. 2x # 3y # 4z " 2 9. p # 4r " !73x # 2y ! z " 17 5x ! 2y # 3z " 0 p ! 3q " !84x ! 3y # 2z " 17 x ! 5y ! 2z " !4 q # r " 1(5, 1, 0) (2, 2, !2) (1, 3, !2)
10. 4x # 4y ! 2z " 8 11. d # 3e # f " 0 12. 4x # y # 5z " !93x ! 5y # 3z " 0 !d # 2e # f " !1 x ! 4y ! 2z " !2 2x # 2y ! z " 4 4d # e ! f " 1 2x # 3y ! 2z " 21infinitely many (1, !1, 2) (2, 3, !4)
13. 5x # 9y # z " 20 14. 2x # y ! 3z " !3 15. 3x # 3y # z " 102x ! y ! z " !21 3x # 2y # 4z " 5 5x # 2y # 2z " 75x # 2y # 2z " !21 !6x ! 3y # 9z " 9 3x ! 2y # 3z " !9(!7, 6, 1) infinitely many (1, 3, !2)
16. 2u # v # w " 2 17. x # 5y ! 3z " !18 18. x ! 2y # z " !1!3u # 2v # 3w " 7 3x ! 2y # 5z " 22 !x # 2y ! z " 6 !u ! v # 2w " 7 !2x ! 3y # 8z " 28 !4y # 2z " 1(0, !1, 3) (1, !2, 3) no solution
19. 2x ! 2y ! 4z " !2 20. x ! y # 9z " !27 21. 2x ! 5y ! 3z " 73x ! 3y ! 6z " !3 2x ! 4y ! z " !1 !4x # 10y # 2z " 6!2x # 3y # z " 7 3x # 6y ! 3z " 27 6x ! 15y ! z " !19(4, 5, 0) (2, 2, !3) (1, 2, !5)
22. The sum of three numbers is 6. The third number is the sum of the first and secondnumbers. The first number is one more than the third number. Find the numbers.4, !1, 3
23. The sum of three numbers is !4. The second number decreased by the third is equal tothe first. The sum of the first and second numbers is !5. Find the numbers. !3, !2, 1
24. SPORTS Alexandria High School scored 37 points in a football game. Six points areawarded for each touchdown. After each touchdown, the team can earn one point for theextra kick or two points for a 2-point conversion. The team scored one fewer 2-pointconversions than extra kicks. The team scored 10 times during the game. How manytouchdowns were made during the game? 5
Practice (Average)
Solving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
Reading to Learn MathematicsSolving Systems of Equations in Three Variables
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
© Glencoe/McGraw-Hill 147 Glencoe Algebra 2
Less
on
3-5
Pre-Activity How can you determine the number and type of medals U.S.Olympians won?
Read the introduction to Lesson 3-5 at the top of page 138 in your textbook.
At the 1996 Summer Olympics in Atlanta, Georgia, the United States won101 medals. The U.S. team won 12 more gold medals than silver and 7fewer bronze medals than silver. Using the same variables as those in theintroduction, write a system of equations that describes the medals won forthe 1996 Summer Olympics.g # s # b " 101; g " s # 12; b " s ! 7
Reading the Lesson
1. The planes for the equations in a system of three linear equations in three variablesdetermine the number of solutions. Match each graph description below with thedescription of the number of solutions of the system. (Some of the items on the right maybe used more than once, and not all possible types of graphs are listed.)
a. three parallel planes I. one solution
b. three planes that intersect in a line II. no solutions
c. three planes that intersect in one point III. infinite solutions
d. one plane that represents all three equations
2. Suppose that three classmates, Monique, Josh, and Lilly, are studying for a quiz on thislesson. They work together on solving a system of equations in three variables, x, y, andz, following the algebraic method shown in your textbook. They first find that z " 3, thenthat y " !2, and finally that x " !1. The students agree on these values, but disagreeon how to write the solution. Here are their answers:
Monique: (3, !2, !1) Josh: (!2, !1, 3) Lilly: (!1, !2, 3)
a. How do you think each student decided on the order of the numbers in the orderedtriple? Sample answer: Monique arranged the values in the order inwhich she found them. Josh arranged them from smallest to largest.Lilly arranged them in alphabetical order of the variables.
b. Which student is correct? Lilly
Helping You Remember
3. How can you remember that obtaining the equation 0 " 0 indicates a system withinfinitely many solutions, while obtaining an equation such as 0 " 8 indicates a systemwith no solutions? 0 " 0 is always true, while 0 " 8 is never true.
III
I
III
II
© Glencoe/McGraw-Hill 148 Glencoe Algebra 2
BilliardsThe figure at the right shows a billiard table.The object is to use a cue stick to strike the ball at point C so that the ball will hit the sides (or cushions) of the table at least once before hitting the ball located at point A. In playing the game, you need to locate point P.
Step 1 Find point B so that " and $ . B is called the reflected
image of C in .
Step 2 Draw .
Step 3 intersects at the desired point P.
For each billiards problem, the cue ball at point C must strike the indicatedcushion(s) and then strike the ball at point A. Draw and label the correct path for the cue ball using the process described above.
1. cushion 2. cushion
3. cushion , then cushion 4. cushion , then cushion
C
A
RK
T S
C
ARK
T S
RSKTRSTS
C
ARK
T S
C
A
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RSKR
STAB
AB
STCHBH
STBC
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Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
3-53-5
©G
lencoe/McG
raw-H
illA
2G
lencoe Algebra 2
Answ
ers(Lesson 3-1)
Study Guide and InterventionSolving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 119 Glencoe Algebra 2
Less
on
3-1
Graph Systems of Equations A system of equations is a set of two or moreequations containing the same variables. You can solve a system of linear equations bygraphing the equations on the same coordinate plane. If the lines intersect, the solution isthat intersection point.
Solve the system of equations by graphing. x ! 2y " 4x # y " !2
Write each equation in slope-intercept form.x ! 2y " 4 → y " ! 2
x # y " !2 → y " !x ! 2
The graphs appear to intersect at (0, !2).
CHECK Substitute the coordinates into each equation.x ! 2y " 4 x # y " !2
0 ! 2(!2) ! 4 0 # (!2) ! !24 " 4 ✓ !2 " !2 ✓
The solution of the system is (0, !2).
Solve each system of equations by graphing.
1. y " ! # 1 2. y " 2x ! 2 3. y " ! # 3
y " ! 4 (6, !1) y " !x # 4 (2, 2) y " (4, 1)
4. 3x ! y " 0 5. 2x # " !7 6. ! y " 2
x ! y " !2 (1, 3) # y " 1 (!4, 3) 2x ! y " !1 (!2, !3)
(–2, –3)
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ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 120 Glencoe Algebra 2
Classify Systems of Equations The following chart summarizes the possibilities forgraphs of two linear equations in two variables.
Graphs of Equations Slopes of Lines Classification of System Number of Solutions
Lines intersect Different slopes Consistent and independent One
Lines coincide (same line)Same slope, same
Consistent and dependent Infinitely many y-intercept
Lines are parallelSame slope, different
Inconsistent None y-intercepts
Graph the system of equations x ! 3y " 6and describe it as consistent and independent, 2x ! y " !3consistent and dependent, or inconsistent.
Write each equation in slope-intercept form.
x ! 3y " 6 → y " x ! 2
2x ! y " !3 → y " 2x # 3
The graphs intersect at (!3, !3). Since there is one solution, the system is consistent and independent.
Graph the system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
1. 3x # y " !2 2. x # 2y " 5 3. 2x ! 3y " 06x # 2y " 10 3x ! 15 " !6y consistent 4x ! 6y " 3inconsistent and dependent inconsistent
4. 2x ! y " 3 5. 4x # y " !2 6. 3x ! y " 2x # 2y " 4 consistent 2x # " !1 consistent x # y " 6 consistent and independent and dependent and independent
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Study Guide and Intervention (continued)
Solving Systems of Equations by Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
ExercisesExercises
ExampleExample
©G
lencoe/McG
raw-H
illA
3G
lencoe Algebra 2
Answers
Answ
ers(Lesson 3-1)
Skills PracticeSolving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill 121 Glencoe Algebra 2
Less
on
3-1
Solve each system of equations by graphing.
1. x " 2 2. y " !3x # 6 3. y " 4 ! 3x
y " 0 (2, 0) y " 2x ! 4 (2, 0) y " ! x ! 1 (2, !2)
4. y " 4 ! x 5. y " !2x # 2 6. y " x
y " x ! 2 (3, 1) y " x ! 5 (3, !4) y " !3x # 4 (1, 1)
7. x # y " 3 8. x ! y " 4 9. 3x ! 2y " 4x ! y " 1 (2, 1) 2x ! 5y " 8 (4, 0) 2x ! y " 1 (!2, !5)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
10. y " !3x 11. y " x ! 5 12. 2x ! 5y " 10y " !3x # 2 !2x # 2y " !10 3x # y " 15
inconsistent consistent and consistent and dependent independent
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1$2
© Glencoe/McGraw-Hill 122 Glencoe Algebra 2
Solve each system of equations by graphing.
1. x ! 2y " 0 2. x # 2y " 4 3. 2x # y " 3
y " 2x ! 3 (2, 1) 2x ! 3y " 1 (2, 1) y " x ! (3, !3)
4. y ! x " 3 5. 2x ! y " 6 6. 5x ! y " 4y " 1 (!2, 1) x # 2y " !2 (2, !2) !2x # 6y " 4 (1, 1)
Graph each system of equations and describe it as consistent and independent,consistent and dependent, or inconsistent.
7. 2x ! y " 4 8. y " !x ! 2 9. 2y ! 8 " x
x ! y " 2 x # y " !4 y " x # 4
consistent and indep. inconsistent consistent and dep.SOFTWARE For Exercises 10–12, use the following information.Location Mapping needs new software. Software A costs $13,000 plus $500 per additionalsite license. Software B costs $2500 plus $1200 per additional site license.
10. Write two equations that represent the cost of each software. A: y " 13,000 # 500x,B: y " 2500 # 1200x
11. Graph the equations. Estimate the break-even point of the software costs.15 additional licenses
12. If Location Mapping plans to buy 10 additional site licenses, which software will cost less? B
(15, 20,500)
A
B
Software Costs
Additional Licenses
Tota
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Practice (Average)
Solving Systems of Equations By Graphing
NAME ______________________________________________ DATE ____________ PERIOD _____
3-13-1
© Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 3-1)
Rea
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o L
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Lesson 3-1
Pre-
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the
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Tip
on
page
110
of
your
tex
tboo
k sa
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hat
whe
n yo
u so
lve
a sy
stem
of
equa
tion
s by
gra
phin
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d fi
nd a
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nt o
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ters
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e tw
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the
ori
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l equ
atio
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hy is
it n
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gh t
och
eck
the
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f th
e eq
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ons?
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ple
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o be
a s
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of t
he s
yste
m,t
he o
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air
mus
tm
ake
both
of
the
equa
tions
tru
e.
2.U
nder
eac
h sy
stem
gra
phed
bel
ow,w
rite
all
of t
he f
ollo
win
g w
ords
tha
t ap
ply:
cons
iste
nt,
inco
nsis
tent
,dep
ende
nt,a
nd i
ndep
ende
nt.
inco
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tent
cons
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cons
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depe
nden
tin
depe
nden
t
Hel
pin
g Y
ou
Rem
emb
er
3.L
ook
up t
he w
ords
con
sist
ent
and
inco
nsis
tent
in a
dic
tion
ary.
How
can
the
mea
ning
of
thes
e w
ords
hel
p yo
u di
stin
guis
h be
twee
n co
nsis
tent
and
inco
nsis
tent
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tem
s of
equa
tion
s?S
ampl
e an
swer
:One
mea
ning
of
cons
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ntis
“be
ing
in a
gree
men
t,”or
“com
patib
le,”
whi
le o
ne m
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ng o
f in
cons
iste
ntis
“no
t be
ing
inag
reem
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or “
inco
mpa
tible
.”W
hen
a sy
stem
is c
onsi
sten
t,th
eeq
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ns a
re c
ompa
tible
bec
ause
bot
h ca
n be
tru
e at
the
sam
e tim
e (f
orth
e sa
me
valu
es o
f x
and
y).W
hen
a sy
stem
is in
cons
iste
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ns a
re in
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patib
le b
ecau
se t
hey
can
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r be
tru
e at
the
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me
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©G
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4G
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lgeb
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Inve
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The
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e va
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Are
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be
the
valu
e of
the
mut
ual f
und
afte
r 11
yea
rs o
f in
vest
men
t?$3
5,03
0
5.W
hat
will
be
the
valu
e of
the
ban
k ac
coun
t af
ter
11 y
ears
of
inve
stm
ent?
$36,
468
6.W
hen
will
the
mut
ual f
und
and
the
bank
acc
ount
hav
e eq
ual v
alue
?af
ter
11.6
7 ye
ars
of in
vest
men
t7.
Whi
ch in
vest
men
t ha
s th
e gr
eate
st p
ayof
f aft
er 1
1 ye
ars
of in
vest
men
t?th
e m
utua
l fun
d
A B25 203035 10 5 0
15
12
34
56
78
9
Amount Invested (thousands)
Year
s In
vest
ed
x
y
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-1
3-1
© Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-2)
Stu
dy
Gu
ide
and I
nte
rven
tion
Sol
ving
Sys
tem
s of
Equ
atio
ns A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
5G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Sub
stit
uti
on
To s
olve
a s
yste
m o
f lin
ear
equa
tion
s by
su
bsti
tuti
on,f
irst
sol
ve f
or o
neva
riab
le in
ter
ms
of t
he o
ther
in o
ne o
f th
e eq
uati
ons.
The
n su
bsti
tute
thi
s ex
pres
sion
into
the
othe
r eq
uati
on a
nd s
impl
ify.
Use
su
bsti
tuti
on t
o so
lve
the
syst
em o
f eq
uat
ion
s.2x
!y
"9
x#
3y"
!6
Solv
e th
e fi
rst
equa
tion
for
yin
ter
ms
of x
.2x
!y
"9
Firs
t equ
atio
n!
y"
!2x
#9
Sub
trac
t 2x
from
bot
h si
des.
y"
2x!
9M
ultip
ly b
oth
side
s by
!1.
Subs
titu
te t
he e
xpre
ssio
n 2x
!9
for
yin
to t
he s
econ
d eq
uati
on a
nd s
olve
for
x.
x#
3y"
!6
Sec
ond
equa
tion
x#
3(2x
!9)
"!
6S
ubst
itute
2x
!9
for
y.x
#6x
!27
"!
6D
istr
ibut
ive
Pro
pert
y7x
!27
"!
6S
impl
ify.
7x"
21A
dd 2
7 to
eac
h si
de.
x"
3D
ivid
e ea
ch s
ide
by 7
.
Now
,sub
stit
ute
the
valu
e 3
for
xin
eit
her
orig
inal
equ
atio
n an
d so
lve
for
y.2x
!y
"9
Firs
t equ
atio
n2(
3) !
y"
9R
epla
ce x
with
3.
6 !
y"
9S
impl
ify.
!y
"3
Sub
trac
t 6 fr
om e
ach
side
.y
"!
3M
ultip
ly e
ach
side
by
!1.
The
sol
utio
n of
the
sys
tem
is (
3,!
3).
Sol
ve e
ach
sys
tem
of
lin
ear
equ
atio
ns
by u
sin
g su
bsti
tuti
on.
1.3x
#y
"7
2.2x
#y
"5
3.2x
#3y
"!
34x
#2y
"16
3x!
3y"
3x
#2y
"2
(!1,
10)
(2,1
)(!
12,7
)
4.2x
!y
"7
5.4x
!3y
"4
6.5x
#y
"6
6x!
3y"
142x
#y
"!
83
!x
"0
no s
olut
ion
(!2,
!4)
(3,!
9)
7.x
#8y
"!
28.
2x!
y"
!4
9.x
!y
"!
2x
!3y
"20
4x#
y"
12x
!3y
"2
(14,
!2)
!!,3
"(!
8,!
6)
10.x
!4y
"4
11.x
#3y
"2
12.2
x#
2y"
42x
#12
y"
134x
#12
y"
8x
!2y
"0
!5,"
infin
itely
man
y!
,"
2 $ 34 $ 3
1 $ 4
1 $ 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill12
6G
lenc
oe A
lgeb
ra 2
Elim
inat
ion
To s
olve
a s
yste
m o
f lin
ear
equa
tion
s by
eli
min
atio
n,a
dd o
r su
btra
ct t
heeq
uati
ons
to e
limin
ate
one
of t
he v
aria
bles
.You
may
fir
st n
eed
to m
ulti
ply
one
or b
oth
of t
heeq
uati
ons
by a
con
stan
t so
tha
t on
e of
the
var
iabl
es h
as t
he s
ame
(or
oppo
site
) co
effi
cien
t in
one
equa
tion
as
it h
as in
the
oth
er.
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.2x
!4y
"!
263x
!y
"!
24
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sol
ving
Sys
tem
s of
Equ
atio
ns A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Mul
tipl
y th
e se
cond
equ
atio
n by
4.T
hen
subt
ract
the
equa
tion
s to
elim
inat
e th
e y
vari
able
.2x
!4y
"!
262x
!4y
"!
263x
!y
"!
24M
ultip
ly b
y 4.
12x
!4y
"!
96!
10x
"70
x"
!7
Rep
lace
xw
ith
!7
and
solv
e fo
r y.
2x!
4y"
!26
2(!
7) !
4y"
!26
!14
!4y
"!
26!
4y"
!12
y"
3T
he s
olut
ion
is (!
7,3)
.
Mul
tipl
y th
e fi
rst
equa
tion
by
3 an
d th
e se
cond
equa
tion
by
2.T
hen
add
the
equa
tion
s to
elim
inat
e th
e y
vari
able
.3x
!2y
"4
Mul
tiply
by
3.9x
!6y
"12
5x#
3y"
!25
Mul
tiply
by
2.10
x#
6y"
!50
19x
"!
38x
"!
2
Rep
lace
xw
ith
!2
and
solv
e fo
r y.
3x!
2y"
43(
!2)
!2y
"4
!6
!2y
"4
!2y
"10
y"
!5
The
sol
utio
n is
(!2,
!5)
.
Use
th
e el
imin
atio
n m
eth
od t
o so
lve
the
syst
em o
f eq
uat
ion
s.3x
!2y
"4
5x#
3y"
!25
Exer
cises
Exer
cises
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g el
imin
atio
n.
1.2x
!y
"7
2.x
!2y
"4
3.3x
#4y
"!
104.
3x!
y"
123x
#y
"8
!x
#6y
"12
x!
4y"
25x
#2y
"20
(3,!
1)(1
2,4)
(!2,
!1)
(4,0
)
5.4x
!y
"6
6.5x
#2y
"12
7.2x
#y
"8
8.7x
#2y
"!
1
2x!
"4
!6x
!2y
"!
143x
#y
"12
4x!
3y"
!13
no s
olut
ion
(2,1
)in
finite
ly m
any
(!1,
3)
9.3x
#8y
"!
610
.5x
#4y
"12
11.!
4x#
y"
!12
12.5
m#
2n"
!8
x!
y"
97x
!6y
"40
4x#
2y"
64m
#3n
"2
(6,!
3)(4
,!2)
!,!
2 "(!
4,6)
5 $ 2
3 $ 2y $ 2
© Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 3-2)
Skil
ls P
ract
ice
Sol
ving
Sys
tem
s of
Equ
atio
ns A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
7G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g su
bsti
tuti
on.
1.m
#n
"20
2.x
#3y
"!
33.
w!
z"
1m
!n
"!
4(8
,12)
4x#
3y"
6(3
,!2)
2w#
3z"
12(3
,2)
4.3r
#s
"5
5.2b
#3c
"!
46.
x!
y"
12r
!s
"5
(2,!
1)b
#c
"3
(13,
!10
)2x
#3y
"12
(3,2
)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g el
imin
atio
n.
7.2p
!q
"5
8.2j
!k
"3
9.3c
!2d
"2
3p#
q"
5(2
,!1)
3j#
k"
2(1
,!1)
3c#
4d"
50(6
,8)
10.2
f#
3g"
911
.!2x
#y
"!
112
.2x
!y
"12
f!
g"
2(3
,1)
x#
2y"
3(1
,1)
2x!
y"
6no
sol
utio
n
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g ei
ther
su
bsti
tuti
on o
r el
imin
atio
n.
13.!
r#
t"
514
.2x
!y
"!
515
.x!
3y"
!12
!2r
#t
"4
(1,6
)4x
#y
"2
!!,4
"2x
#y
"11
(3,5
)
16.2
p!
3q"
617
.6w
!8z
"16
18.c
#d
"6
!2p
#3q
"!
6(3
,0)
3w!
4z"
8c
!d
"0
(3,3
)in
finite
ly m
any
19.2
u#
4v"
!6
20.3
a#
b"
!1
21.2
x#
y"
6u
#2v
"3
no s
olut
ion
!3a
#b
"5
(!1,
2)3x
!2y
"16
(4,!
2)
22.3
y!
z"
!6
23.c
#2d
"!
224
.3r
!2s
"1
!3y
!z
"6
(!2,
0)!
2c!
5d"
3(!
4,1)
2r!
3s"
9(!
3,!
5)
25.T
he s
um o
f tw
o nu
mbe
rs is
12.
The
dif
fere
nce
of t
he s
ame
two
num
bers
is !
4.F
ind
the
num
bers
.4,
8
26.T
wic
e a
num
ber
min
us a
sec
ond
num
ber
is !
1.T
wic
e th
e se
cond
num
ber
adde
d to
thr
ee t
imes
the
fir
st n
umbe
r is
9.F
ind
the
two
num
bers
.1,
3
1 $ 2
©G
lenc
oe/M
cGra
w-H
ill12
8G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g su
bsti
tuti
on.
1.2x
#y
"4
2.x
!3y
"9
3.g
#3h
"8
no3x
#2y
"1
(7,!
10)
x#
2y"
!1
(3,!
2)g
#h
"9
solu
tion
4.2a
!4b
"6
infin
itely
5.2m
#n
"6
6.4x
!3y
"!
6!
a#
2b"
!3
man
y5m
#6n
"1
(5,!
4)!
x!
2y"
7(!
3,!
2)
7.u
!2v
"8.
x!
3y"
169.
w#
3z"
1
!u
#2v
"5
no s
olut
ion
4x!
y"
9(1
,!5)
3w!
5z"
!4
!!,
"S
olve
eac
h s
yste
m o
f eq
uat
ion
s by
usi
ng
elim
inat
ion
.
10.2
r#
s"
511
.2m
!n
"!
112
.6x
#3y
"6
3r!
s"
20(5
,!5)
3m#
2n"
30(4
,9)
8x#
5y"
12(!
1,4)
13.3
j!k
"10
14.2
x!
y"
!4
no
15.2
g#
h"
64j
!k
"16
(6,8
)!
4x#
2y"
6so
lutio
n3g
!2h
"16
(4,!
2)
16.2
t#
4v"
6in
finite
ly17
.3x
!2y
"12
18.
x#
3y"
11!
t!
2v"
!3
man
y2x
#y
"14
(6,3
)8x
!5y
"17
(4,3
)
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by u
sin
g ei
ther
su
bsti
tuti
on o
r el
imin
atio
n.
19.8
x#
3y"
!5
20.8
q!
15r
"!
4021
.3x
!4y
"12
infin
itely
10x
#6y
"!
13!
,!3 "
4q#
2r"
56(1
0,8)
x!
y"
man
y
22.4
b!
2d"
5no
23.s
#3y
"4
24.4
m!
2p"
0!
2b#
d"
1so
lutio
ns
"1
(1,1
)!
3m#
9p"
5!
,"
25.5
g#
4k"
1026
.0.5
x#
2y"
527
.h!
z"
3no
!3g
!5k
"7
(6,!
5)x
!2y
"!
8(!
2,3)
!3h
#3z
"6
solu
tion
SPO
RTS
For
Exe
rcis
es 2
8 an
d 2
9,u
se t
he
foll
owin
g in
form
atio
n.
Las
t ye
ar t
he v
olle
ybal
l tea
m p
aid
$5 p
er p
air
for
sock
s an
d $1
7 pe
r pa
ir f
or s
hort
s on
ato
tal p
urch
ase
of $
315.
Thi
s ye
ar t
hey
spen
t $3
42 t
o bu
y th
e sa
me
num
ber
of p
airs
of
sock
san
d sh
orts
bec
ause
the
soc
ks n
ow c
ost
$6 a
pai
r an
d th
e sh
orts
cos
t $1
8.
28.W
rite
a s
yste
m o
f tw
o eq
uati
ons
that
rep
rese
nts
the
num
ber
of p
airs
of
sock
s an
d sh
orts
boug
ht e
ach
year
.5x
#17
y"
315,
6x#
18y
"34
2
29.H
ow m
any
pair
s of
soc
ks a
nd s
hort
s di
d th
e te
am b
uy e
ach
year
?so
cks:
12,s
hort
s:15
2 $ 31 $ 3
4 $ 34 $ 9
1 $ 31 $ 2
2 $ 3
1 $ 2
1 $ 21 $ 2
1 $ 2
1 $ 3
Pra
ctic
e (A
vera
ge)
Sol
ving
Sys
tem
s of
Equ
atio
ns A
lgeb
raic
ally
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
© Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-2)
Rea
din
g t
o L
earn
Math
emati
csS
olvi
ng S
yste
ms
of E
quat
ions
Alg
ebra
ical
ly
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
©G
lenc
oe/M
cGra
w-H
ill12
9G
lenc
oe A
lgeb
ra 2
Lesson 3-2
Pre-
Act
ivit
yH
ow c
an s
yste
ms
of e
quat
ion
s be
use
d t
o m
ake
con
sum
erd
ecis
ion
s?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 3-
2 at
the
top
of
page
116
in y
our
text
book
.
•H
ow m
any
mor
e m
inut
es o
f lo
ng d
ista
nce
tim
e di
d Yo
land
a us
e in
Febr
uary
tha
n in
Jan
uary
?13
min
utes
•H
ow m
uch
mor
e w
ere
the
Febr
uary
cha
rges
tha
n th
e Ja
nuar
y ch
arge
s?$1
.04
•U
sing
you
r an
swer
s fo
r th
e qu
esti
ons
abov
e,ho
w c
an y
ou f
ind
the
rate
per
min
ute?
Find
$1.
04 %
13.
Rea
din
g t
he
Less
on
1.Su
ppos
e th
at y
ou a
re a
sked
to
solv
e th
e sy
stem
of
equa
tion
s 4x
!5y
"7
at t
he r
ight
by
the
subs
titu
tion
met
hod.
3x#
y"
!9
The
fir
st s
tep
is t
o so
lve
one
of t
he e
quat
ions
for
one
var
iabl
e in
ter
ms
of t
he o
ther
.To
mak
e yo
ur w
ork
as e
asy
as p
ossi
ble,
whi
ch e
quat
ion
wou
ld y
ou s
olve
for
whi
ch v
aria
ble?
Exp
lain
.S
ampl
e an
swer
:Sol
ve t
he s
econ
d eq
uatio
n fo
r y
beca
use
in t
hat
equa
tion
the
vari
able
yha
s a
coef
ficie
nt o
f 1.
2.Su
ppos
e th
at y
ou a
re a
sked
to
solv
e th
e sy
stem
of
equa
tion
s 2x
#3y
"!
2at
the
rig
ht b
y th
e el
imin
atio
n m
etho
d.7x
!y
"39
To m
ake
your
wor
k as
eas
y as
pos
sibl
e,w
hich
var
iabl
e w
ould
yo
u el
imin
ate?
Des
crib
e ho
w y
ou w
ould
do
this
.S
ampl
e an
swer
:Elim
inat
e th
e va
riab
le y
;mul
tiply
the
sec
ond
equa
tion
by3
and
then
add
the
res
ult
to t
he f
irst
equ
atio
n.
Hel
pin
g Yo
u R
emem
ber
3.T
he s
ubst
itut
ion
met
hod
and
elim
inat
ion
met
hod
for
solv
ing
syst
ems
both
hav
e se
vera
lst
eps,
and
it m
ay b
e di
ffic
ult
to r
emem
ber
them
.You
may
be
able
to
rem
embe
r th
emm
ore
easi
ly if
you
not
ice
wha
t th
e m
etho
ds h
ave
in c
omm
on.W
hat
step
is t
he s
ame
inbo
th m
etho
ds?
Sam
ple
answ
er:A
fter
fin
ding
the
val
ue o
f on
e of
the
var
iabl
es,y
ou f
ind
the
valu
e of
the
oth
er v
aria
ble
by s
ubst
itutin
g th
e va
lue
you
have
foun
din
one
of
the
orig
inal
equ
atio
ns.
©G
lenc
oe/M
cGra
w-H
ill13
0G
lenc
oe A
lgeb
ra 2
Usi
ng C
oord
inat
esF
rom
one
obs
erva
tion
poi
nt,t
he li
ne o
f si
ght
to a
dow
ned
plan
e is
giv
en b
y y
"x
!1.
Thi
s eq
uati
on d
escr
ibes
the
dis
tanc
e fr
om t
he o
bser
vati
on p
oint
to
the
pla
ne in
a s
trai
ght
line.
Fro
m a
noth
er o
bser
vati
on p
oint
,the
line
of
sigh
t is
giv
en b
y x
#3y
"21
.Wha
t ar
e th
e co
ordi
nate
s of
the
poi
nt a
t w
hich
th
e cr
ash
occu
rred
?
Solv
e th
e sy
stem
of
equa
tion
s!y
"x
!1
.x
#3y
"21
x#
3y"
21x
#3(
x!
1) "
21S
ubst
itute
x!
1 fo
r y.
x#
3x!
3 "
214x
"24
x"
6
x#
3y"
216
#3y
"21
Sub
stitu
te 6
for
x.
3y"
15y
"5
The
coo
rdin
ates
of
the
cras
h ar
e (6
,5).
Sol
ve t
he
foll
owin
g.
1.T
he li
nes
of s
ight
to
a fo
rest
fir
e ar
e as
fol
low
s.
Fro
m R
ange
r St
atio
n A
:3x
#y
"9
Fro
m R
ange
r St
atio
n B
:2x
#3y
"13
Fin
d th
e co
ordi
nate
s of
the
fir
e.(2
,3)
2.A
n ai
rpla
ne is
tra
velin
g al
ong
the
line
x!
y"
!1
whe
n it
see
s an
othe
r ai
rpla
ne t
rave
ling
alon
g th
e lin
e 5x
#3y
"19
.If
they
con
tinu
e al
ong
the
sam
e lin
es,a
t w
hat
poin
t w
ill t
heir
flig
ht p
aths
cro
ss?
(2,3
)
3.T
wo
min
e sh
afts
are
dug
alo
ng t
he p
aths
of
the
follo
win
g eq
uati
ons.
x!
y"
1400
2x#
y"
1300
If t
he s
haft
s m
eet
at a
dep
th o
f 20
0 fe
et,w
hat
are
the
coor
dina
tes
of t
he
poin
t at
whi
ch t
hey
mee
t?(9
00,!
500)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-2
3-2
© Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 3-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Sol
ving
Sys
tem
s of
Ineq
ualit
ies
by G
raph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
1G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Gra
ph S
yste
ms
of In
equa
litie
sTo
sol
ve a
sys
tem
of i
nequ
alit
ies,
grap
h th
e in
equa
litie
sin
the
sam
e co
ordi
nate
pla
ne.T
he s
olut
ion
set
is r
epre
sent
ed b
y th
e in
ters
ecti
on o
f the
gra
phs.
Sol
ve t
he
syst
em o
f in
equ
alit
ies
by g
rap
hin
g.y
&2x
!1
and
y'
#2
The
sol
utio
n of
y%
2x!
1 is
Reg
ions
1 a
nd 2
.
The
sol
utio
n of
y&
#2
is R
egio
ns 1
and
3.
The
inte
rsec
tion
of
thes
e re
gion
s is
Reg
ion
1,w
hich
is
the
solu
tion
set
of
the
syst
em o
f in
equa
litie
s.
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.x
!y
%2
2.3x
!2y
%!
13.
y
%1
x#
2y'
1x
#4y
'!
12x
&2
4.y
'!
35.
y(
#2
6.y
'!
#1
y(
2xy
(!
2x#
1y
(3x
!1
7.x
#y
'4
8.x
#3y
(3
9.x
!2y
&6
2x!
y&
2x
!2y
'4
x#
4y(
!4
x
y
Ox
y
O
x
y
O
x
y
Ox
y
O
x
y
O
x $ 4x $ 3
x $ 2
x
y
Ox
y
Ox
y
O
x $ 3x $ 3
x
y
O
Regi
on 3
Regi
on 1
Regi
on 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
2G
lenc
oe A
lgeb
ra 2
Fin
d V
erti
ces
of
a Po
lyg
on
al R
egio
nSo
met
imes
the
gra
ph o
f a
syst
em o
fin
equa
litie
s fo
rms
a bo
unde
d re
gion
.You
can
fin
d th
e ve
rtic
es o
f th
e re
gion
by
a co
mbi
nati
onof
the
met
hods
use
d ea
rlie
r in
thi
s ch
apte
r:gr
aphi
ng,s
ubst
itut
ion,
and/
or e
limin
atio
n.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y 5x
#4y
(20
,y(
2x#
3,an
d x
!3y
(4.
Gra
ph t
he b
ound
ary
of e
ach
ineq
ualit
y.T
he in
ters
ecti
ons
of t
he
boun
dary
line
s ar
e th
e ve
rtic
es o
f a
tria
ngle
.T
he v
erte
x (4
,0)
can
be d
eter
min
ed f
rom
the
gra
ph.T
o fi
nd t
heco
ordi
nate
s of
the
sec
ond
and
thir
d ve
rtic
es,s
olve
the
tw
o sy
stem
s of
equ
atio
ns
y"
2x#
3an
dy
"2x
#3
5x#
4y"
20x
!3y
"4
x
y
O
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sol
ving
Sys
tem
s of
Ineq
ualit
ies
by G
raph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
For
the
firs
t sy
stem
of
equa
tion
s,re
wri
te t
he f
irst
equa
tion
in s
tand
ard
form
as
2x!
y"
!3.
The
nm
ulti
ply
that
equ
atio
n by
4 a
nd a
dd t
o th
e se
cond
equa
tion
.2x
!y
"!
3M
ultip
ly b
y 4.
8x!
4y"
!12
5x#
4y"
20(#
)5x
#4y
"20
13x
"8
x"
The
n su
bsti
tute
x"
in o
ne o
f th
e or
igin
al e
quat
ions
an
d so
lve
for
y.
2 "#!
y"
!3
!y
"!
3
y"
The
coo
rdin
ates
of
the
seco
nd v
erte
x ar
e "
,4#.
3 $ 138 $ 13
55 $ 13
16 $ 138 $ 13
8 $ 13
8 $ 13
For
the
seco
nd s
yste
m o
feq
uati
ons,
use
subs
titu
tion
.Su
bsti
tute
2x
#3
for
yin
the
seco
nd e
quat
ion
to g
etx
!3(
2x#
3) "
4x
!6x
!9
"4
!5x
"13
x"
!
The
n su
bsti
tute
x"
!in
the
firs
t eq
uati
on t
o so
lve
for
y.
y"
2 "!##
3
y"
!#
3
y"
!
The
coo
rdin
ates
of
the
thir
d
vert
ex a
re "!
2,!
2#.
1 $ 53 $ 5
11 $ 526 $ 5
13 $ 5
13 $ 5
13 $ 5
Exer
cises
Exer
cises
Thu
s,th
e co
ordi
nate
s of
the
thr
ee v
erti
ces
are
(4,0
),"
,4#,a
nd "!
2,!
2#.
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
1.y
%!
3x#
72.
x&
!3
3.y
(!
x#
3
y(
x(2
,1),
(!4,
!2)
,y
(!
x#
3(!
3,4)
,y
&x
#1
(!2,
4),(
2,2)
,
y&
!2
(3,!
2)y
&x
!1
(!3,
!4)
,(3,
2)y
(3x
#10
!!3
,!"
4 $ 53 $ 5
1 $ 21 $ 3
1 $ 2
1 $ 2
1 $ 53 $ 5
3 $ 138 $ 13
Exam
ple
Exam
ple
© Glencoe/McGraw-Hill A9 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-3)
Skil
ls P
ract
ice
Sol
ving
Sys
tem
s of
Ineq
ualit
ies
by G
raph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
3G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.x
(1
2.x
'!
33.
x%
2y
'!
1y
'!
3x
&4
no s
olut
ion
4.y
'x
5.y
(!
4x6.
x!
y'
!1
y'
!x
y'
3x!
23x
!y
%4
7.y
(3
8.y
(!
2x#
39.
x!
y%
4x
#2y
(12
y%
x!
22x
#y
(4
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
10.y
(0
11.y
(3
!x
12.x
'!
2x
(0
y'
3y
&x
!2
y'
!x
!1
x&
!5
x#
y%
2 (!
2,4)
,(0
,0),
(0,!
1),(
!1,
0)(0
,3),
(!5,
3),(
!5,
8)(!
2,!
4),(
2,0)
x
y
Ox
y
Ox
y
O2
2
x
y
Ox
y
Ox
y
O
x
y
Ox
y
Ox
y
O
©G
lenc
oe/M
cGra
w-H
ill13
4G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
ineq
ual
itie
s by
gra
ph
ing.
1.y
#1
(!
x2.
x&
!2
3.y
%2x
!3
y'
12y
'3x
#6
y%
!x
#2
4.x
#y
&!
25.
y
%1
6.3y
&4x
3x!
y'
!2
y(
x!
12x
!3y
&!
6
Fin
d t
he
coor
din
ates
of
the
vert
ices
of
the
figu
re f
orm
ed b
y ea
ch s
yste
m o
fin
equ
alit
ies.
7.y
'1
!x
8.x
!y
%2
9.y
'2x
!2
y%
x!
1x
#y
%2
2x#
3y'
6x
%3
x'
!2
y(
4
(1,0
),(3
,2),
(3,!
2)(!
2,4)
,(!
2,!
4),(
2,0)
(!3,
4), !
,1",(
3,4)
DR
AM
AF
or E
xerc
ises
10
and
11,
use
th
e fo
llow
ing
info
rmat
ion
.T
he d
ram
a cl
ub is
sel
ling
tick
ets
to it
s pl
ay.A
n ad
ult
tick
et
cost
s $1
5 an
d a
stud
ent
tick
et c
osts
$11
.The
aud
itor
ium
will
seat
300
tic
ket-
hold
ers.
The
dra
ma
club
wan
ts t
o co
llect
at
leas
t $3
630
from
tic
ket
sale
s.
10.W
rite
and
gra
ph a
sys
tem
of
four
ineq
ualit
ies
that
de
scri
be h
ow m
any
of e
ach
type
of
tick
et t
he c
lub
mus
t se
ll to
mee
ts it
s go
al.
x)
0,y
)0,
x#
y&
300,
15x
#11
y)
3630
11.L
ist
thre
e di
ffer
ent
com
bina
tion
s of
tic
kets
sol
d th
atsa
tisf
y th
e in
equa
litie
s.S
ampl
e an
swer
:25
0 ad
ult
and
50 s
tude
nt,2
00 a
dult
and
100
stud
ent,
145
adul
t an
d 14
8 st
uden
t
Play
Tic
kets
Ad
ult
Tic
kets
Student Tickets
100
020
030
040
0
400
350
300
250
200
150
100 50
3 $ 2
x
y O
y
xO
y
xO
x
y
Ox
y
Ox
y
O
1 $ 2
Pra
ctic
e (A
vera
ge)
Sol
ving
Sys
tem
s of
Ineq
ualit
ies
by G
raph
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
© Glencoe/McGraw-Hill A10 Glencoe Algebra 2
Answers (Lesson 3-3)
Rea
din
g t
o L
earn
Math
emati
csS
olvi
ng S
yste
ms
of In
equa
litie
s by
Gra
phin
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
©G
lenc
oe/M
cGra
w-H
ill13
5G
lenc
oe A
lgeb
ra 2
Lesson 3-3
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e w
het
her
you
r bl
ood
pre
ssu
re i
s in
a
nor
mal
ran
ge?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 3-
3 at
the
top
of
page
123
in y
our
text
book
.
Sati
sh is
37
year
s ol
d.H
e ha
s a
bloo
d pr
essu
re r
eadi
ng o
f 13
5/99
.Is
his
bloo
d pr
essu
re w
ithi
n th
e no
rmal
ran
ge?
Exp
lain
.S
ampl
e an
swer
:No;
his
syst
olic
pre
ssur
e is
nor
mal
,but
his
dias
tolic
pre
ssur
e is
too
hig
h.It
shou
ld b
e be
twee
n 60
and
90.
Rea
din
g t
he
Less
on
1.W
itho
ut a
ctua
lly d
raw
ing
the
grap
h,de
scri
be t
he b
ound
ary
lines
x
(
3fo
r th
e sy
stem
of
ineq
ualit
ies
show
n at
the
rig
ht.
y
%5
Two
dash
ed v
ertic
al li
nes
(x"
3 an
d x
"!
3) a
nd t
wo
solid
hor
izon
tal
lines
(y
"!
5 an
d y
"5)
2.T
hink
abo
ut h
ow t
he g
raph
wou
ld lo
ok f
or t
he s
yste
m g
iven
abo
ve.W
hat
will
be
the
shap
e of
the
sha
ded
regi
on?
(It
is n
ot n
eces
sary
to
draw
the
gra
ph.S
ee if
you
can
imag
ine
it w
itho
ut d
raw
ing
anyt
hing
.If
this
is d
iffi
cult
to
do,m
ake
a ro
ugh
sket
ch t
ohe
lp y
ou a
nsw
er t
he q
uest
ion.
)a
rect
angl
e
3.W
hich
sys
tem
of
ineq
ualit
ies
mat
ches
the
gra
ph s
how
n at
the
rig
ht?
BA
.x
!y
%!
2B
.x
!y
'!
2x
!y
&2
x!
y(
2
C.
x#
y%
!2
D.
x!
y&
!2
x#
y&
2x
!y
%2
Hel
pin
g Y
ou
Rem
emb
er
4.To
gra
ph a
sys
tem
of
ineq
ualit
ies,
you
mus
t gr
aph
two
or m
ore
boun
dary
line
s.W
hen
you
grap
h ea
ch o
f th
ese
lines
,how
can
the
ineq
ualit
y sy
mbo
ls h
elp
you
rem
embe
rw
heth
er t
o us
e a
dash
ed o
r so
lid li
ne?
Use
a d
ashe
d lin
e if
the
ineq
ualit
y sy
mbo
l is
'or
(,b
ecau
se t
hese
sym
bols
do
not
incl
ude
equa
lity
and
the
dash
ed li
ne r
emin
ds y
ou t
hat
the
line
itsel
f is
not
incl
uded
in t
he g
raph
.Use
a s
olid
line
if t
he s
ymbo
lis
)or
&,b
ecau
se t
hese
sym
bols
incl
ude
equa
lity
and
tell
you
that
the
line
itsel
f is
incl
uded
in t
he g
raph
.
x
y
O
©G
lenc
oe/M
cGra
w-H
ill13
6G
lenc
oe A
lgeb
ra 2
Trac
ing
Str
ateg
yT
ry t
o tr
ace
over
eac
h of
the
fig
ures
bel
ow w
itho
ut t
raci
ng t
he s
ame
segm
ent
twic
e.
The
fig
ure
at t
he le
ft c
anno
t be
tra
ced,
but
the
one
at t
he r
ight
can
.The
rul
e is
tha
t a
figu
re is
tra
ceab
le if
it h
as n
o m
ore
than
tw
o po
ints
whe
re a
n od
d nu
mbe
r of
se
gmen
ts m
eet.
The
fig
ure
at t
he le
ft h
as t
hree
seg
men
ts m
eeti
ng a
t ea
ch o
f th
e fo
ur c
orne
rs.H
owev
er,t
he f
igur
e at
the
rig
ht h
as o
nly
two
poin
ts,L
and
Q,w
here
an
odd
num
ber
of s
egm
ents
mee
t.
Det
erm
ine
if e
ach
fig
ure
can
be
trac
ed w
ith
out
trac
ing
the
sam
e se
gmen
t tw
ice.
If i
t ca
n,t
hen
nam
e th
e st
arti
ng
poi
nt
and
nam
e th
e se
gmen
ts i
n
the
ord
er t
hey
sh
ould
be
trac
ed.
1.2.
yes;
X;X#
Y#,Y#F#
,F#X#,
X#G#,G#
F#,ye
s;E
;E#D#
,D#A#
,A#E#,
E#B#,B#
F#,F#C#
,C#K#
,F #E#
,E#H#
,H#X#,
X#W#,W#
H#,H#
G#K#
F#,F#J#
,J#H#
,H#F#,
F#E#,E#
H#,H#
G#,G#
E#
3.
not
trac
eabl
e
T SRU
PV
A
EF
DK
BC
GJ
H
EF
HG
XW
Y
K M
JL
PQ
AB
DC
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-3
3-3
© Glencoe/McGraw-Hill A11 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-4)
Stu
dy
Gu
ide
an
d I
nte
rven
tion
Line
ar P
rogr
amm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill13
7G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Max
imu
m a
nd
Min
imu
m V
alu
esW
hen
a sy
stem
of
linea
r in
equa
litie
s pr
oduc
es a
boun
ded
poly
gona
l reg
ion,
the
max
imum
or m
inim
umva
lue
of a
rel
ated
fun
ctio
n w
ill o
ccur
at a
ver
tex
of t
he r
egio
n.
Gra
ph
th
e sy
stem
of
ineq
ual
itie
s.N
ame
the
coor
din
ates
of
the
vert
ices
of
the
feas
ible
reg
ion
.Fin
d t
he
max
imu
m a
nd
min
imu
m v
alu
es o
f th
efu
nct
ion
f(x
,y)
!3x
"2y
for
this
pol
ygon
al r
egio
n.
y#
4y
#$
x"
6
y%
x$
y#
6x"
4F
irst
fin
d th
e ve
rtic
es o
f th
e bo
unde
d re
gion
.Gra
ph
the
ineq
ualit
ies.
The
pol
ygon
for
med
is a
qua
drila
tera
l wit
h ve
rtic
es a
t (0
,4),
(2,4
),(5
,1),
and
(!1,
!2)
.Use
the
tab
le t
o fi
nd t
hem
axim
um a
nd m
inim
um v
alue
s of
f(x
,y) "
3x#
2y.
The
max
imum
val
ue is
17
at (
5,1)
.The
min
imum
val
ue is
!7
at (
!1,
!2)
.
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.y
$2
2.y
$!
23.
x#
y$
21
%x
%5
y$
2x!
44y
%x
#8
y%
x#
3x
!2y
$!
1y
$2x
!5
f(x,
y) "
3x!
2yf(
x,y)
"4x
!y
f(x,
y) "
4x#
3y
vert
ices
:(1,
2),(
1,4)
,ve
rtic
es:(
$5,
$2)
,ve
rtic
es (
0,2)
,(4,
3),
(5,8
),(5
,2);
max
:11;
(3,2
),(1
,$2)
;!
,$";m
ax:2
5;m
in:6
m
in;$
5m
ax:1
0;m
in:$
181 & 3
7 & 3
x
y
Ox
y
O
x
y
O
(x,y
)3x
"2y
f(x,
y)
(0, 4
)3(
0) #
2(4)
8
(2, 4
)3(
2) #
2(4)
14
(5, 1
)3(
5) #
2(1)
17
(!1,
!2)
3(!
1) #
2(!
2)!
7
x
y O
3 & 21 & 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill13
8G
lenc
oe A
lgeb
ra 2
Rea
l-W
orl
d P
rob
lem
sW
hen
solv
ing
lin
ear
pro
gram
min
gpr
oble
ms,
use
the
follo
win
g pr
oced
ure.
1.D
efin
e va
riab
les.
2.W
rite
a s
yste
m o
f in
equa
litie
s.3.
Gra
ph t
he s
yste
m o
f in
equa
litie
s.4.
Fin
d th
e co
ordi
nate
s of
the
ver
tice
s of
the
fea
sibl
e re
gion
.5.
Wri
te a
n ex
pres
sion
to
be m
axim
ized
or
min
imiz
ed.
6.Su
bsti
tute
the
coo
rdin
ates
of
the
vert
ices
in t
he e
xpre
ssio
n.7.
Sele
ct t
he g
reat
est
or le
ast
resu
lt t
o an
swer
the
pro
blem
.
A p
ain
ter
has
exa
ctly
32
un
its
of y
ello
w d
ye a
nd
54
un
its
of g
reen
dye
.He
pla
ns
to m
ix a
s m
any
gall
ons
as p
ossi
ble
of c
olor
A a
nd
col
or B
.Eac
hga
llon
of
colo
r A
req
uir
es 4
un
its
of y
ello
w d
ye a
nd
1 u
nit
of
gree
n d
ye.E
ach
gall
on o
f co
lor
B r
equ
ires
1 u
nit
of
yell
ow d
ye a
nd
6 u
nit
s of
gre
en d
ye.F
ind
th
em
axim
um
nu
mbe
r of
gal
lon
s h
e ca
n m
ix.
Step
1D
efin
e th
e va
riab
les.
x"
the
num
ber
of g
allo
ns o
f co
lor
A m
ade
y"
the
num
ber
of g
allo
ns o
f co
lor
B m
ade
Step
2W
rite
a s
yste
m o
f in
equa
litie
s.Si
nce
the
num
ber
of g
allo
ns m
ade
cann
ot b
ene
gati
ve,x
$0
and
y$
0.T
here
are
32
unit
s of
yel
low
dye
;eac
h ga
llon
of
colo
r A
req
uire
s 4
unit
s,an
d ea
ch g
allo
n of
co
lor
B r
equi
res
1 un
it.
So 4
x#
y%
32.
Sim
ilarl
y fo
r th
e gr
een
dye,
x#
6y%
54.
Step
s 3
and
4G
raph
the
sys
tem
of
ineq
ualit
ies
and
find
the
coo
rdin
ates
of
the
vert
ices
of
the
feas
ible
reg
ion.
The
ver
tice
s of
the
fea
sibl
e re
gion
are
(0,
0),(
0,9)
,(6,
8),a
nd (
8,0)
.St
eps
5–7
Fin
d th
e m
axim
um n
umbe
r of
gal
lons
,x
#y,
that
he
can
mak
e.T
he m
axim
um n
umbe
r of
gal
lons
the
pai
nter
can
mak
e is
14,
6 ga
llons
of
colo
r A
and
8 g
allo
ns o
f co
lor
B.
1.FO
OD
A d
elic
ates
sen
has
8 po
unds
of
plai
n sa
usag
e an
d 10
pou
nds
of g
arlic
-fla
vore
dsa
usag
e.T
he d
eli w
ants
to
mak
e as
muc
h br
atw
urst
as
poss
ible
.Eac
h po
und
of
brat
wur
st r
equi
res
poun
d of
pla
in s
ausa
ge a
nd
poun
d of
gar
lic-f
lavo
red
saus
age.
Fin
d th
e m
axim
um n
umbe
r of
pou
nds
of b
ratw
urst
tha
t ca
n be
mad
e.10
lb2.
MA
NU
FAC
TUR
ING
Mac
hine
A c
an p
rodu
ce 3
0 st
eeri
ng w
heel
s pe
r ho
ur a
t a
cost
of
$16
per
hour
.Mac
hine
B c
an p
rodu
ce 4
0 st
eeri
ng w
heel
s pe
r ho
ur a
t a
cost
of
$22
per
hour
.A
t le
ast
360
stee
ring
whe
els
mus
t be
mad
e in
eac
h 8-
hour
shi
ft.W
hat
is t
he le
ast
cost
invo
lved
in m
akin
g 36
0 st
eeri
ng w
heel
s in
one
shi
ft?
$194
2 & 3
1 & 43 & 4
(x,y
)x
"y
f(x,
y)
(0, 0
)0
#0
0
(0, 9
)0
#9
9
(6, 8
)6
#8
14
(8, 0
)8
#0
8
Co
lor
A (
gal
lon
s)
Color B (gallons)
510
1520
2530
3540
4550
550
40 35 30 25 20 15 10 5
( 6, 8
)
( 8, 0
)( 0
, 9)
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Line
ar P
rogr
amm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A12 Glencoe Algebra 2
Answers (Lesson 3-4)
Skil
ls P
ract
ice
Line
ar P
rogr
amm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill13
9G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.x
'2
2.x
'1
3.x
'0
x%
5y
%6
y'
0y
'1
y'
x!
2y
%7
!x
y%
4f(
x,y)
"x
!y
f(x,
y) "
3x#
yf(
x,y)
"x
#y
max
.:9,
min
.:3
max
.:2,
min
.:!
5m
ax.:
21,m
in.:
0
4.x
'!
15.
y%
2x6.
y'
!x
!2
x#
y%
6y
'6
!x
y'
3x#
2f(
x,y)
"x
#2y
y%
6y
%x
#4
f(x,
y) "
4x#
3yf(
x,y)
"!
3x#
5y
max
.:13
,no
min
.no
max
.,m
in.:
20m
ax.:
22,m
in.:
!2
7.M
AN
UFA
CTU
RIN
GA
bac
kpac
k m
anuf
actu
rer
prod
uces
an
inte
rnal
fra
me
pack
and
an
exte
rnal
fra
me
pack
.Let
xre
pres
ent
the
num
ber
of in
tern
al f
ram
e pa
cks
prod
uced
inon
e ho
ur a
nd le
t y
repr
esen
t th
e nu
mbe
r of
ext
erna
l fra
me
pack
s pr
oduc
ed in
one
hou
r.T
hen
the
ineq
ualit
ies
x#
3y%
18,2
x#
y%
16,x
'0,
and
y'
0 de
scri
be t
he c
onst
rain
tsfo
r m
anuf
actu
ring
bot
h pa
cks.
Use
the
pro
fit
func
tion
f(x)
"50
x#
80y
and
the
cons
trai
nts
give
n to
det
erm
ine
the
max
imum
pro
fit
for
man
ufac
turi
ng b
oth
back
pack
sfo
r th
e gi
ven
cons
trai
nts.
$620
( 1, 5
)
( –1,
–1)
( –3,
1)
x
y
O
( 3, 6
)
( 2, 4
)
x
y
O
( –1,
7)
x
y
O
x
y
O
( 0, 7
)
( 0, 0
)( 7
, 0)
x
y
O
( 1, 6
)
( 1, –
1)
( 8, 6
)
x
y
O
( 2, 4
)
( 2, 1
)( 5
, 1)
( 5, 4
)
©G
lenc
oe/M
cGra
w-H
ill14
0G
lenc
oe A
lgeb
ra 2
Gra
ph
eac
h s
yste
m o
f in
equ
alit
ies.
Nam
e th
e co
ord
inat
es o
f th
e ve
rtic
es o
f th
efe
asib
le r
egio
n.F
ind
th
e m
axim
um
an
d m
inim
um
val
ues
of
the
give
n f
un
ctio
n f
orth
is r
egio
n.
1.2x
!4
%y
2.3x
!y
%7
3.x
'0
!2x
!4
%y
2x!
y'
3y
'0
y%
2y
'x
!3
y%
6f(
x,y)
"!
2x#
yf(
x,y)
"x
!4y
y%
!3x
#15
f(x,
y) "
3x#
y
max
.:8,
min
.:!
4m
ax.:
12,m
in.:
!16
max
.:15
,min
.:0
4.x
%0
5.y
%3x
#6
6.2x
#3y
'6
y%
04y
#3x
%3
2x!
y%
24x
#y
'!
7x
'!
2x
'0
f(x,
y) "
!x
!4y
f(x,
y) "
!x
#3y
y'
0f(
x,y)
"x
#4y
#3
max
.:28
,min
.:0
max
.:,n
o m
in.
no m
ax.,
min
.:
PRO
DU
CTI
ON
For
Exe
rcis
es 7
–9,u
se t
he
foll
owin
g in
form
atio
n.
A g
lass
blo
wer
can
for
m 8
sim
ple
vase
s or
2 e
labo
rate
vas
es in
an
hour
.In
a w
ork
shif
t of
no
mor
e th
an 8
hou
rs,t
he w
orke
r m
ust
form
at
leas
t 40
vas
es.
7.L
et s
repr
esen
t th
e ho
urs
form
ing
sim
ple
vase
s an
d e
the
hour
s fo
rmin
g el
abor
ate
vase
s.W
rite
a s
yste
m o
f in
equa
litie
s in
volv
ing
the
tim
e sp
ent
on e
ach
type
of
vase
.s
)0,
e)
0,s
#e
&8,
8s#
2e)
408.
If t
he g
lass
blo
wer
mak
es a
pro
fit
of $
30 p
er h
our
wor
ked
on t
he s
impl
e va
ses
and
$35
per
hour
wor
ked
on t
he e
labo
rate
vas
es,w
rite
a f
unct
ion
for
the
tota
l pro
fit
on t
he v
ases
.f(
s,e)
"30
s#
35e
9.F
ind
the
num
ber
of h
ours
the
wor
ker
shou
ld s
pend
on
each
typ
e of
vas
e to
max
imiz
epr
ofit
.Wha
t is
tha
t pr
ofit
?4
h on
eac
h;$2
60
17 $ 234 $ 5
x
y
(3 – 2, 1)
( 0, 2
) O
(–7 – 5, 9 – 5)
( –2,
0)
x
y O
x
y ( 0, –
7)
(–7 – 4, 0)
( 0, 0
)O
( 3, 6
) ( 5, 0
)
( 0, 6
)
( 0, 0
)x
y
O
( 2, –
1)
( 4, 5
)
( 0, –
3)
x
y
O
( –3,
2)
( 3, 2
)
( 0, –
4)
x
y
O
Pra
ctic
e (A
vera
ge)
Line
ar P
rogr
amm
ing
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
© Glencoe/McGraw-Hill A13 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-4)
Rea
din
g t
o L
earn
Math
emati
csLi
near
Pro
gram
min
g
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
©G
lenc
oe/M
cGra
w-H
ill14
1G
lenc
oe A
lgeb
ra 2
Lesson 3-4
Pre-
Act
ivit
yH
ow i
s li
nea
r p
rogr
amm
ing
use
d i
n s
ched
uli
ng
wor
k?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 3-
4 at
the
top
of
page
129
in y
our
text
book
.
Nam
e tw
o or
mor
e fa
cts
that
indi
cate
tha
t yo
u w
ill n
eed
to u
se in
equa
litie
sto
mod
el t
his
situ
atio
n.S
ampl
e an
swer
:The
buo
y te
nder
can
car
ry u
p to
8 n
ew b
uoys
.Th
ere
seem
s to
be
a lim
it of
24
hour
s on
the
tim
e th
e cr
ew h
asat
sea
.The
cre
w w
ill w
ant
to r
epai
r or
rep
lace
the
max
imum
num
ber
of b
uoys
pos
sibl
e.
Rea
din
g t
he
Less
on
1.C
ompl
ete
each
sen
tenc
e.
a.W
hen
you
find
the
fea
sibl
e re
gion
for
a li
near
pro
gram
min
g pr
oble
m,y
ou a
re s
olvi
ng
a sy
stem
of
linea
r ca
lled
.The
poi
nts
in t
he f
easi
ble
regi
on a
re
of t
he s
yste
m.
b.T
he c
orne
r po
ints
of
a po
lygo
nal r
egio
n ar
e th
e of
the
fe
asib
le r
egio
n.
2.A
pol
ygon
al r
egio
n al
way
s ta
kes
up o
nly
a lim
ited
par
t of
the
coo
rdin
ate
plan
e.O
ne w
ayto
thi
nk o
f th
is is
to
imag
ine
a ci
rcle
or
rect
angl
e th
at t
he r
egio
n w
ould
fit
insi
de.I
n th
eca
se o
f a
poly
gona
l reg
ion,
you
can
alw
ays
find
a c
ircl
e or
rec
tang
le t
hat
is la
rge
enou
ghto
con
tain
all
the
poin
ts o
f th
e po
lygo
nal r
egio
n.W
hat
wor
d is
use
d to
des
crib
e a
regi
onth
at c
an b
e en
clos
ed in
thi
s w
ay?
Wha
t w
ord
is u
sed
to d
escr
ibe
a re
gion
tha
t is
too
larg
eto
be
encl
osed
in t
his
way
?bo
unde
d;un
boun
ded
3.H
ow d
o yo
u fi
nd t
he c
orne
r po
ints
of
the
poly
gona
l reg
ion
in a
line
ar p
rogr
amm
ing
prob
lem
?Yo
u so
lve
a sy
stem
of
two
linea
r eq
uatio
ns.
4.W
hat
are
som
e ev
eryd
ay m
eani
ngs
of t
he w
ord
feas
ible
that
rem
ind
you
of t
hem
athe
mat
ical
mea
ning
of
the
term
feas
ible
reg
ion?
Sam
ple
answ
er:p
ossi
ble
or a
chie
vabl
e
Hel
pin
g Y
ou
Rem
emb
er
5.L
ook
up t
he w
ord
cons
trai
ntin
a d
icti
onar
y.If
mor
e th
an o
ne d
efin
itio
n is
giv
en,c
hoos
eth
e on
e th
at s
eem
s cl
oses
t to
the
idea
of
a co
nstr
aint
in a
line
ar p
rogr
amm
ing
prob
lem
.H
ow c
an t
his
defi
niti
on h
elp
you
to r
emem
ber
the
mea
ning
of c
onst
rain
tas
it is
use
d in
this
less
on?
Sam
ple
answ
er:A
con
stra
int
is a
res
tric
tion
or li
mita
tion.
The
cons
trai
nts
in a
line
ar p
rogr
amm
ing
prob
lem
are
res
tric
tions
on
the
vari
able
s th
at t
rans
late
into
ineq
ualit
y st
atem
ents
.
vert
ices
solu
tions
cons
trai
nts
ineq
ualit
ies
©G
lenc
oe/M
cGra
w-H
ill14
2G
lenc
oe A
lgeb
ra 2
Com
pute
r C
ircu
its a
nd L
ogic
Com
pute
rs o
pera
te a
ccor
ding
to
the
law
s of
logi
c.T
he c
ircu
its
of a
com
pute
r ca
n be
des
crib
ed u
sing
logi
c.
1.W
ith s
witc
h A
open
, no
curr
ent f
low
s. T
he v
alue
0 is
ass
igne
d to
an
open
sw
itch.
2.W
ith s
witc
h A
clos
ed, c
urre
nt fl
ows.
The
val
ue 1
is a
ssig
ned
to a
clo
sed
switc
h.
3.W
ith s
witc
hes
Aan
d B
ope
n, n
o cu
rren
t flo
ws.
Thi
s ci
rcui
t ca
n be
des
crib
ed b
y th
e co
njun
ctio
n, A
)B
.
4.In
this
circ
uit,
curr
ent f
low
s if
eith
er A
or B
is c
lose
d. T
his
circ
uit
can
be d
escr
ibed
by
the
disj
unct
ion,
A#
B.
Tru
th t
able
s ar
e us
ed t
o de
scri
be t
he f
low
of
curr
ent
in a
cir
cuit
.T
he t
able
at
the
left
des
crib
es t
he c
ircu
it in
dia
gram
4.A
ccor
ding
to
the
tab
le,t
he o
nly
tim
e cu
rren
t do
es n
ot f
low
thr
ough
the
ci
rcui
t is
whe
n bo
th s
wit
ches
A a
nd B
are
ope
n.
Dra
w a
cir
cuit
dia
gram
for
eac
h o
f th
e fo
llow
ing.
1.(A
)B
) #
C2.
(A #
B)
)C
3.(A
#B
) )
(C #
D)
4.(A
)B
) #
(C )
D)
5.C
onst
ruct
a t
ruth
tab
le f
or t
he
follo
win
g ci
rcui
t.
B C
A
A CDB
A BDC
A
BC
AB
C
A A A B
AB
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-4
3-4
AB
C(B
#C
)A
(B #
C)
00
00
00
01
10
01
11
00
10
10
10
11
11
10
11
11
11
11
00
00
AB
A#
B
00
00
11
10
11
11
© Glencoe/McGraw-Hill A14 Glencoe Algebra 2
Answers (Lesson 3-5)
Stu
dy
Gu
ide
and I
nte
rven
tion
Sol
ving
Sys
tem
s of
Equ
atio
ns in
Thr
ee V
aria
bles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
3G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Syst
ems
in T
hre
e V
aria
ble
sU
se t
he m
etho
ds u
sed
for
solv
ing
syst
ems
of li
near
equa
tion
s in
tw
o va
riab
les
to s
olve
sys
tem
s of
equ
atio
ns in
thr
ee v
aria
bles
.A s
yste
m o
fth
ree
equa
tion
s in
thr
ee v
aria
bles
can
hav
e a
uniq
ue s
olut
ion,
infi
nite
ly m
any
solu
tion
s,or
no s
olut
ion.
A s
olut
ion
is a
n or
der
ed t
rip
le.
Sol
ve t
his
sys
tem
of
equ
atio
ns.
3x#
y!
z"
!6
2x!
y#
2z"
84x
#y
!3z
"!
21
Step
1U
se e
limin
atio
n to
mak
e a
syst
em o
f tw
o eq
uati
ons
in t
wo
vari
able
s.3x
#y
!z
"!
6F
irst e
quat
ion
2x!
y#
2z"
8S
econ
d eq
uatio
n(#
) 2x
!y
#2z
"8
Sec
ond
equa
tion
(#) 4
x#
y!
3z"
!21
Thi
rd e
quat
ion
5x#
z"
2A
dd to
elim
inat
e y.
6x!
z
"!
13A
dd to
elim
inat
e y.
Step
2So
lve
the
syst
em o
f tw
o eq
uati
ons.
5x#
z"
2(#
) 6x
!z
"!
1311
x"
!11
Add
to e
limin
ate
z.x
"!
1D
ivid
e bo
th s
ides
by
11.
Subs
titu
te !
1 fo
r x
in o
ne o
f th
e eq
uati
ons
wit
h tw
o va
riab
les
and
solv
e fo
r z.
5x#
z"
2E
quat
ion
with
two
varia
bles
5(!
1) #
z"
2R
epla
ce x
with
!1.
!5
#z
"2
Mul
tiply
.z
"7
Add
5 to
bot
h si
des.
The
res
ult
so f
ar is
x"
!1
and
z"
7.
Step
3Su
bsti
tute
!1
for
xan
d 7
for
zin
one
of
the
orig
inal
equ
atio
ns w
ith
thre
e va
riab
les.
3x#
y!
z"
!6
Orig
inal
equ
atio
n w
ith th
ree
varia
bles
3(!
1) #
y!
7 "
!6
Rep
lace
xw
ith !
1 an
d z
with
7.
!3
#y
!7
"!
6M
ultip
ly.
y"
4S
impl
ify.
The
sol
utio
n is
(!1,
4,7)
.
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
#3y
!z
"0
2.2x
!y
#4z
"11
3.x
!2y
#z
"8
x!
2y!
4z"
14x
#2y
!6z
"!
112x
#y
!z
"0
3x#
y!
8z"
173x
!2y
!10
z"
113x
!6y
#3z
"24
(4,!
3,!
1)!2,
!5,
"in
finite
ly m
any
solu
tions
4.3x
!y
!z
"5
5.2x
!4y
!z
"10
6.x
!6y
#4z
"2
3x#
2y!
z"
114x
!8y
!2z
"16
2x#
4y!
8z"
166x
!3y
#2z
"!
123x
#y
#z
"12
x!
2y"
5
!,2
,!5 "
no s
olut
ion
!6,,!
"1 $ 4
1 $ 22 $ 3
1 $ 2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill14
4G
lenc
oe A
lgeb
ra 2
Real
-Wor
ld P
robl
ems
Th
e L
ared
o S
por
ts S
hop
sol
d 1
0 ba
lls,
3 ba
ts,a
nd
2 b
ases
for
$99
on
Mon
day
.On
Tu
esd
ay t
hey
sol
d 4
bal
ls,8
bat
s,an
d 2
bas
es f
or $
78.O
n W
edn
esd
ayth
ey s
old
2 b
alls
,3 b
ats,
and
1 b
ase
for
$33.
60.W
hat
are
th
e p
rice
s of
1 b
all,
1 ba
t,an
d 1
bas
e?
Fir
st d
efin
e th
e va
riab
les.
x"
pric
e of
1 b
all
y"
pric
e of
1 b
atz
"pr
ice
of 1
bas
e
Tra
nsla
te t
he in
form
atio
n in
the
pro
blem
into
thr
ee e
quat
ions
.
10x
#3y
#2z
"99
4x#
8y#
2z"
782x
#3y
#z
"33
.60
Stu
dy
Gu
ide
and I
nte
rven
tion
(c
onti
nued
)
Sol
ving
Sys
tem
s of
Equ
atio
ns in
Thr
ee V
aria
bles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5
Exam
ple
Exam
ple
Subt
ract
the
sec
ond
equa
tion
fro
m t
he f
irst
equa
tion
to
elim
inat
e z.
10x
#3y
#2z
"99
(!)
4x#
8y#
2z"
786x
!5y
"21
Mul
tipl
y th
e th
ird
equa
tion
by
2 an
dsu
btra
ct f
rom
the
sec
ond
equa
tion
.4x
#8y
#2z
"78
(!) 4
x#
6y#
2z"
67.2
02y
"10
.80
y"
5.40
Subs
titu
te 5
.40
for
yin
the
equ
atio
n 6x
!5y
"21
.
6x!
5(5.
40)
"21
6x"
48x
"8
Subs
titu
te 8
for
xan
d 5.
40 f
or y
in o
ne o
fth
e or
igin
al e
quat
ions
to
solv
e fo
r z.
10x
#3y
#2z
"99
10(8
) #
3(5.
40)
#2z
"99
80 #
16.2
0 #
2z"
992z
"2.
80z
"1.
40So
a b
all c
osts
$8,
a ba
t $5
.40,
and
a ba
se $
1.40
.
1.FI
TNES
S TR
AIN
ING
Car
ly is
tra
inin
g fo
r a
tria
thlo
n.In
her
tra
inin
g ro
utin
e ea
ch w
eek,
she
runs
7 t
imes
as
far
as s
he s
wim
s,an
d sh
e bi
kes
3 ti
mes
as
far
as s
he r
uns.
One
wee
ksh
e tr
aine
d a
tota
l of
232
mile
s.H
ow f
ar d
id s
he r
un t
hat
wee
k?56
mile
s
2.EN
TERT
AIN
MEN
TA
t th
e ar
cade
,Rya
n,Sa
ra,a
nd T
im p
laye
d vi
deo
raci
ng g
ames
,pin
ball,
and
air
hock
ey.R
yan
spen
t $6
for
6 r
acin
g ga
mes
,2 p
inba
ll ga
mes
,and
1 g
ame
of a
irho
ckey
.Sar
a sp
ent
$12
for
3 ra
cing
gam
es,4
pin
ball
gam
es,a
nd 5
gam
es o
f air
hoc
key.
Tim
spen
t $1
2.25
for
2 ra
cing
gam
es,7
pin
ball
gam
es,a
nd 4
gam
es o
f air
hoc
key.
How
muc
h di
dea
ch o
f th
e ga
mes
cos
t?R
acin
g ga
me:
$0.5
0;pi
nbal
l:$0
.75;
air
hock
ey:$
1.50
3.FO
OD
A n
atur
al f
ood
stor
e m
akes
its
own
bran
d of
tra
il m
ix o
ut o
f dr
ied
appl
es,r
aisi
ns,
and
pean
uts.
One
pou
nd o
f th
e m
ixtu
re c
osts
$3.
18.I
t co
ntai
ns t
wic
e as
muc
h pe
anut
sby
wei
ght
as a
pple
s.O
ne p
ound
of
drie
d ap
ples
cos
ts $
4.48
,a p
ound
of
rais
ins
$2.4
0,an
da
poun
d of
pea
nuts
$3.
44.H
ow m
any
ounc
es o
f ea
ch in
gred
ient
are
con
tain
ed in
1 p
ound
of t
he t
rail
mix
?3
oz o
f ap
ples
,7 o
z of
rai
sins
,6 o
z of
pea
nuts
Exer
cises
Exer
cises
© Glencoe/McGraw-Hill A15 Glencoe Algebra 2
An
swer
s
Answers (Lesson 3-5)
Skil
ls P
ract
ice
Sol
ving
Sys
tem
s of
Equ
atio
ns in
Thr
ee V
aria
bles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
5G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2a
#c
"!
10(5
,!5,
!20
)2.
x#
y#
z"
3(0
,2,1
)b
!c
"15
13x
#2z
"2
a!
2b#
c"
!5
!x
!5z
"!
5
3.2x
#5y
#2z
"6
(!3,
2,1)
4.x
#4y
!z
"1
no s
olut
ion
5x!
7y"
!29
3x!
y#
8z"
0z
"1
x#
4y!
z"
10
5.!
2z"
!6
(2,!
1,3)
6.3x
!2y
#2z
"!
2(!
2,1,
3)2x
#3y
!z
"!
2x
#6y
!2z
"!
2x
#2y
#3z
"9
x#
2y"
0
7.!
x!
5z"
!5
(0,0
,1)
8.!
3r#
2t"
1(1
,!6,
2)y
!3x
"0
4r#
s!
2t"
!6
13x
#2z
"2
r#
s#
4t"
3
9.x
!y
#3z
"3
no s
olut
ion
10.5
m#
3n#
p"
4(!
2,3,
5)!
2x#
2y!
6z"
63m
#2n
"0
y!
5z"
!3
2m!
n#
3p"
8
11.2
x#
2y#
2z"
!2
infin
itely
man
y12
.x#
2y!
z"
4(1
,2,1
)2x
#3y
#2z
"4
3x!
y#
2z"
3x
#y
#z
"!
1!
x#
3y#
z"
6
13.3
x!
2y#
z"
1(5
,7,0
)14
.3x
!5y
#2z
"!
12in
finite
ly m
any
!x
#y
!z
"2
x#
4y!
2z"
85x
#2y
#10
z"
39!
3x#
5y!
2z"
12
15.2
x#
y#
3z"
!2
(!1,
3,!
1)16
.2x
!4y
#3z
"0
(3,0
,!2)
x!
y!
z"
!3
x!
2y!
5z"
133x
!2y
#3z
"!
125x
#3y
!2z
"19
17.!
2x#
y#
2z"
2(1
,!2,
3)18
.x!
2y#
2z"
!1
infin
itely
man
y3x
#3y
#z
"0
x#
2y!
z"
6x
#y
#z
"2
!3x
#6y
!6z
"3
19.T
he s
um o
f th
ree
num
bers
is 1
8.T
he s
um o
f th
e fi
rst
and
seco
nd n
umbe
rs is
15,
and
the
firs
t nu
mbe
r is
3 t
imes
the
thi
rd n
umbe
r.F
ind
the
num
bers
.9,
6,3
©G
lenc
oe/M
cGra
w-H
ill14
6G
lenc
oe A
lgeb
ra 2
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.2x
!y
#2z
"15
2.x
!4y
#3z
"!
273.
a#
b"
3!
x#
y#
z"
32x
#2y
!3z
"22
!b
#c
"3
3x!
y#
2z"
184z
"!
16a
#2c
"10
(3,1
,5)
(1,4
,!4)
(2,1
,4)
4.3m
!2n
#4p
"15
5.2g
#3h
!8j
"10
6.2x
#y
!z
"!
8m
!n
#p
"3
g!
4h"
14x
!y
#2z
"!
3m
#4n
!5p
"0
!2g
!3h
#8j
"5
!3x
#y
#2z
"5
(3,3
,3)
no s
olut
ion
(!2,
!3,
1)
7.2x
!5y
#z
"5
8.2x
#3y
#4z
"2
9.p
#4r
"!
73x
#2y
!z
"17
5x!
2y#
3z"
0p
!3q
"!
84x
!3y
#2z
"17
x!
5y!
2z"
!4
q#
r"
1(5
,1,0
)(2
,2,!
2)(1
,3,!
2)
10.4
x#
4y!
2z"
811
.d#
3e#
f"
012
.4x
#y
#5z
"!
93x
!5y
#3z
"0
!d
#2e
#f
"!
1x
!4y
!2z
"!
2 2x
#2y
!z
"4
4d#
e!
f"
12x
#3y
!2z
"21
infin
itely
man
y(1
,!1,
2)(2
,3,!
4)
13.5
x#
9y#
z"
2014
.2x
#y
!3z
"!
315
.3x
#3y
#z
"10
2x!
y!
z"
!21
3x#
2y#
4z"
55x
#2y
#2z
"7
5x#
2y#
2z"
!21
!6x
!3y
#9z
"9
3x!
2y#
3z"
!9
(!7,
6,1)
infin
itely
man
y(1
,3,!
2)
16.2
u#
v#
w"
217
.x#
5y!
3z"
!18
18.x
!2y
#z
"!
1!
3u#
2v#
3w"
73x
!2y
#5z
"22
!x
#2y
!z
"6
!u
!v
#2w
"7
!2x
!3y
#8z
"28
!4y
#2z
"1
(0,!
1,3)
(1,!
2,3)
no s
olut
ion
19.2
x!
2y!
4z"
!2
20.x
!y
#9z
"!
2721
.2x
!5y
!3z
"7
3x!
3y!
6z"
!3
2x!
4y!
z"
!1
!4x
#10
y#
2z"
6!
2x#
3y#
z"
73x
#6y
!3z
"27
6x!
15y
!z
"!
19(4
,5,0
)(2
,2,!
3)(1
,2,!
5)
22.T
he s
um o
f th
ree
num
bers
is 6
.The
thi
rd n
umbe
r is
the
sum
of
the
firs
t an
d se
cond
num
bers
.The
fir
st n
umbe
r is
one
mor
e th
an t
he t
hird
num
ber.
Fin
d th
e nu
mbe
rs.
4,!
1,3
23.T
he s
um o
f th
ree
num
bers
is !
4.T
he s
econ
d nu
mbe
r de
crea
sed
by t
he t
hird
is e
qual
to
the
firs
t.T
he s
um o
f th
e fi
rst
and
seco
nd n
umbe
rs is
!5.
Fin
d th
e nu
mbe
rs.
!3,
!2,
1
24.S
PORT
SA
lexa
ndri
a H
igh
Scho
ol s
core
d 37
poi
nts
in a
foo
tbal
l gam
e.Si
x po
ints
are
awar
ded
for
each
tou
chdo
wn.
Aft
er e
ach
touc
hdow
n,th
e te
am c
an e
arn
one
poin
t fo
r th
eex
tra
kick
or
two
poin
ts f
or a
2-p
oint
con
vers
ion.
The
tea
m s
core
d on
e fe
wer
2-p
oint
conv
ersi
ons
than
ext
ra k
icks
.The
tea
m s
core
d 10
tim
es d
urin
g th
e ga
me.
How
man
yto
uchd
owns
wer
e m
ade
duri
ng t
he g
ame?
5
Pra
ctic
e (A
vera
ge)
Sol
ving
Sys
tem
s of
Equ
atio
ns in
Thr
ee V
aria
bles
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5
© Glencoe/McGraw-Hill A16 Glencoe Algebra 2
Answers (Lesson 3-5)
Rea
din
g t
o L
earn
Math
emati
csS
olvi
ng S
yste
ms
of E
quat
ions
in T
hree
Var
iabl
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5
©G
lenc
oe/M
cGra
w-H
ill14
7G
lenc
oe A
lgeb
ra 2
Lesson 3-5
Pre-
Act
ivit
yH
ow c
an y
ou d
eter
min
e th
e n
um
ber
and
typ
e of
med
als
U.S
.O
lym
pia
ns
won
?
Rea
d th
e in
trod
ucti
on t
o L
esso
n 3-
5 at
the
top
of
page
138
in y
our
text
book
.
At
the
1996
Sum
mer
Oly
mpi
cs in
Atl
anta
,Geo
rgia
,the
Uni
ted
Stat
es w
on10
1 m
edal
s.T
he U
.S.t
eam
won
12
mor
e go
ld m
edal
s th
an s
ilver
and
7fe
wer
bro
nze
med
als
than
silv
er.U
sing
the
sam
e va
riab
les
as t
hose
in t
hein
trod
ucti
on,w
rite
a s
yste
m o
f eq
uati
ons
that
des
crib
es t
he m
edal
s w
on f
orth
e 19
96 S
umm
er O
lym
pics
.g
#s
#b
"10
1;g
"s
#12
;b"
s!
7
Rea
din
g t
he
Less
on
1.T
he p
lane
s fo
r th
e eq
uati
ons
in a
sys
tem
of
thre
e lin
ear
equa
tion
s in
thr
ee v
aria
bles
dete
rmin
e th
e nu
mbe
r of
sol
utio
ns.M
atch
eac
h gr
aph
desc
ript
ion
belo
w w
ith
the
desc
ript
ion
of t
he n
umbe
r of
sol
utio
ns o
f th
e sy
stem
.(So
me
of t
he it
ems
on t
he r
ight
may
be u
sed
mor
e th
an o
nce,
and
not
all p
ossi
ble
type
s of
gra
phs
are
liste
d.)
a.th
ree
para
llel p
lane
s I.
one
solu
tion
b.th
ree
plan
es t
hat
inte
rsec
t in
a li
ne
II.
no s
olut
ions
c.th
ree
plan
es t
hat
inte
rsec
t in
one
poi
nt
III.
infi
nite
sol
utio
ns
d.on
e pl
ane
that
rep
rese
nts
all t
hree
equ
atio
ns
2.Su
ppos
e th
at t
hree
cla
ssm
ates
,Mon
ique
,Jos
h,an
d L
illy,
are
stud
ying
for
a q
uiz
on t
his
less
on.T
hey
wor
k to
geth
er o
n so
lvin
g a
syst
em o
f eq
uati
ons
in t
hree
var
iabl
es,x
,y,a
ndz,
follo
win
g th
e al
gebr
aic
met
hod
show
n in
you
r te
xtbo
ok.T
hey
firs
t fi
nd t
hat
z"
3,th
enth
at y
"!
2,an
d fi
nally
tha
t x
"!
1.T
he s
tude
nts
agre
e on
the
se v
alue
s,bu
t di
sagr
eeon
how
to
wri
te t
he s
olut
ion.
Her
e ar
e th
eir
answ
ers:
Mon
ique
:(3,
!2,
!1)
Josh
:(!
2,!
1,3)
Lill
y:(!
1,!
2,3)
a.H
ow d
o yo
u th
ink
each
stu
dent
dec
ided
on
the
orde
r of
the
num
bers
in t
he o
rder
edtr
iple
?S
ampl
e an
swer
:Mon
ique
arr
ange
d th
e va
lues
in t
he o
rder
inw
hich
she
foun
d th
em.J
osh
arra
nged
the
m f
rom
sm
alle
st t
o la
rges
t.Li
lly a
rran
ged
them
in a
lpha
betic
al o
rder
of
the
vari
able
s.
b.W
hich
stu
dent
is c
orre
ct?
Lilly
Hel
pin
g Y
ou
Rem
emb
er
3.H
ow c
an y
ou r
emem
ber
that
obt
aini
ng t
he e
quat
ion
0 "
0 in
dica
tes
a sy
stem
wit
hin
fini
tely
man
y so
luti
ons,
whi
le o
btai
ning
an
equa
tion
suc
h as
0 "
8 in
dica
tes
a sy
stem
wit
h no
sol
utio
ns?
0 "
0 is
alw
ays
true
,whi
le 0
"8
is n
ever
tru
e.
III
I
III
II
©G
lenc
oe/M
cGra
w-H
ill14
8G
lenc
oe A
lgeb
ra 2
Bill
iard
sT
he fi
gure
at
the
righ
t sh
ows
a bi
lliar
d ta
ble.
The
obj
ect
is t
o us
e a
cue
stic
k to
str
ike
the
ball
at p
oint
Cso
tha
t th
e ba
ll w
ill h
it t
he s
ides
(o
r cu
shio
ns) o
f the
tab
le a
t le
ast
once
bef
ore
hitt
ing
the
ball
loca
ted
at p
oint
A.I
n pl
ayin
g th
e ga
me,
you
need
to
loca
te p
oint
P.
Ste
p 1
Fin
d po
int
Bso
tha
t "
and
$
.Bis
cal
led
the
refle
cted
im
age
of C
in
.
Ste
p 2
Dra
w
.
Ste
p 3
inte
rsec
ts
at t
he d
esir
ed p
oint
P.
For
eac
h b
illi
ard
s p
robl
em,t
he
cue
ball
at
poi
nt
Cm
ust
str
ike
the
ind
icat
edcu
shio
n(s
) an
d t
hen
str
ike
the
ball
at
poi
nt
A.D
raw
an
d l
abel
th
e co
rrec
t p
ath
fo
r th
e cu
e ba
ll u
sin
g th
e p
roce
ss d
escr
ibed
abo
ve.
1.cu
shio
n 2.
cush
ion
3.cu
shio
n ,t
hen
cush
ion
4.cu
shio
n ,t
hen
cush
ion C
A
B1
B2
P1
P2
RK T
S
C
A
B1
B2
P1
P2R
K TS
RS
KT
RS
TS
C
A
B
R P
K TS
C
A
B
RK T
S
RS
KR
ST
AB
AB
ST
CH
BH
ST
BC
KR
C
A
B
P
HT
S
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
ATE
____
____
____
PE
RIO
D__
___
3-5
3-5