Chapter 2 Resource Masters -...

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Chapter 2Resource Masters

New York, New York Columbus, Ohio Woodland Hills, California Peoria, Illinois

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StudentWorksTM This CD-ROM includes the entire Student Edition along with theStudy Guide, Practice, and Enrichment masters.

TeacherWorksTM All of the materials found in this booklet are included for viewing and printing in the Advanced Mathematical Concepts TeacherWorksCD-ROM.

Copyright © The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce thematerial contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Advanced Mathematical Concepts.Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-869129-X Advanced Mathematical ConceptsChapter 2 Resource Masters

1 2 3 4 5 6 7 8 9 10 XXX 11 10 09 08 07 06 05 04

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© Glencoe/McGraw-Hill iii Advanced Mathematical Concepts

Vocabulary Builder . . . . . . . . . . . . . . . . . vii-x

Lesson 2-1Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 45Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Lesson 2-2Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 48Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Lesson 2-3Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 51Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Lesson 2-4Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 54Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Lesson 2-5Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 57Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Lesson 2-6Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 60Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Lesson 2-7Study Guide . . . . . . . . . . . . . . . . . . . . . . . . . . 63Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 2 AssessmentChapter 2 Test, Form 1A . . . . . . . . . . . . . . 67-68Chapter 2 Test, Form 1B . . . . . . . . . . . . . . 69-70Chapter 2 Test, Form 1C . . . . . . . . . . . . . . 71-72Chapter 2 Test, Form 2A . . . . . . . . . . . . . . 73-74Chapter 2 Test, Form 2B . . . . . . . . . . . . . . 75-76Chapter 2 Test, Form 2C . . . . . . . . . . . . . . 77-78Chapter 2 Extended Response

Assessment . . . . . . . . . . . . . . . . . . . . . . . . 79Chapter 2 Mid-Chapter Test . . . . . . . . . . . . . . 80Chapter 2 Quizzes A & B . . . . . . . . . . . . . . . . 81Chapter 2 Quizzes C & D. . . . . . . . . . . . . . . . 82Chapter 2 SAT and ACT Practice . . . . . . . 83-84Chapter 2 Cumulative Review . . . . . . . . . . . . 85

SAT and ACT Practice Answer Sheet,10 Questions . . . . . . . . . . . . . . . . . . . . . . . A1

SAT and ACT Practice Answer Sheet,20 Questions . . . . . . . . . . . . . . . . . . . . . . . A2

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A3-A16

Contents

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© Glencoe/McGraw-Hill iv Advanced Mathematical Concepts

A Teacher’s Guide to Using theChapter 2 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file theresources you use most often. The Chapter 2 Resource Masters include the corematerials needed for Chapter 2. These materials include worksheets, extensions,and assessment options. The answers for these pages appear at the back of thisbooklet.

All of the materials found in this booklet are included for viewing and printing inthe Advanced Mathematical Concepts TeacherWorks CD-ROM.

Vocabulary Builder Pages vii-x include a student study tool that presents the key vocabulary terms from the chapter. Students areto record definitions and/or examples for eachterm. You may suggest that students highlight orstar the terms with which they are not familiar.

When to Use Give these pages to studentsbefore beginning Lesson 2-1. Remind them toadd definitions and examples as they completeeach lesson.

Study Guide There is one Study Guide master for each lesson.

When to Use Use these masters as reteaching activities for students who need additional reinforcement. These pages can alsobe used in conjunction with the Student Editionas an instructional tool for those students whohave been absent.

Practice There is one master for each lesson.These problems more closely follow the structure of the Practice section of the StudentEdition exercises. These exercises are ofaverage difficulty.

When to Use These provide additional practice options or may be used as homeworkfor second day teaching of the lesson.

Enrichment There is one master for eachlesson. These activities may extend the conceptsin the lesson, offer a historical or multiculturallook at the concepts, or widen students’perspectives on the mathematics they are learning. These are not written exclusively for honors students, but are accessible for usewith all levels of students.

When to Use These may be used as extracredit, short-term projects, or as activities fordays when class periods are shortened.

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© Glencoe/McGraw-Hill v Advanced Mathematical Concepts

Assessment Options

The assessment section of the Chapter 2Resources Masters offers a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessments

Chapter Tests• Forms 1A, 1B, and 1C Form 1 tests contain

multiple-choice questions. Form 1A isintended for use with honors-level students,Form 1B is intended for use with average-level students, and Form 1C is intended foruse with basic-level students. These testsare similar in format to offer comparabletesting situations.

• Forms 2A, 2B, and 2C Form 2 tests arecomposed of free-response questions. Form2A is intended for use with honors-levelstudents, Form 2B is intended for use withaverage-level students, and Form 2C isintended for use with basic-level students.These tests are similar in format to offercomparable testing situations.

All of the above tests include a challengingBonus question.

• The Extended Response Assessmentincludes performance assessment tasks thatare suitable for all students. A scoringrubric is included for evaluation guidelines.Sample answers are provided for assessment.

Intermediate Assessment• A Mid-Chapter Test provides an option to

assess the first half of the chapter. It iscomposed of free-response questions.

• Four free-response quizzes are included tooffer assessment at appropriate intervals inthe chapter.

Continuing Assessment• The SAT and ACT Practice offers

continuing review of concepts in variousformats, which may appear on standardizedtests that they may encounter. This practiceincludes multiple-choice, quantitative-comparison, and grid-in questions. Bubble-in and grid-in answer sections are providedon the master.

• The Cumulative Review provides studentsan opportunity to reinforce and retain skillsas they proceed through their study ofadvanced mathematics. It can also be usedas a test. The master includes free-responsequestions.

Answers• Page A1 is an answer sheet for the SAT and

ACT Practice questions that appear in theStudent Edition on page 65. Page A2 is ananswer sheet for the SAT and ACT Practicemaster. These improve students’ familiaritywith the answer formats they mayencounter in test taking.

• The answers for the lesson-by-lesson masters are provided as reduced pages withanswers appearing in red.

• Full-size answer keys are provided for theassessment options in this booklet.

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primarily skillsprimarily conceptsprimarily applications

BASIC AVERAGE ADVANCED

Study Guide

Vocabulary Builder

Parent and Student Study Guide (online)

Practice

Enrichment

4

5

3

2

Five Different Options to Meet the Needs of Every Student in a Variety of Ways

1

Chapter 2 Leveled Worksheets

Glencoe’s leveled worksheets are helpful for meeting the needs of everystudent in a variety of ways. These worksheets, many of which are foundin the FAST FILE Chapter Resource Masters, are shown in the chartbelow.

• Study Guide masters provide worked-out examples as well as practiceproblems.

• Each chapter’s Vocabulary Builder master provides students theopportunity to write out key concepts and definitions in their ownwords.

• Practice masters provide average-level problems for students who are moving at a regular pace.

• Enrichment masters offer students the opportunity to extend theirlearning.

© Glencoe/McGraw-Hill vi Advanced Mathematical Concepts

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Reading to Learn MathematicsVocabulary Builder

NAME _____________________________ DATE _______________ PERIOD ________

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 2.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term.

© Glencoe/McGraw-Hill vii Advanced Mathematical Concepts

Vocabulary Term Foundon Page Definition/Description/Example

additive identity matrix

alternate optimal solution

column matrix

consistent

constraints

dependent

determinant

dilation

dimensions

element

(continued on the next page)

Chapter

2

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© Glencoe/McGraw-Hill viii Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder (continued)

NAME _____________________________ DATE _______________ PERIOD ________

Vocabulary Term Foundon Page Definition/Description/Example

elimination method

equal matrices

identity matrix for multiplication

image

inconsistent

independent

infeasible

inverse matrix

linear programming

m � n matrix

matrix

(continued on the next page)

Chapter

2

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© Glencoe/McGraw-Hill ix Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder (continued)

NAME _____________________________ DATE _______________ PERIOD ________

Vocabulary Term Foundon Page Definition/Description/Example

minor

nth order

ordered triple

polygonal convex set

pre-image

reflection

reflection matrix

rotation

rotation matrix

row matrix

scalar

(continued on the next page)

Chapter

2

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© Glencoe/McGraw-Hill x Advanced Mathematical Concepts

Reading to Learn MathematicsVocabulary Builder (continued)

NAME _____________________________ DATE _______________ PERIOD ________

Vocabulary Term Foundon Page Definition/Description/Example

solution

square matrix

substitution method

system of equations

system of linear inequalities

transformations

translation

translation matrix

unbounded

vertex matrix

Vertex Theorem

zero matrix

Chapter

2

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© Glencoe/McGraw-Hill 45 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

2-1

Solving Systems of Equations in Two VariablesOne way to solve a system of equations in two variables is bygraphing. The intersection of the graphs is called the solutionto the system of equations.

Example 1 Solve the system of equations by graphing.3x � y � 10x � 4y � 12

First rewrite each equation of the system inslope-intercept form by solving for y.

3x � y � 10 y � 3x � 10

x � 4y � 12 y � ��14�x � 3

The solution to the system is (4, 2).

Systems of linear equations can also be solved algebraically using the elimination method or the substitution method.

Example 2Use the elimination method to solvethe system of equations.2x � 3y � �215x � 6y � 15To solve this system, multiply each sideof the first equation by 2, and add thetwo equations to eliminate y. Then solvethe resulting equation.2(2x � 3y) � 2(�21) → 4x � 6y � �424x � 6y � �425x � 6y � 159x � �27

x � �3Now substitute �3 for x in either of theoriginal equations.

5x � 6y � 155(�3) � 6y � 15 x � �3

6y � 30y � 5

The solution is (�3, 5).

Example 3Use the substitution method tosolve the system of equations.x � 7y � 32x � y � �7The first equation is stated in terms of x,so substitute 7y � 3 for x in the secondequation.

2x � y � �72(7y � 3) � y � �7

13y � �13y � �1

Now solve for x by substituting �1 for yin either of the original equations.x � 7y � 3x � 7(�1) � 3 y � �1x � �4The solution is (�4, �1).

becomes

A consistent system has at least one solution. A system having exactly one solution is independent.If a system has infinitely many solutions, the system is dependent.Systems that have no solution are inconsistent.

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© Glencoe/McGraw-Hill 46 Advanced Mathematical Concepts

Solving Systems of Equations in Two Variables

State whether each system is consistent and independent,consistent and dependent, or inconsistent.

1. �x � y � �4 2. 2x � 5y � 83x � 3y � 12 15y � 6x � �24

Solve each system of equations by graphing.

3. x � y � 6 4. x � y � 62x � 3y � 12 3x � y � 6

Solve each system of equations algebraically.

5. x � y � 4 6. 3x � 4y � 10 7. 4x � 3y � 153x � 2y � 7 �3x � 4y � 8 2x � y � 5

8. 4x � 5y � 11 9. 2x � 3y � 19 10. 2x � y � 63x � 2y � �9 7x � y � 9 x � y � 6

11. Real Estate AMC Homes, Inc. is planning to build three- and four-bedroomhomes in a housing development called Chestnut Hills. Consumer demandindicates a need for three times as many four-bedroom homes as for three-bedroomhomes. The net profit from each three-bedroom home is $16,000 and from eachfour-bedroom home, $17,000. If AMC Homes must net a total profit of $13.4 millionfrom this development, how many homes of each type should they build?

PracticeNAME _____________________________ DATE _______________ PERIOD ________

2-1

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© Glencoe/McGraw-Hill 47 Advanced Mathematical Concepts

EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

2-1

Set Theory and Venn DiagramsSet theory, which was developed by the nineteenth century Germanlogician and mathematician, Georg Cantor, became a fundamentalunifying principle in the study of mathematics in the middle of thetwentieth century. The use of sets permits the precise description ofmathematical concepts.

The intersection of two sets determines the elements common to thetwo sets. Thus, the intersection of two lines in a system of equationsrefers to the point or points that are common to the sets of pointsbelonging to each of the lines. We can use Venn diagrams, which arenamed for the British logician, John Venn, to visually represent theintersection of two sets.Example Let U � the set of all points in the Cartesian coordinate plane.

Let A � the set of all points that satisfy the equation x � 4.Let B � the set of all points that satisfy the equation y � �2.Draw a Venn diagram to represent U, A, and B.

The shaded region represents theintersection of sets A and B, writ-ten A � B. In this example,A � B � {(4, –2)}.

Since the solution sets of two parallel lines have no points in com-mon, the sets are called disjoint sets. In a Venn diagram, such setsare drawn as circles that do not overlap.

A � B � �.

Use the diagram at the right to answer the following questions.

Let A = the set of all points that satisfy the equation for line p.

Let B = the set of all points that satisfy the equation for line q.

1. In which numbered region do the points that satisfy only theequation for line p lie?

2. In which numbered region do the points that satisfy only theequation for line q lie?

3. In which numbered region do the points that satisfy the equationsfor neither line p nor line q lie?

4. If the equation of line p is 2x � y � 4 and the equation of line q is3x � y � 6, which point lies in region 2?

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© Glencoe/McGraw-Hill 48 Advanced Mathematical Concepts

Solving Systems of Equations in Three VariablesYou can solve systems of three equations in three variablesusing the same algebraic techniques you used to solvesystems of two equations.

Example Solve the system of equations by elimination.

4x � y � 2z � �82x � 3y � 4z � 4�7x � 2y � 3z � 2

Choose a pair of equations and then eliminate one of thevariables. Because the coefficient of y is 1 in the firstequation, eliminate y from the second and third equations.

To eliminate y using the first andsecond equations, multiply both sides ofthe first equation by 3.

3(4x � y � 2z) � 3(�8)12x � 3y � 6z � �24

Then add the second equation to thatresult.

12x � 3y � 6z � �242x � 3y � 4z � 4

14x � 2z � �20

Now you have two linear equations in two variables. Solvethis system. Eliminate z by multiplying both sides of thesecond equation by 2. Then add the two equations.

2(�15x � z) � 2(18) 14x � 2z � �20�30x � 2z � 36 �30x � 2z � 36

�16x � 16x � �1 The value of x is �1.

By substituting the value of x into oneof the equations in two variables, we cansolve for the value of z.

14x � 2z � �2014(�1) � 2z � �20 x � �1

z � �3

The value of z is �3.

The solution is x � �1, y � 2, z � �3. This can be written asthe ordered triple (�1, 2, �3).

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

2-2

To eliminate y using the first and thirdequations, multiply both sides of the firstequation by �2.

�2(4x � y � 2z) � �2(�8)�8x � 2y � 4z � 16

Then add the third equation to thatresult.

�8x � 2y � 4z � 16�7x � 2y � 3z � 2

�15x �z � 18

Finally, use one of the original equations to find the value of y.

4x � y � 2z � �84(�1) � y � 2(�3) � �8 x � �1, z � �3

y � 2

The value of y is 2.

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© Glencoe/McGraw-Hill 49 Advanced Mathematical Concepts

PracticeNAME _____________________________ DATE _______________ PERIOD ________

Solving Systems of Equations in Three VariablesSolve each system of equations.

1. x � y � z � �1 2. x � y � 5x � y � z � 3 3x � z � 23x � 2y � z � �4 4y � z � 8

3. 3x � 5y � z � 8 4. 2x � 3y � 3z � 24y � z � 10 10x � 6y � 3z � 07x � y � 4 4x � 3y � 6z � 2

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

7. x � z � 5 8. 2x � 4y � 2z � 9y � 3z � 12 4x � 6y � 2z � �92x � y � 7 x � y � 3z � �4

9. Business The president of Speedy Airlines has discovered that her competitor, Zip Airlines, has purchased 13 new airplaines from Commuter Aviation for a total of $15.9 million. She knows that Commuter Aviation produces three types of planes and that type A sells for $1.1 million, type B sells for $1.2 million, and type C sells for $1.7 million. The president ofSpeedy Airlines also managed to find out that Zip Airlines purchased 5 more type A planes than type C planes. How many planes of each type did Zip Airlines purchase?

2-2

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© Glencoe/McGraw-Hill 50 Advanced Mathematical Concepts

EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

2-2

Graph ColoringThe student council is scheduling a volleyball tournament for teamsfrom all four classes. They want each class team to play every otherclass team exactly once. How should they schedule the tournament?

If we call the teams A, B, C, and D, all of the possible games among thefour teams can be represented as AB, AC, AD, BC, BD, and CD.

Draw a graph to represent the problem. Then color the graph.To color a graph means to color the vertices of that graph so that notwo vertices connected by an edge have the same color.

Let the vertices represent the possible games. Let the edges representgames that cannot be scheduled at the same time. For example, if teamB is playing team C, then team C cannot playteam D.

Choose a vertex at which to begin.

• Color AB red. Since CD is not connected to AB,color it red as well. All of the other vertices areconnected to AB, so do not color them red.

• Color AC blue. Since BD is not connected toAC, color it blue.

• Color AD green. Since BC is not connected toAD, color it green.

The colored graph shows that pairs of games can bescheduled as follows

AB with CD; AC with BD; AD with BC

The chromatic number of a graph is the least number of colors necessary to color the graph. The chromatic number of the graph above is 3.

1. What does the chromatic number of the graph in the example aboverepresent?

2. Draw and color a graph to represent the same type of tournament,but with 6 teams playing.

3. What is the chromatic number for your graph?

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© Glencoe/McGraw-Hill 51 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

Modeling Real-World Data with MatricesA matrix is a rectangular array of terms called elements.A matrix with m rows and n columns is an m � n matrix.The dimensions of the matrix are m and n.

Example 1 Find the values of x and y for which

� � � � � is true.

Since the corresponding elements mustbe equal, we can express the equalmatrices as two equations.

3x � 5 � y � 9y � �4x

Solve the system of equations by usingsubstitution.

To add or subtract matrices, the dimensions of the matricesmust be the same.

To multiply two matrices, the number of columns in the first matrixmust be equal to the number of rows in the second matrix. Theproduct of two matrices is found by multiplying columns and rows.

Example 3 Find each product if X � � � and Y � � �.a. XY

XY � � � or � �b. YX

YX � � � or � �2415

326

8(3) � 0(2)9(3) � (�6)(2)

8(4) � 0(5)9(4) � (�6)(5)

�18�12

5958

4(0) � 3(�6)5(0) � 2(�6)

4(8) � 3(9)5(8) � 2(9)

0�6

89

32

45

y � 9�4x

3x � 5y

2-3

3x � 5 � y � 93x � 5 � (�4x) � 9 Substitute �4x for y.

x � �2

y � �4xy � �4(�2) Substitute �2 for x.y � 8

The matrices are equal if x � �2 and y � 8.

Example 2 Find C � D if C � � � and D � � �.C � D � C � (�D)

� � � � � �� � �� � �1

�425

6 � (�5)4 � (�8)

3 � (�1)�2 � 7

�5�8

�17

64

3�2

58

1�7

64

3�2

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© Glencoe/McGraw-Hill 52 Advanced Mathematical Concepts

Modeling Real-World Data with Matrices

Find the values of x and y for which each matrix equation istrue.

1. � � � 2. � � �

Use matrices A, B, and C to find each sum, difference, orproduct.

A � � � B � � � C � � �3. A � B 4. A � B

5. B � A 6. �2A

7. CA 8. AB

9. AA 10. CB

11. (CA)B 12. C(AB)

13. Entertainment On one weekend, the Goxfield Theater reported the followingticket sales for three first-run movies, asshown in the matrix at the right. If theticket prices were $6 for each adult and$4 for each child, what were the weekend sales for each movie.

�914

1012

8�6

143

31

�2

2�1

5

6�2

2

5�7

4

�124

y3x

2x � 34y

2y � 42x

xy

PracticeNAME _____________________________ DATE _______________ PERIOD ________

2-3

Adults Children

� �523785

2456

102125473652

Movie 1Movie 2Movie 3

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© Glencoe/McGraw-Hill 53 Advanced Mathematical Concepts

EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

2-3

1. Find the elementary matrix that willinterchange rows 1 and 3 of matrix A.

3. Find the elementary matrix that willmultiply the elements of the first rowof matrix A by –2 and add the resultsto the corresponding elements of thethird row.

2. Find the elementary matrix that willmultiply row 2 of matrix A by 5.

4. Find the elementary matrix that willinterchange rows 2 and 3 of matrix Aand multiply row 1 by –2.

5. Find the elementary matrix that will multiply the elements of the second row ofmatrix A by –3 and add the results to the corresponding elements of row 1.

Elementary Matrix TransformationsElementary row transformations can be made by multiplying a matrix onthe left by the appropriate transformation matrix. For example, to interchange rows 2 and 3 in a 3 � 3 matrix, multiply the

matrix on the left by the matrix � �.

Example Let C � � �. Multiply on the left by � �.

� � � �� � �More complicated row transformations can also be made by a matrix multiplier.

Example Find the elementary matrix that, when matrix A is multiplied by it on the left, row 1 will be replaced by the sum of 2 times row 1 and row 3.

Let A � � �.Find M such that M � � �

� �. Try M � � �.

Check to see that the product MA meets the required conditions.

2 0 10 1 0 0 0 1

2a11 � a31 2a12 � a32 2a13 � a33a21 a22 a23a31 a32 a33

a11 a12 a13a21 a22 a23a31 a32 a33

a11 a12 a13a21 a22 a23a31 a32 a33

–5 3 110 –6 –2

1 5 0

–5 3 11 5 0

10 –6 �2

1 0 00 0 10 1 0

1 0 00 0 10 1 0

�5 3 11 5 0

10 �6 �2

1 0 00 0 10 1 0

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© Glencoe/McGraw-Hill 54 Advanced Mathematical Concepts

Modeling Motion with MatricesYou can use matrices to perform many transformations, such astranslations (slides), reflections (flips), rotations (turns), anddilations (enlargements or reductions).Example 1 Suppose quadrilateral EFGH with vertices

E(�1, 5), F(3, 4), G(4, 0), and H(�2, 1) istranslated 2 units left and 3 units down.

The vertex matrix for the quadrilateral is � �.The translation matrix is � �.Adding the two matrices gives the coordinates of thevertices of the translated quadrilateral.

� � � � � � � �Graphing the pre-image and the image of thetranslated quadrilateral on the same axes, wecan see the effect of the translation.

The chart below summarizes the matrices needed to produce specific reflections or rotations. All rotations are counterclockwise about the origin.

Example 2 A triangle has vertices A(�1, 2), B(4, 4), and C(3, �2).Find the coordinates of the image of the triangle after a rotation of 90° counterclockwise about the origin.

The vertex matrix is � �.Multiply it by the 90° rotation matrix.

� � · � � � � �Example 3 Trapezoid WXYZ has vertices W(2, 1),

X(1, �2), Y(�1, �2), and Z(�2, 1). Find thecoordinates of dilated trapezoid W′X′Y′Z′ fora scale factor of 2.5.Perform scalar multiplication on the vertexmatrix for the trapezoid.

23

�44

�2�1

3�2

44

�12

�10

01

3�2

44

�12

�4�2

2�3

11

�32

�2�3

�2�3

�2�3

�2�3

�21

40

34

�15

�2�3

�2�3

�2�3

�2�3

�21

40

34

�15

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

2-4

Reflections Rx-axis � � � Ry-axis � � � Ry�x � � �

Rotations Rot90 � � � Rot180 � � � Rot270 � � �10

0�1

0�1

�10

�10

01

10

01

01

�10

0�1

10

y

O x

A

BB´

C

2.5� � �� ��52.5

�2.5�5

2.5�5

52.5

�21

�1�2

1�2

21

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© Glencoe/McGraw-Hill 55 Advanced Mathematical Concepts

PracticeNAME _____________________________ DATE _______________ PERIOD ________

Modeling Motion with Matrices

Use scalar multiplication to determine the coordinates of thevertices of each dilated figure. Then graph the pre-image and theimage on the same coordinate grid.

Use matrices to determine the coordinates of the vertices of eachtranslated figure. Then graph the pre-image and the image on thesame coordinate grid.

Use matrices to determine the coordinates of the vertices of eachreflected figure. Then graph the pre-image and the image on thesame coordinate grid.

Use matrices to determine the coordinates of the vertices of eachrotated figure. Then graph the pre-image and the image on thesame coordinate grid.

2-4

1. triangle with vertices A(1, 2), B(2, �1), and C(�2, 0); scale factor 2

2. quadrilateral with vertices E(�2, �7),F (4, �3), G(0, 1),and H(�4, �2); scalefactor 0.5

3. square with vertices W(1, �3), X(�4, �2),Y(�3, 3), and Z(2, 2) translated 2 units right and 3 units down

4. triangle with vertices J(3, 1), K(2, �4), and L(0, �2) translated 4 units left and 2 units up

5. �MNP with vertices M(�3, 4), N(3, 1), and P(�4, �3) reflected over the y-axis

6. a rhombus with vertices Q(2, 3),R(4, �1), S(�1, �2),and T(�3, 2) reflected over the line y � x

7. quadrilateral CDFGwith vertices C(�2, 3),D(3, 4), F(3, �1), andG(�3, �4) rotated 90°

8. Pentagon VWXYZwith vertices V(1, 3),W(4, 2), X(3, �2),Y(�1, �4), Z(�2, 1) rotated 180°

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© Glencoe/McGraw-Hill 56 Advanced Mathematical Concepts

EnrichmentNAME _____________________________ DATE _______________ PERIOD ________

2-4

Map ColoringEach of the graphs below is drawn on a plane with no accidental crossings of its edges.This kind of graph is called a planar graph. A planar graph separates the plane intoregions. A very famous rule called the four-color theorem states that every planar graphcan be colored with at most four colors, without any adjacent regions having the samecolor.

Graph A Graph B Graph C Graph Done region two regions three regions three regionsone color two colors two colors three colors

If there are four or more regions, two, three, or four colors may be required.

Example Color the map at the right using theleast number of colors.

Look for the greatest number of isolatedstates or regions. Virginia, Georgia, andMississippi are isolated. Color them pink.Alabama and North Carolina are also isolated. Color them blue.South Carolina and Tennessee are remaining. Because they do not share a common edge, they may be colored with thesame color. Color them both green.Thus, three colors are needed to color thismap.

Color each map with the least number of colors.

1. 2.

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© Glencoe/McGraw-Hill 57 Advanced Mathematical Concepts

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

2-5

Determinants and Multiplicative Inverses of MatricesEach square matrix has a determinant. The determinant of

a 2 � 2 matrix is a number denoted by or det� �.Its value is a1b2 � a2b1.

Example 1 Find the value of .

� 5(1) � 3(�2) or 11

The minor of an element can be found by deleting the row and column containing the element.

The minor of a1 is .

Example 2 Find the value of ·

� 8 � 9 � 3 � 8(6) � 9(19) � 3(11) or �90

The multiplicative inverse of a matrix is defined as follows.

If A � � � and � 0, then A�1 � � �.

Example 3 Solve the system of equations by using matrix equations.5x � 4y � �33x � 5y � �24

Write the system as a matrix equation.

� � · � � � � �To solve the matrix equation,first find the inverse of the coefficient matrix.

� � � ��317�� ��4

5�5�3

�45

�5�3

�3�24

xy

4�5

53

�b1a1

b2�a2

b1b2

a1a2

b1b2

a1a2

52

3�1

74

3�1

74

52

374

952

83

�1

374

952

83

�1

c2c3

b2b3

�21

53

�21

53

b1b2

a1a2

b1b2

a1a2

Now multiply each side of the matrix equation by the inverse and solve.

��317�� � �

The solution is (�3, 3).

�45

�5�3

c1c2c3

b1b2b3

a1a2a3

���317� � � � � ��3

�24�4

5�5�3

4�5

53

1

� � � � �xy

4�5

53

� � � � ��33

xy

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© Glencoe/McGraw-Hill 58 Advanced Mathematical Concepts

Determinants and Multiplicative Inverses of Matrices

Find the value of each determinant.

1. � � 2. � �

3. � � 4. � �Find the inverse of each matrix, if it exists.

5. � � 6. � �

Solve each system by using matrix equations.

7. 2x � 3y � 17 8. 4x � 3y � �163x � y � 9 2x � 5y � 18

Solve each matrix equation.

9. � � · � � � � �, if the inverse is ��16

�� �

10. � � · � � � � �, if the inverse is ��18

� � �11. Landscaping Two dump truck have capacities of 10 tons and 12 tons.

They make a total of 20 round trips to haul 226 tons of topsoil for a landscaping project. How many round trips does each truck make?

122

�14

�101

13

2�1�5

�201

xyz

421

�2�4�3

531

�715

�11�1

7

�11

�1

�83

�7

xyz

31

�2

�12

�3

21

�1

24

510

85

3�1

152

3�1�4

2�3

1

045

�11

�3

125

�59

37

3�12

�28

PracticeNAME _____________________________ DATE _______________ PERIOD ________

2-5

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© Glencoe/McGraw-Hill 59 Advanced Mathematical Concepts

NAME _____________________________ DATE _______________ PERIOD ________

Area of a TriangleDeterminants can be used to find the area of a triangle onthe coordinate plane. For a triangle with vertices P1(x1,y1), P2(x2, y2), and P3(x3, y3), the area is given by thefollowing formula.

Area � � �The sign is chosen so that the result is positive.

Example Find the area of the triangle with vertices A(3, 8), B(�2, 5), and C(0, �1).

Area � � � � �3� � � 8� � � 1� ��

� [3(5 � (–1)) � 8(–2 � 0) � 1(2 � 0)]

� [36] � 18

The area is 18 square units.

Find the area of the triangle having vertices with the given coordinates.

1. A(0, 6), B(0, �4), C(0, 0) 2. A(5, 1), B(–3, 7), C(–2, –2)

3. A(8, –2), B(0, �4), C(–2, 10) 4. A(12, 4), B(–6, 4), C(3, 16)

5. A(–1, –3), B(7, –5), C(–3, 9) 6. A(1.2, 3.1), B(5.7, 6.2),C(8.5, 4.4)

7. What is the sign of the determinant in the formula when the points are taken inclockwise order? in counterclockwise order?

1�2

1�2

–2 50 �1

–2 10 1

5 1–1 1

1�2

3 8 1–2 5 10 �1 1

1�2

x1 y1 1x2 y2 1x3 y3 1

1�2

Enrichment2-5

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© Glencoe/McGraw-Hill 60 Advanced Mathematical Concepts

Solving Systems of Linear InequalitiesTo solve a system of linear inequalities, you must find theordered pairs that satisfy all inequalities. One way to do thisis to graph the inequalities on the same coordinate plane. Theintersection of the graphs contains points with ordered pairsin the solution set. If the graphs of the inequalities do notintersect, then the system has no solution.

Example 1 Solve the system of inequalitiesby graphing.

The shaded region represents the solution to thesystem of inequalities.

A system of more than two linear inequalities can have asolution that is a bounded set of points called a polygonalconvex set.

Example 2 Solve the system of inequalities by graphing and name the coordinates of the vertices of the polygonal convex set.

The region shows points that satisfy all threeinequalities. The region is a triangle whosevertices are the points at (0, 0), (0, 5), and (8, 0).

Example 3 Find the maximum and minimum values of ƒ(x, y) � x � 2y � 1 for the polygonal convex set determined by the following inequalities.x � 0 y � 0 2x � y � 4 x � y � 3

First, graph the inequalities and find thecoordinates of the vertices of the resultingpolygon.

The coordinates of the vertices are (0, 0),(2, 0), (1, 2), and (0, 3).

Then, evaluate the function ƒ(x, y) � x � 2y � 1at each vertex.ƒ(0, 0) � 0 � 2(0) � 1 � 1 ƒ(1, 2)� 1 � 2(2) � 1 � 6ƒ(2, 0) � 2 � 2(0) � 1 � 3 ƒ(0, 3) � 0 � 2(3) � 1 � 7

Thus, the maximum value of the function is 7,and the minimum value is 1.

Study GuideNAME _____________________________ DATE _______________ PERIOD ________

2-6

�x � 2y � 2x � 4

x � 0y � 05x � 8y � 40

x

y

O

(4, 3)

(2, 0)

(1, 2)

(0, 3)2 + = 4x y

x y+ = 3

1 2 3

1

2

y

O x

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© Glencoe/McGraw-Hill 61 Advanced Mathematical Concepts

Practice

NAME _____________________________ DATE _______________ PERIOD ________

Solving Systems of Linear InequalitiesSolve each system of inequalities by graphing.1. �4x � 7y �21; 3x � 7y � 28 2. x � 3; y � 5; x � y 1

Solve each system of inequalities by graphing. Name thecoordinates of the vertices of the polygonal convex set.3. x 0; y 0; y x � 4; 7x � 6y � 54 4. x 0; y � 2 0; 5x � 6y � 18

Find the maximum and minimum values of each function forthe polygonal convex set determined by the given system ofinequalities.5. 3x � 2y 0 y 0 6. y � �x � 8 4x � 3y �3

3x � 2y � 24 ƒ(x, y) � 7y � 3x x � 8y 8 ƒ(x, y) � 4x � 5y

7. Business Henry Jackson, a recent college graduate,plans to start his own business manufacturing bicycle tires. Henry knows that his start-up costs are going to be $3000 and that each tire will cost him at least $2 to manufacture. In order to remain competitive, Henry cannot charge more than $5 per tire. Draw a graph to show when Henry will make a profit.

2-6

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© Glencoe/McGraw-Hill 62 Advanced Mathematical Concepts

Reading Mathematics: Reading the Graphof a System of InequalitiesLike all other kinds of graphs, the graph of a system of linear inequalities is useful to us only as long as we are able to “read” theinformation presented on it.

The line of each equation separates the xy-planeinto two half-planes. Shading is used to tell whichhalf-plane satisfies the inequality bounded by theline. Thus, the shading below the boundary line y � 4 indicates the solution set of y � 4. By drawing horizontal or vertical shadinglines or lines with slopes of 1 or �1, and by usingdifferent colors, you will make it easier to “read”your graph.

The shading in between the boundary lines x 0 andx � 5, indicate that the inequality 0 � x � 5 is equivalent to the intersection x 0 � x � 5.

Similarly, the solution set of the system, indicated by the cross-hatching, is the intersection of the solutions of all the inequalities inthe system. We can use set notation to write a description of this solution set:

{(x, y)|0 � x � 5 � y � 4 � x � 2y 7}.

We read this “ The set of all ordered pairs, (x, y), such that x is greaterthan or equal to 0 and x is less than or equal to 5 and y is less than orequal to 4 and x � 2y is greater than or equal to 7.”

The graph also shows where any maximum and minimum values ofthe function f occur. Notice, however, that these values occur only atpoints belonging to the set containing the union of the vertices of theconvex polygonal region.

Use the graph at the right to show which numbered region or regions belong to the graph of the solution set of each system.

1. x 0 2. 0 � x � 4y � x y � x

x � y 4

3. The domain of f(x, y) � 2x � y is the xy-plane. If we graphed this function, wherewould we represent its range?

Enrichment

NAME _____________________________ DATE _______________ PERIOD ________

2-6

0 � x � 5y � 4

x � 2y 7

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© Glencoe/McGraw-Hill 63 Advanced Mathematical Concepts

Study Guide

NAME _____________________________ DATE _______________ PERIOD ________

Linear ProgrammingThe following example outlines the procedure used to solvelinear programming problems.

Example The B & W Leather Company wants to addhandmade belts and wallets to its product line. Eachbelt nets the company $18 in profit, and each walletnets $12. Both belts and wallets require cutting andsewing. Belts require 2 hours of cutting time and 6 hours of sewing time. Wallets require 3 hours ofcutting time and 3 hours of sewing time. If thecutting machine is available 12 hours a week and thesewing machine is available 18 hours per week, whatmix of belts and wallets will produce the most profitwithin the constraints?

Define Let b � the number of belts.variables. Let w � the number of wallets.

Write b 0inequalities. w 0

2b � 3w � 12 cutting6b � 3w � 18 sewing

Graph the system.

Write an Since the profit on belts is $18 and theequation. profit on wallets is $12, the profit function

is B(b, w) � 18b � 12w.

Substitute B(0, 0) � 18(0) � 12 (0) � 0values. B(0, 4) � 18(0) � 12(4) � 48

B(1.5, 3) � 18(1.5) � 12(3) � 63B(3, 0) � 18(3) � 12(0) � 54

Answer the The B & W Company will maximize profitproblem. if it makes and sells 1.5 belts for every

3 wallets.

When constraints of a linear programming problem cannot besatisfied simultaneously, then infeasibility is said to occur.

The solution of a linear programming problem is unbounded if theregion defined by the constraints is infinitely large.

2-7

(3, 0)

(0, 4) (1.5, 3)

w

O b

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© Glencoe/McGraw-Hill 64 Advanced Mathematical Concepts

Linear Programming

Graph each system of inequalities. In a problem asking youto find the maximum value of ƒ(x, y), state whether thesituation is infeasible, has alternate optimal solutions, or isunbounded. In each system, assume that x � 0 and y � 0unless stated otherwise.1. �2y 2x � 36 2. 2x � 2y 10

x � y 30 2x � y 8ƒ(x, y) � 3x � 3y ƒ(x, y) � x � y

Solve each problem, if possible. If not possible, statewhether the problem is infeasible, has alternate optimalsolutions, or is unbounded.3. Nutrition A diet is to include at least 140 milligrams of

Vitamin A and at least 145 milligrams of Vitamin B. Theserequirements can be obtained from two types of food. Type Xcontains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. If type X food costs $12 per pound and type Y food costs $8 perpound how many pounds of each type of food should bepurchased to satisfy the requirements at the minimum cost?

4. Manufacturing The Cruiser Bicycle Company makes two styles of bicycles: the Traveler, which sells for $200, and theTourester, which sells for $600. Each bicycle has the same frame and tires, but the assembly and painting time required for the Traveler is only 1 hour, while it is 3 hours for the Tourister. There are 300 frames and 360 hours of labor available for production. How many bicycles of each modelshould be produced to maximize revenue?

Practice

NAME _____________________________ DATE _______________ PERIOD ________

2-7

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© Glencoe/McGraw-Hill 65 Advanced Mathematical Concepts

Enrichment

NAME _____________________________ DATE _______________ PERIOD ________

Convex PolygonsYou have already learned that over a closed convex polygonal region,the maximum and minimum values of any linear function occur at thevertices of the polygon. To see why the values of the function at anypoint on the boundary of the region must be between the values at thevertices, consider the convex polygon with vertices P and Q.

Let W be a point on PQ__

.If W lies between P and Q, let �PP

WQ� � w.

Then 0 � w � 1 and the coordinates of W are((1 � w)x1 � wx2

, (1 � w)y1 � wy2). Now consider the function f(x, y) � 3x � 5y.

f (W) � 3[(1�w)x1 � wx2] � 5[(1�w)y1 � wy2]� (1�w)(3x1) � 3wx2 � (1�w)(�5y1) � 5wy2� (1�w)(3x1 � 5y1) � w(3x2 � 5y2)� (1�w)f(P) � wf(Q)

This means that f(W) is between f(P) and f(Q), or that the greatest and least values of f(x, y) must occur at P or Q.

Example If f (x, y) � 3x � 2y, find the maximum value of thefunction over the shaded region at the right.

The maximum value occurs at the vertex (6, 3). Theminimum value occurs at (0, 0). The values of f(x, y)at W1 and W2 are between the maximum and minimum values.

f(Q) � f(6, 3) � 24f(W1) � f(2, 1) � 8f(W2) � f(5, 2.5) � 20f(P) � f(0, 0) � 0

Let P and Q be vertices of a closed convex polygon, and let W lie onPQ__

. Let f(x, y) = ax + by.

1. If f(Q) � f(P), what is true of f ? of f(W)?

2. If f(Q) � f(P), find an equation of the line containing P and Q.

2-7

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BLANK

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© Glencoe/McGraw-Hill 67 Advanced Mathematical Concepts

Chapter 2 Test, Form 1A

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. Which system is inconsistent? 1. ________A. y � 0.5x B. 2x � y � 8 C. x � y � 0 D. y � �1

x � 2y � 1 4x � y � 5 2x � y x � 2

2. Which system of equations is shown by the graph? 2.A. �3x � y � 1 B. 3x � y � 1

x � y � 3 x � y � 3C. x � y � 3 D. �6x � 3y � �3

�x � y � 1 3x � y � 1

3. Solve algebraically. 2x � 7y � 5 3. ________6x � y � 5

A. ���815�, �

54

�� B. (1, �1) C. (6, �1) D. ��34

�, �12

��4. Solve algebraically. 6x � 3y � 8z � 4 4. ________

�x � 9y � 2z � 24x � 6y � 4z � 3

A. ��52

�, �53

�, ��34

�� B. (1, �2, �1) C. ��12

�, �13

�, �14

�� D. �5, �2, �54

��5. Find 2 A � B if A � � � and B � � �. 5. ________

6. Find the values of x and y for which � � � � � is true. 6. ________

A. (�1, �1) B. (�2, �4) C. (5, 1) D. (�1, 4)

7. Find DE if D � � � and E � � �. 7. ________

8. A triangle with vertices A(�3, 4), B(3, 1) and C(�1, �5) is rotated 90° 8. ________counterclockwise about the origin. Find the coordinates of A′, B′, and C′.A. A′(�3, �4) B. A′(3, �4) C. A′(4, 3) D. A′(�4, �3)

B′(3, �1) B′(�3, �1) B′(1, �3) B′(�1, 3)C′(�1, 5) C′(1, 5) C′(�5, 1) C′(5, �1)

9. Find the value of � �. 9. ________

A. 26 B. 22 C. �16 D. 20

�11

�3

2�2

1

130

�244

10

�3

6�1

4�7

�25

x � 13x

2y � 5y � 2

�15

�42

�3�2

61

Chapter

2

A. � � B. � � C. � � D. � ��71

84

�4�14

20�2

�5�9

160

�2�7

10�1

A. � � B. � � C. � � D. � ��18�4

0�28

�2�10

8�42

�2044

�10�28

�4

�20

�18

44�42

�208

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© Glencoe/McGraw-Hill 68 Advanced Mathematical Concepts

10. Find the inverse of � �, if it exists. 10. ________

A. does not exist B. ��17

�� � C. �17

�� � D. �17

�� �11. Which product represents the solution to the system? 11. ________

12. Which is the graph of the system y 0, 2x � 3y � 9, and x � 2 y? 12. ________A. B. C. D.

13. Find the minimum value of ƒ(x, y) � 2x � y � 2 for the polygonal 13. ________convex set determined by this system of inequalities.x 1 x � 3 y 0 �

12

�x � y � 5

A. 0 B. �3 C. 3 D. �0.5

14. Describe the linear programming situation for this system of 14. ________inequalities where you are asked to find the maximum value of ƒ(x, y) � x � y. x 0 y 0 6x � 3y � 18 x � 3y � 9A. infeasible B. unboundedC. an optimal solution D. alternate optimal solutions

15. A farmer can plant a combination of two different crops on 20 acres 15. ________of land. Seed costs $120 per acre for crop A and $200 per acre for crop B. Government restrictions limit acreage of crop A to 15 acres,but do not limit crop B. Crop A will take 15 hours of labor per acre at a cost of $5.60 per hour, and crop B will require 10 hours of labor per acre at $5.00 per hour. If the expected income from crop A is $600 per acre and from crop B is $520 per acre, how many acres of crop A should be planted in order to maximize profit?A. 5 acres B. 20 acres C. 15 acres D. infeasible

Bonus The system below forms a polygonal convex set. Bonus: ________x � 0 y �x, if �6 � x � 0y � 10 2x � 3y 6, if �12 � x � �6What is the area of the closed figure?

A. 60 units2 B. 52 units2 C. 54 units2 D. 120 units2

�4�1

�31

�13

14

�13

14

11

3�4

Chapter 2 Test, Form 1A (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

A. ��217� �

247�� � � �1

14B. ���

73

� �43

�� � � �114

C. ��73

� ��43

� � � � �114

D. ���217� ��

247�� � � �1

14

�y � 7x � 14�x � 4y � 1

��13

� �13

�13

� ��13

� ��247� ��

217�

1�27

7�27

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© Glencoe/McGraw-Hill 69 Advanced Mathematical Concepts

Chapter 2 Test, Form 1B

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. Which term(s) describe(s) this system? 4x � y � 8 1. ________3x � 2y � �5

A. dependent B. consistent and dependentC. consistent and independent D. inconsistent

2. Which system of equations is shown by the graph? 2.A. 3x � y � 8 B. y � x

x � 1 3x � �4yC. y � 2x � 1 D. x � y � 4

x � y � 4 3x � y � 8

3. Solve algebraically. 5x � y � 16 3. ________2x � 3y � 3

A. (4, 4) B. (�1, 3) C. (9, �5) D. (3, �1)

4. Solve algebraically. x � 3y � 3z � 0 4. ________2x � 5y � 5z � 1�x � 5y � 6z � �9

A. (3, 4, �3) B. (3, 0, 1) C. (�2, 4, 3) D. (0, �3, 3)

5. Find A � B if A � � � and B � � �. 5. ________

A. � � B. � � C. � � D. � �6. Find the values of x and y for which � � � � � is true. 6. ________

A. �0, ��35�� B. �0, �53�� C. �0, �35�� D. ��2, �53��7. Find DE if D � [5 2] and E � � �. 7. ________

A. � � B. [57] C. [45 �12] D. [33]

8. A triangle with vertices A(�3, 4), B(3, 1), and C(�1, �5) is translated 8. ________5 units left and 3 units up. Find the coordinates of A′, B′, and C′.A. A′(�2, 7) B. A′(�8, 7) C. A′(�8, 7) D. A′(�8, 1)

B′(8, 4) B′(2, �4) B′(�2, 4) B′(�2, �2)C′(�7, �2) C′(6, 2) C′(�6, �2) C′(�6, �8)

9. Find the value of | |. 9. ________

A. 39 B. 37 C. �37 D. 33

�17

52

45�12

9�6

x � 50

10

3y2x6y

�24

�6

�144

�2�2�6

1�2

4

�2�2

6

�1�2

4

24

�8

�1144

�2�3

1

6�3

0

01

�7

�514

Chapter

2

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© Glencoe/McGraw-Hill 70 Advanced Mathematical Concepts

10. Find the inverse of � �, if it exists. 10. ________

A. does not B. ��112�� �C. ��1

12�� � D. ��1

12�� �

exist

11. Which product represents the solution to the system? 3x � 4y � �2 11. ________2x � 5y � 6

A. � �·� � B. �17�� �·� � C. � �·� � D. �17�� �·� �12. Which is the graph of the system? y � 0 x y 12. ________

x � 0 x � y �4A. B. C. D.

13. Find the maximum and minimum values of ƒ(x, y) � 2y � x for the 13. ________polygonal convex set determined by this system of inequalities.x 1 y 0 x � 4 � yA. minimum: 1; maximum: 4 B. minimum: 0; maximum: 5C. minimum: 1; maximum: 7 D. minimum: 4; maximum: 8

14. Describe the linear programming situation for this system of 14. ________inequalities.x � 1 y 0 3x � y � 5A. infeasible B. unboundedC. an optimal solution D. alternate optimal solutions

15. A farm supply store carries 50-lb bags of both grain pellets and 15. ________grain mash for pig feed. It can store 600 bags of pig feed. At leasttwice as many of its customers prefer the mash to the pellets. Thestore buys the pellets for $3.75 per bag and sells them for $6.00. Itbuys the mash for $2.50 per bag and sells it for $4.00. If the storeorders no more than $1400 worth of pig feed, how many bags ofmash should the store order to make the most profit?A. 160 bags B. 320 bags C. 200 bags D. 400 bags

Bonus Find the value of c for which the system is consistent and Bonus: ________dependent.3x � y � 54.5y � c � 9x

A. no value B. �8 C. 9.5 D. 7.5

�26

�43

5�2

�26

�23

5�4

�26

25

34

�26

25

34

6�2

3�1

�6�2

31

63

�2�1

�6�2

31

Chapter 2 Test, Form 1B (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

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© Glencoe/McGraw-Hill 71 Advanced Mathematical Concepts

Chapter 2 Test, Form 1C

NAME _____________________________ DATE _______________ PERIOD ________

Write the letter for the correct answer in the blank at the right ofeach problem.

1. Which term(s) describe(s) this system? 3x � 2y � 7 1. ________�9x � 6y � �10

A. dependent B. consistent and dependentC. consistent and independent D. inconsistent

2. Solve by graphing. x � y � �1 2. ________x � 2y � 5

A. B. C. D.

3. Solve algebraically. y � 2x � 7 3. ________y � 5 � x

A. (8, �3) B. (2, 3) C. (4, 1) D. (�2, 3)4. Solve algebraically. x � y � z � 5 4. ________

�2x � y � z � �63x � 3y � 2z � �5

A. (1, 0, 4) B. (0, �1, 6) C. (�1, �2, 4) D. (�3, 0, 2)

5. Find A � B if A � � � and B � � �. 5. ________

6. Find the values of x and y for which � � � � � is true. 6. ________

A. (2, 3) B. (4, 1) C. (�1, 6) D. (1, 4)

7. Find EF if E � � � and F � � �. 7. ________

8. A triangle of vertices A(�3, 4), B(3, 1), and C(�1, �5) is dilated by a 8. ________scale factor of 2. Find the coordinates of A′, B′, C′.A. A′(�3, 4) B. A′(�3, 8) C. A′(�6, 8) D. A′(�1.5, 2)

B′(3, �1) B′(3, 2) B′(6, 2) B′(1.5, 0.5)C′(�1, 5) C′(�1, �10) C′(�2, �10) C′(�0.5, �2.5)

9. Find the value of � �. 9. ________

A. 6 B. �3 C. 14 D. �6

12

54

12

3�1

�21

�12

5 � x2y � 5

y3x

�1�3

24

�58

1�4

30

2�1

Chapter

2

A. � � B. � � C. � � D. � �74

�7

�350

0�7

54

�37

07

54

3�7

�747

360

A. � � B. � � C. � � D. � ��54

�15

�2�3

2�2

�22

�3�2

�30

�57

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© Glencoe/McGraw-Hill 72 Advanced Mathematical Concepts

10. Find the inverse of � �, if it exists. 10. ________

11. Which product represents the solution to the system? � � � � � � � � 11. ________

12. Which is the graph of the system x 0 y 0 x � 3y � 6? 12. ________A. B. C. D.

13. Find the maximum value of ƒ(x, y) � y � x � 1 for the polygonal 13. ________convex set determined by this system of inequalities.x 0 y 0 2x � y � 4A. 1 B. 8 C. 5 D. �2

14. Which term best describes the linear programming 14. ________situation represented by the graph?A. infeasibleB. unboundedC. an optimal solutionD. alternate optimal solutions

15. Chase Quinn wants to expand his cut-flower business. He has 12 15. ________additional acres on which he intends to plant lilies and gladioli. He can plant at most 7 acres of gladiolus bulbs and no more than 11 acres of lilies. In addition, the number of acres planted to gladioli Gcan be no more than twice the number of acres planted to lilies L.The inequality L � 2G 10 represents his labor restrictions. If his profits are represented by the function ƒ(L, G) � 300L � 200G, how many acres of lilies should he plant to maximize his profit?A. 1 acre B. 11 acres C. 0 acres D. 9.5 acres

Bonus Solve the system|y| 2 and|x|� 1 by graphing. Bonus: ________

A. no solution B. C. D.

32

xy

14

12

10

73

Chapter 2 Test, Form 1C (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

A. does not exist B. ��13

�� � C. ��13

�� � D. ��14

�� ��17

0�3

10

73

�17

0�3

A. �12

�� � � � � B. �12

�� � � � �C. � � � � � D. � � � � �3

2�1

14

�232

14

12

32

�11

4�2

32

�14

1�2

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© Glencoe/McGraw-Hill 73 Advanced Mathematical Concepts

Chapter 2 Test, Form 2A

NAME _____________________________ DATE _______________ PERIOD ________

1. Identify the system shown by the 1. __________________graph as consistent and independent,consistent and dependent, or inconsistent.

2. Solve by graphing. 2.6 � y � �3x2x � 6y � 4 � 0

3. Solve algebraically. 4x � y � �3 3. __________________5x � 2y � 1

4. Solve algebraically. �3x � y � z � 2 4. __________________5x � 2y � 4z � 21x � 3y � 7z � �10

Use matrices J, K, and L to evaluate each expression. If the matrixdoes not exist, write impossible.

J � � � K � � � L � � � 5. __________________

6. __________________

5. �2K � L 6. 3K � J 7. JL 7. __________________

8. For what values of x, y, and z is � ��� � true? 8. __________________

9. A triangle with vertices A(�3, 4), B(5, �2), and C(7, �4) 9. __________________is rotated 90° countrclockwise about the origin. Find the coordinates of A′, B′, and C′.

10. Find the value of � �. 10. __________________

11. If it exists, find A�1 if A � � �. 11. __________________71

�35

1�5

3

�321

47

�1

y0

�4y

2 � 2x�3z � 4x

9z � 6

3�1

2�2

�47

�10

52

�83

4�3

1

�526

Chapter

2

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© Glencoe/McGraw-Hill 74 Advanced Mathematical Concepts

12. Write the matrix product that represents the solution to 12. __________________the system.4x � 3y � �15x � 2y � 3

Solve each system of inequalities by graphing and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each set.13. y � 5 13.

x � 33y � 4x 0ƒ(x, y) � 2x � y �1

14. y 0 14.0 � x � 4�x � y � 6ƒ(x, y) � 3x � 5y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.15. The BJ Electrical Company needs to hire master electricians

and apprentices for a one-week project. Master electricians receive a salary of $750 per week and apprentices receive $350 per week. As part of its contract, the company has agreed to hire at least 30 workers. The local Building Safety Council recommends that each master electrician allow 15. __________________3 hours for inspection time during the project. This project should require at least 24 hours of inspection time. How many workers of each type should be hired to meet the safety requirements, but minimize the payroll?

16. A company makes two models of light fixtures, A and B,each of which must be assembled and packed. The time required to assemble model A is 12 minutes, and model B takes 18 minutes. It takes 2 minutes to package model A and 1 minute to package model B. Each week, 240 hours are available for assembly time and 20 hours for packing.a. If model A sells for $1.50 and model B sells for $1.70, 16a. _________________

how many of each model should be made to obtain the maximum weekly profit?

b. What is the maximum weekly profit? 16b. _________________

Bonus Use a matrix equation to find the value of x for the system.ax � by � cdx � ey � ƒ Bonus: __________________

Chapter 2 Test, Form 2A (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

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© Glencoe/McGraw-Hill 75 Advanced Mathematical Concepts

Chapter 2 Test, Form 2B

NAME _____________________________ DATE _______________ PERIOD ________

1. Identify the system shown by the 1. __________________graph as consistent and independent,consistent and dependent, or inconsistent.

2. Solve by graphing. 5x � 2y � 8 2.x � y � 3

3. Solve algebraically. 3x � 2y � 1 3. __________________2x � 3y � 18

4. Solve. 2x � y � z � �3 4. __________________y � z � 1 � 0x � y � z � 9

Use matrices D, E, and F to find each sum or product. 5. __________________

D � � � E � � � F � � � 6. __________________

5. E � D 6. 3F 7. DF 7. __________________

8. For what values of x and y is [5 �3x] � [�4x 5y] true? 8. __________________

9. A triangle with vertices A(3, �3), B(1, 4), and C(�2, �1) 9. __________________is reflected over the line with equation y � x. Find the coordinates of A′, B′, and C′.

10. Find the value of | |. 10. __________________

11. Given A � � �, find A�1, if it exists. 11. __________________�6

432

�7�2

35

1�6

54

�23

�416

�302

15

�4

�273

Chapter

2

2

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© Glencoe/McGraw-Hill 76 Advanced Mathematical Concepts

12. Write the matrix product that represents the solution 12. __________________to the system.3x � 2y � 55x � 4y � 7

Solve each system of inequalities by graphing and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each function.

13. x � y 13. __________________y � 0x �4ƒ(x, y) � 7x � y

14. y 0 14. __________________x �1y � x � 32x � y � 6ƒ(x, y) � 5x � 3y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.

15. A manufacturer of garden furniture makes a Giverny 15. __________________bench and a Kensington bench. The company needs toproduce at least 15 Giverny benches a day and at least 20 Kensington benches a day. It must also meet the demandfor at least twice as many Kensington benches as Givernybenches. The company can produce no more than 60 benchesa day. If each Kensington sells for $250 and each Givernysells for $325, how many of each kind of bench should beproduced for maximum daily income?

16. André Gagné caters small dinner parties on weekends. 16. __________________Because of an increase in demand for his work, he needs tohire more chefs and waiters. He will have to pay each chef$120 per weekend and each waiter $70 per weekend. Heneeds at least 2 waiters for each chef he hires. Find themaximum amount André will need to spend to hire extrahelp for one weekend.

Bonus Find an equation of the line that passes through Bonus: __________________P(2, 1) and through the intersection of�32

x� � �4y

� � 7 and �5x� � �

23y� � �73�.

Chapter 2 Test, Form 2B (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

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© Glencoe/McGraw-Hill 77 Advanced Mathematical Concepts

Chapter 2 Test, Form 2C

NAME _____________________________ DATE _______________ PERIOD ________

1. Determine whether the system 1. __________________shown by the graph is consistant and independent,consistent and dependent, or inconsistent.

2. Solve by graphing. 2.x � y � 3x � y � 1

3. Solve algebraically. 4x � 3y � 5 3. __________________4x � 3y � 10

4. Solve algebraically. x � 2y � z � �5 4. __________________3x � 2y � z � 32x � y � 2z � �7

Use matrices A, B, and C to find the sum or product.

A � � � B � � � C � � � 5. __________________

6. __________________

5. A � B 6. �2B 7. AC 7. __________________

8. For what values of x and y is � � � � � true? 8. __________________

9. A triangle with vertices A(1, 2), B(2, �4),and C(�2, 0) 9. __________________is translated 2 units left and 3 units up. Find the coordinates A′, B′, and C′.

10. Find the value of � �. 10. __________________

11. If it exists, find A�1 if A � � �. 11. __________________03

21

2�7

�35

5yx � 7

xy � 3

1�6

25

�30

64

�1

8�5

3

�25

�1

�306

Chapter

2

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© Glencoe/McGraw-Hill 78 Advanced Mathematical Concepts

12. Write the matrix product that represents the solution to the system. 3x � 4y � 5

x � 2y � 10 12. __________________

Solve each system of inequalities by graphing, and name thevertices of each polygonal convex set. Then, find the maximumand minimum values for each function.13. x 0 13.

y 0x � y � 5ƒ(x, y) � 5x � 3y

14. x 0 14.0 � y � 42x � y � 6ƒ(x, y) � 3x � 2y

Solve each problem, if possible. If not possible, state whether the problem is infeasible, has alternate optimal solutions, or isunbounded.15. The members of the junior class at White Mountain High 15. __________________

School are selling ice-cream and frozen yogurt cones in the school cafeteria to raise money for their prom. The students have enough ice cream for 50 cones and enough frozen yogurt for 80 cones. They have 100 cones available. If they plan to sell each ice-cream cone for $2 and each frozen yogurt cone for $1, and they sell all 100 cones, what is the maximum amount they can expect to make?

16. Ginny Dettore custom-sews bridal gowns and bridesmaids’ 16. __________________dresses on a part-time basis. Each dress sells for $200, and each gown sells for $650. It takes her 2 weeks to produce a bridesmaid’s dress and 5 weeks to produce a bridal gown.She accepts orders for at least three times as many bridesmaids’ dresses as she does bridal gowns. In the next 22 weeks, what is the maximum amount of money she can expect to earn?

Bonus Find the coordinates of the vertices of the Bonus: __________________triangle determined by the graphs of 4x � 3y � 1 � 0, 4x � 3y � 17 � 0, and 4x � 9y � 13 � 0.

Chapter 2 Test, Form 2C (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

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© Glencoe/McGraw-Hill 79 Advanced Mathematical Concepts

Chapter 2 Open-Ended Assessment

NAME _____________________________ DATE _______________ PERIOD ________

Instructions: Demonstrate your knowledge by giving clear, concisesolutions to each problem. Be sure to include all relevantdrawings and justify your answers. You may show your solution inmore than one way or investigate beyond the requirements of theproblem.

1. Stores A and B sell cassettes, CDs, and videotapes. The figures given are in the thousands.

Total Units Total UnitsSales Sold Sales Sold

� �Store A Store B

a. Find A � B and write one or two sentences describing the meaning of the sum.

b. Find B � A and write one or two sentences describing the meaning of the difference.

c. Find 2B. Interpret its meaning, and describe at least one situation in which a merchant may be interested in this product.

d. If C = [2 3 3], find CB and write a word problem to illustrate a possible meaning of the product.

e. Does CB � BC? Why or why not?

2. A builder builds two styles of houses: the Executive, on which he makes a profit of $30,000, and the Suburban, on which he makes a $25,000 profit. His construction crew can complete no more than 10 houses each year, and he wishes to build no more than 6 of the Executive per year.a. How many houses of each style should he build to maximize

profit? Explain why your answer makes sense.

b. If he chooses to limit the number of Executives he builds each year to 8, how many houses of each style should he build if he wishes to maximize profit? Explain your answer.

c. If he keeps the original limit on Executives, but raises the limit on the total number of houses he builds, what would be the effect on the number of each style built to maximize profit? Why?

200150175

$1000$1500$2000

CassettesA � Videotapes

CDs

Chapter

2

� �350250250

$1750$2500$3000

CassettesB � Videotapes

CDs

Page 46: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts

1. Identify the system shown by the graph as consistent and independent, consistent and dependent, or inconsistent 1. __________________

2. Solve by graphing. x � 2y � 7 2.y � �x � 2

3. Solve algebraically. 4x � 2y � 6 3. __________________3x � 4y � 10

4. Solve algebraically. x � 2y � z � 3 4. __________________2x � 3y � 2z � �1x � 3y � 2z � 1

Use matrices A, B, and C to evaluate each expression. If the matrix does not exist, write impossible.

A � � � B � � � C � � � 5. __________________

6. __________________

5. A � (�B) 6. �4C 7. BC 7. __________________

8. For what values of x and y is � � � � � true? 8. __________________

9. A quadrilateral with vertices A(�3, 5), B(1, 2), C(6, 3) 9. __________________and D(�4, �5) is translated 2 units left and 7 units down. Find the coordinates of A′, B′, C′, and D′.

10. Find the coordinates of a triangle with vertices E(3, 4), 10. __________________F(5, �1), and G(�2, 3) after a rotation of 180°counterclockwise about the origin.

y � 2x � 6�14

�3xy

3y � 1

23

�41

0�3

21

3�5

42

5�1

Mid-Chapter Test (Lessons 2-1 through 2-4)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2

Page 47: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

1. Identify the system shown by the graph 1. __________________as consistent and independent, consistent and dependent, or inconsistent.

2. Solve by graphing. y � �54�x � 3 2.x � 2y � 8

Solve each system of equations algebraically.

3. 2x � y � �2 4. 3x � 5y � 21 3. __________________4x � 2y � �4 x � y � 5

4. __________________5. 2x � y � 3z � 8

x � 2y � 2z � 3 5. __________________5x � y � z � 1

Chapter 2 Quiz B (Lessons 2-3 and 2-4)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 2 Quiz A (Lessons 2-1 and 2-2)

NAME _____________________________ DATE _______________ PERIOD ________

© Glencoe/McGraw-Hill 81 Advanced Mathematical Concepts

Chapter

2

Chapter

2Use matrices A, B, and C to evaluate each expression. If thematrix does not exist, write impossible.

A � � � B � � � C � � �1. __________________

2. __________________

1. A � B 2. CB 3. 2B � A 3. __________________

4. For what values of x and y is � � � � � true? 4. __________________

5. Each of the following transformations is applied to a 5a. _________________triangle with vertices A(4, 1), B(�5, �3), and C(�2, 6).Find the coordinates of A′, B′, and C′ in each case. 5b. _________________a. translation right 3 units and down 5 unitsb. reflection over the line y � x 5c. _________________c. rotation of 90° counterclockwise about the origin

2y � xx

x � 62y � 1

�426

35

�8

�32

24

9�8

�64

Page 48: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

1. Find the value of � �. 1. __________________

2. Find the multiplicative inverse of � �, if it exists. 2. __________________

3. Solve the system 6x � 4y � 0 and 3x � y � 1 by using a 3. __________________matrix equation.

Use the system x � 0, y � 0, y � 4, 2x � y � 4and the function f(x, y) � 3x � 2y for Exercises 4 and 5.

4. Solve the system by graphing, and name the vertices of 4.the polygonal convex set.

5. Find the minimum and maximum values of the function. 5. __________________

2�4

�48

612

�10

�2

4�3

5

Chapter 2 Quiz D (Lessons 2-7)

NAME _____________________________ DATE _______________ PERIOD ________

Chapter 2 Quiz C (Lessons 2-5 and 2-6)

NAME _____________________________ DATE _______________ PERIOD ________

© Glencoe/McGraw-Hill 82 Advanced Mathematical Concepts

Chapter

2

Chapter

21. Carpentry Emilio has a small carpentry shop where he 1. __________________

makes large and small bookcases. His profit on a large bookcase is $80 and on a small bookcase is $50. It takes Emilio 6 hours to make a large bookcase and 2 hours to make a small one. He can spend only 24 hours each week on his carpentry work. He must make at least two of each size each week. What is his maximum weekly profit?

2. Painting Michael Thomas, the manager of a paint store, 2. __________________is mixing paint for a spring sale. There are 32 units of yellow dye, 54 units of brown dye, and an unlimited supply of base paint available. Mr. Thomas plans to mix as many gallons as possible of Autumn Wheat and Harvest Brown paint. Each gallon of Autumn Wheat requires 4 units of yellow dye and 1 unit of brown dye. Each gallon of Harvest Brown paint requires 1 unit of yellow dye and 6 units of brown dye. Find the maximum number of gallons of paint that Mr. Thomas can mix.

Page 49: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill 83 Advanced Mathematical Concepts

SAT and ACT Practice

NAME _____________________________ DATE _______________ PERIOD ________Chapter

2After working each problem, record thecorrect answer on the answer sheetprovided or use your own paper.

1. If x � 3 � 2(1 � x) � 1, then what isthe value of x?A. �8B. �2C. 2D. 5E. 8

2. If �ab� � 0.625, then �ab� is equal to which

of the following?A. 1.60B. 2.67C. 2.70D. 3.33E. 4.25

3. Points A, B, C, and D are arranged on a line in that order. If AC � 13,BD � 12, and AD � 21, then BC � ?A. 12B. 9C. 8D. 4E. 3

4. A group of z people buys x widgetsat a price of y dollars each. If the people divide the cost of this purchaseevenly, then how much must each person pay in dollars?A. �

xzy�

B. �yxz�

C. �xyz�

D. xy � zE. xyz

5. If 7a � 2b � 11 and a � 2b � 5, thenwhat is the value of a?A. �2.0B. �0.5C. 1.4D. 2.0E. It cannot be determined from the

information given.

6. In the figure below, if m � n andb � 125, then c � f � ?

A. 50B. 55C. 110D. 130E. 180

7. Twenty-five percent of 28 is what percent of 4?A. 7%B. 57%C. 70%D. 128%E. 175%

8. Using the figure below, which of thefollowing is equal to a � c?

A. 2bB. b � 90C. d � eD. b � d � eE. 180 � (d � e)

9. At what coordinates does the line 3y � 5 � x � 1 intersect the y-axis?A. (0, �2)B. (0, �1)C. �0, �13��D. (�2, 0)E. (�6, 0)

10. �(n3)6

n�

2(n4)5� �

A. n9

B. n16

C. n19

D. n36

E. n40

b°c°

Page 50: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill 84 Advanced Mathematical Concepts

SAT and ACT Practice (continued)

NAME _____________________________ DATE _______________ PERIOD ________Chapter

211. If m varies directly as n and �

mn� � 5,

then what is the value of m whenn � 2.2?A. 0.44B. 2.27C. 4.10D. 8.20E. 11.00

12. If ��x �� � x2 � 3x, then ���3�� �A. 0B. 6.0C. 18.0D. 3.3E. 9.0

13. How many distinct 3-digit numberscontain only nonzero digits?A. 909B. 899C. 789D. 729E. 504

14. If 2x2 � 5x � 9 � 0, then x could equalwhich of the following?A. �1.12B. 0.76C. 1.54D. 2.63E. 3.71

15. If P�A� and P�B� are tangent to circle Oat A and B, respectively, then which ofthe following must be true?

I. PB POII. �APO � �BPOIII. �APB � �AOB � �PAO � �PBOA. I onlyB. II onlyC. I and II onlyD. II and III onlyE. I, II, and III

16. In the picture below, if the ratio of QR to RS is 2:3 and the ratio of PQ toQS is 1:2, then what is the ratio of PQto RS?

A. 5:6B. 7:6C. 5:4D. 11:6E. 11:5

17–18. Quantitive ComparisonA. if the quantity in Column A is

greaterB. if the quantity in Column B is

greaterC. if the two quantities are equalD. if the relationship cannot be deter-

mined for the information givenn is 12% of m

Column A Column B

17.

r5 � tr 1

18.

19. Grid-In If the average of m, m � 3,m � 10, and 2m � 11 is 4, what is themode?

20. Grid-In A radio station schedules aone-hour block of time to cover a localmusic competition. The station runstwelve 30-second advertisements anduses the remaining time to broadcastlive music. What is the value of �aldivveer

mtiussinicgttiimm

ee�?

6m

t2 � t

50n

r10 � r6

Page 51: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill 85 Advanced Mathematical Concepts

Chapter 2 Cumulative Review (Chapters 1-2)

NAME _____________________________ DATE _______________ PERIOD ________

1. State the domain and range of {(�5, 2), (4, 3), (�2, 0), 1. __________________(�5, 1)}. Then state whether the relation is a function.Write yes or no.

2. If ƒ(x) � �4x2 and g(x) � �2x�, find [ g � ƒ](x). 2. __________________

3. Write the slope-intercept form of the equation of the line 3. __________________that passes through the point (�5, 4) and has a slope of �1.

4. Write the standard form of the equation of the line 4. __________________perpendicular to the line y � 3x � 5 that passes through (�3, 7).

Graph each function.

5. ƒ(x) � 5.

6. ƒ(x) � �x� 6.

7. How many solutions does a consistent and dependentsystem of linear equations have? 7. __________________

Solve each system of equations algebraically.8. 4x � 7y � �2 9. �3x � 2y � 3z � �1 8. __________________

x � 2y � 7 2x � 5y � 3z � �64x � 3y � 3z � 22 9. __________________

10. Find �2A � B if A � � � and B � � �. 10. __________________

11. Find the value of � �. 11. __________________

12. A triangle with vertices A(�2, 5), B(�3, 7) and C(6, �2) 12. __________________is ref lected over the line with equation y = x. Find the coordinates of A′, B′, and C′.

13. Write the matrix product that 13. __________________represents the solution to the system. � �·� � � � �

14. Find the maximum and minimum values of 14. __________________ƒ(x, y) � 3x � y for the polygonal convex setdetermined by x 1, y 0, and x � 0.5y � 2.

15. Champion Lumber converts logs into lumber or plywood. 15. __________________In a given week, the total production cannot exceed 800 units, of which 200 units of lumber and 300 units of plywood are required by regular customers. The profit on a unit of lumber is $20 and on a unit of plywood is $30.Find the number of units of lumber and plywood that should be produced to maximize profit.

�51

xy

31

1�1

5�2

37

14

3�2

�85

�2�1

14

�53

�2 if x � �1�23�x � �13� if x �1

Chapter

2

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BLANK

Page 53: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill A1 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(10 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 9

8

7

6

5

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1

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Page 54: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

© Glencoe/McGraw-Hill A2 Advanced Mathematical Concepts

SAT and ACT Practice Answer Sheet(20 Questions)

NAME _____________________________ DATE _______________ PERIOD ________

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

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Page 55: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

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Page 56: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

Answers (Lesson 2-2)

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rap

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3.W

hat

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chro

mat

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you

r gr

aph

?5

Page 57: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

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Page 58: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

Answers (Lesson 2-4)

© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts

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are

fou

r or

mor

e re

gion

s, t

wo,

th

ree,

or

fou

r co

lors

may

be

requ

ired

.

Exa

mp

leC

olor

th

e m

ap a

t th

e ri

ght

usi

ng

the

leas

t n

um

ber

of

colo

rs.

Loo

k fo

r th

e gr

eate

st n

um

ber

of is

olat

edst

ates

or

regi

ons.

Vir

gin

ia,G

eorg

ia, a

nd

Mis

siss

ippi

are

isol

ated

. Col

or t

hem

pin

k.A

laba

ma

and

Nor

th C

arol

ina

are

also

is

olat

ed. C

olor

th

em b

lue.

Sou

th C

arol

ina

and

Ten

nes

see

are

rem

ain

ing.

Bec

ause

th

ey d

o n

ot s

har

e a

com

mon

edg

e, t

hey

may

be

colo

red

wit

h t

he

sam

e co

lor.

Col

or t

hem

bot

h g

reen

.T

hu

s, t

hre

e co

lors

are

nee

ded

to c

olor

th

ism

ap.

Col

or e

ach

map

wit

h t

he

leas

t n

um

ber

of

colo

rs. S

ee s

tud

ents

’ map

s.1.

2.

Page 59: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

Answers (Lesson 2-5)

© Glencoe/McGraw-Hill A7 Advanced Mathematical Concepts

© G

lenc

oe/M

cGra

w-H

ill59

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Are

a o

f a

Tri

an

gle

Det

erm

inan

ts c

an b

e u

sed

to f

ind

the

area

of

a tr

ian

gle

onth

e co

ordi

nat

e pl

ane.

For

a t

rian

gle

wit

h v

erti

ces

P1(

x 1,y 1)

, P2(

x 2, y

2), a

nd

P3(

x 3, y

3), t

he

area

is g

iven

by

the

foll

owin

g fo

rmu

la.

Are

a �

�T

he

sign

is c

hos

en s

o th

at t

he

resu

lt is

pos

itiv

e.

Exa

mp

leF

ind

th

e ar

ea o

f th

e tr

ian

gle

wit

h v

erti

ces

A(3

, 8),

B(�

2, 5

), a

nd

C(0

,�1)

.

Are

a �

���3 �

�� 8�

�� 1�

��

[3(5

�(–

1))�

8(– 2

�0)

�1(

2�

0)]

[36]

�18

Th

e ar

ea is

18

squ

are

un

its.

Fin

d t

he

area

of

the

tria

ng

le h

avin

g v

erti

ces

wit

h t

he

giv

en c

oord

inat

es.

1.A

(0, 6

), B

(0,�

4), C

(0, 0

)2.

A(5

, 1),

B(–

3, 7

), C

(–2,

– 2)

Po

ints

are

co

lline

ar; t

here

33 u

nits

2

is n

o t

rian

gle

.

3.A

(8,–

2), B

(0,�

4), C

(–2,

10)

4.A

(12,

4),

B(–

6, 4

), C

(3, 1

6)

58 u

nits

210

8 un

its2

5.A

(–1,

– 3),

B(7

,–5)

, C(–

3, 9

)6.

A(1

.2, 3

.1),

B(5

.7, 6

.2),

C(8

.5, 4

.4)

46 u

nits

28.

39 u

nits

2

7.W

hat

is t

he

sign

of

the

dete

rmin

ant

in t

he

form

ula

wh

en t

he

poin

ts a

re t

aken

incl

ockw

ise

orde

r? in

cou

nte

rclo

ckw

ise

orde

r?

neg

ativ

e; p

osi

tive1 � 21 � 2

– 25

0�

1– 2

10

15

1– 1

11 � 2

38

1– 2

51

0�

11

1 � 2

x 1y 1

1x 2

y 21

x 3y 3

11 � 2

Enr

ichm

ent

2-5

© G

lenc

oe/M

cGra

w-H

ill58

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

De

term

ina

nts

an

d M

ult

iplic

ative

In

vers

es

of

Ma

tric

es

Fin

d t

he

valu

e of

eac

h d

eter

min

ant.

1.�

�2.

��

062

3.�

�4.

��

782

Fin

d t

he

inve

rse

of e

ach

mat

rix,

if it

exi

sts.

5.�

�6.

��

� 21 3��

�d

oes

no

t ex

ist

Sol

ve e

ach

sys

tem

by

usi

ng

mat

rix

equ

atio

ns.

7.2x

�3y

�17

8.4x

�3y

��

163x

�y

�9

2x�

5y�

18(4

,�3)

(�1,

4)

Sol

ve e

ach

mat

rix

equ

atio

n.

9.�

�· ����

�, if

the

inve

rse

is�

�1 6� ��(�

4, 3

, 1)

10.

��· ���

��, i

f th

e in

vers

e is

��1 8�

����

1,�

�1 2�,�1 2� �

11.

La

nd

sca

pin

gTw

o du

mp

tru

ck h

ave

capa

citi

es o

f 10

ton

s an

d 12

ton

s.

Th

ey m

ake

a to

tal

of 2

0 ro

un

d tr

ips

to h

aul

226

ton

s of

top

soil

for

a

lan

dsca

pin

g pr

ojec

t. H

ow m

any

rou

nd

trip

s do

es e

ach

tru

ck m

ake?

7 tr

ips

by

the

10-t

on

truc

k, 1

3 tr

ips

by

the

12-t

on

truc

k

12 2�

14

�10 1 13

2�

1�

5

�2 0 1

x y z

4 2 1

�2

�4

�3

5 3 1

�7 1 5

�11 �1 7

�1 1

�1

�8 3

�7

x y z

3 1�

2

�1 2

�3

2 1�

1

�8 3

5 1

2 45 10

8 53

�1

1 5 2

3�

1�

4

2�

3 1

0 4 5

�1 1

�3

1 2 5

�5 9

3 73

�12

�2 8

Pra

ctic

eN

AM

E__

____

____

____

____

____

____

___

DAT

E__

____

____

____

_ P

ER

IOD

____

____

2-5

Page 60: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

Answers (Lesson 2-6)

© Glencoe/McGraw-Hill A8 Advanced Mathematical Concepts

© G

lenc

oe/M

cGra

w-H

ill62

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

Re

ad

ing

Ma

the

ma

tic

s:

Re

ad

ing

th

e G

rap

ho

f a

Sy

ste

m o

f In

eq

ua

litie

sL

ike

all o

ther

kin

ds o

f gr

aph

s, t

he

grap

h o

f a

syst

em o

f li

nea

r in

equ

alit

ies

is u

sefu

l to

us

only

as

lon

g as

we

are

able

to

“rea

d” t

he

info

rmat

ion

pre

sen

ted

on it

.

Th

e li

ne

of e

ach

equ

atio

n s

epar

ates

th

e xy

-pla

ne

into

tw

o h

alf-

pla

nes

. Sh

adin

g is

use

d to

tel

l wh

ich

hal

f-pl

ane

sati

sfie

s th

e in

equ

alit

y bo

un

ded

by t

he

lin

e. T

hu

s, t

he

shad

ing

belo

w t

he

bou

nda

ry

lin

e y

�4

indi

cate

s th

e so

luti

on s

et o

f y

�4.

By

draw

ing

hor

izon

tal o

r ve

rtic

al s

had

ing

lin

es o

r li

nes

wit

h s

lope

s of

1 o

r �

1, a

nd

by u

sin

gdi

ffer

ent

colo

rs, y

ou w

ill m

ake

it e

asie

r to

“re

ad”

you

r gr

aph

.

Th

e sh

adin

g in

bet

wee

n t

he

bou

nda

ry li

nes

x

0 an

dx

�5,

indi

cate

th

at t

he

ineq

ual

ity

0�

x�

5 is

eq

uiv

alen

t to

th

e in

ters

ecti

onx

0

� x

�5.

Sim

ilar

ly, t

he

solu

tion

set

of

the

syst

em, i

ndi

cate

d by

th

e cr

oss-

hat

chin

g, is

th

e in

ters

ecti

on o

f th

e so

luti

ons

of a

ll t

he

ineq

ual

itie

s in

the

syst

em. W

e ca

n u

se s

et n

otat

ion

to

wri

te a

des

crip

tion

of

this

so

luti

on s

et:

{(x,

y)|

0�

x�

5 �

y�

4 �

x�

2y

7}.

We

read

th

is “

Th

e se

t of

all

ord

ered

pai

rs, (

x, y

), s

uch

th

at x

is g

reat

erth

an o

r eq

ual

to

0 an

dx

is le

ss t

han

or

equ

al t

o 5

and

yis

less

th

an o

req

ual

to

4 an

dx

�2y

is g

reat

er t

han

or

equ

al t

o 7.

Th

e gr

aph

als

o sh

ows

wh

ere

any

max

imu

m a

nd

min

imu

m v

alu

es o

fth

e fu

nct

ion

f o

ccu

r. N

otic

e, h

owev

er, t

hat

th

ese

valu

es o

ccu

r on

ly a

tpo

ints

bel

ongi

ng

to t

he

set

con

tain

ing

the

un

ion

of

the

vert

ices

of

the

con

vex

poly

gon

al r

egio

n.

Use

th

e g

rap

h a

t th

e ri

gh

t to

sh

ow w

hic

h n

um

ber

ed r

egio

n o

r re

gio

ns

bel

ong

to

the

gra

ph

of

the

solu

tion

set

of

each

sys

tem

.

1.x

0

2.0

�x

�4

y�

xy

�x

5, 6

, 7x

�y

4

53.

Th

e do

mai

n o

f f(

x, y

)�2x

�y

is t

he

xy-p

lan

e. I

f w

e gr

aph

ed t

his

fu

nct

ion

, wh

ere

wou

ld w

e re

pres

ent

its

ran

ge? O

n a

3rd

axi

s, p

erp

end

icul

ar t

o t

he

xy-p

lane

.

En

ric

hm

en

t

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

2-6

0�

x�

5y

�4

x�

2y

7

© G

lenc

oe/M

cGra

w-H

ill61

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

Pra

ctic

e

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

So

lvin

g S

yst

em

s o

f L

ine

ar

Ine

qu

alit

ies

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s b

y g

rap

hin

g.

1.�

4x�

7y

�21

; 3x

�7y

�28

2.x

�3;

y�

5; x

�y

1

Sol

ve e

ach

sys

tem

of

ineq

ual

itie

s b

y g

rap

hin

g. N

ame

the

coor

din

ates

of

the

vert

ices

of

the

pol

ygon

al c

onve

x se

t.3.

x

0; y

0;

y

x�

4; 7

x�

6y�

544.

x

0; y

�2

0;

5x

�6y

�18

vert

ices

: (0,

0),

(0, 9

), (6

, 2),

(4, 0

)ve

rtic

es: (

0, 3

), (0

,�2)

, (6,

�2)

Fin

d t

he

max

imu

m a

nd

min

imu

m v

alu

es o

f ea

ch f

un

ctio

n f

orth

e p

olyg

onal

con

vex

set

det

erm

ined

by

the

giv

en s

yste

m o

fin

equ

alit

ies.

5.3x

�2y

0

y

0 6.

y�

�x

�8

4x�

3y

�3

3x�

2y�

24ƒ(

x, y

)�7y

�3x

x�

8y

8ƒ(

x, y

)�4x

�5y

max

�30

max

�32

min

��

24m

in�

�13

7.B

usi

nes

sH

enry

Jac

kson

, a r

ecen

t co

lleg

e gr

adu

ate,

pl

ans

to s

tart

his

ow

n b

usi

nes

s m

anu

fact

uri

ng

bicy

cle

tire

s. H

enry

kn

ows

that

his

sta

rt-u

p co

sts

are

goin

g to

be

$30

00 a

nd

that

eac

h t

ire

wil

l co

st h

im a

t le

ast

$2 t

o m

anu

fact

ure

. In

ord

er t

o re

mai

n c

ompe

titi

ve, H

enry

ca

nn

ot c

har

ge m

ore

than

$5

per

tire

. Dra

w a

gra

ph t

o sh

ow w

hen

Hen

ry w

ill

mak

e a

prof

it.

2-6

Page 61: Chapter 2 Resource Masters - rvrhs.enschool.orgrvrhs.enschool.org/ourpages/auto/2015/2/2/45861812/Chapter 2... · alternate optimal solution column matrix consistent constraints dependent

Answers (Lesson 2-7)

© Glencoe/McGraw-Hill A9 Advanced Mathematical Concepts

© G

lenc

oe/M

cGra

w-H

ill65

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

En

ric

hm

en

t

NA

ME

____

____

____

____

____

____

____

_ D

ATE

____

____

____

___

PE

RIO

D__

____

__

Co

nve

x P

oly

go

ns

You

hav

e al

read

y le

arn

ed t

hat

ove

r a

clos

ed c

onve

x po

lygo

nal

reg

ion

,th

e m

axim

um

an

d m

inim

um

val

ues

of

any

lin

ear

fun

ctio

n o

ccu

r at

th

eve

rtic

es o

f th

e po

lygo

n. T

o se

e w

hy

the

valu

es o

f th

e fu

nct

ion

at

any

poin

t on

th

e bo

un

dary

of

the

regi

on m

ust

be

betw

een

th

e va

lues

at

the

vert

ices

, con

side

r th

e co

nve

x po

lygo

n w

ith

ver

tice

s P

and

Q.

Let

W b

e a

poin

t on

PQ__

.If

W li

es b

etw

een

Pan

d Q

, let

�P PW Q��

w.

Th

en 0

�w

�1

and

the

coor

din

ates

of

W a

re

((1

�w

)x1

�w

x 2,(1

�w

)y1

�w

y 2). N

ow c

onsi

der

the

fun

ctio

n f

(x, y

)�3x

�5y

.

f(W

)�

3[(1

�w

)x1

�w

x 2]�

5[(1

�w

)y1

�w

y 2]�

(1�

w)(

3x1)

�3w

x 2�

(1�

w)(

�5y

1)�

5wy 2

�(1

�w

)(3x

1�

5y1)

�w

(3x 2

�5y

2)�

(1�

w)f

(P)�

wf(

Q)

Th

is m

ean

s th

at f

(W)

is b

etw

een

f(P

)an

d f(

Q),

or

that

th

e gr

eate

st a

nd

leas

t va

lues

of

f(x,

y)

mu

st o

ccu

r at

Por

Q.

Exa

mp

leIf

f(x

, y)

�3x

�2y

, fin

d t

he

max

imu

m v

alu

e of

th

efu

nct

ion

ove

r th

e sh

aded

reg

ion

at

the

righ

t.

Th

e m

axim

um

val

ue

occu

rs a

t th

e ve

rtex

(6,

3).

Th

em

inim

um

val

ue

occu

rs a

t (0

, 0).

Th

e va

lues

of

f(x,

y)

at W

1an

d W

2ar

e be

twee

n t

he

max

imu

m a

nd

min

imu

m v

alu

es.

f(Q

)�f(

6, 3

)�24

f(W

1)�

f(2,

1)�

8f(

W2)

�f(

5, 2

.5)�

20f(

P)�

f(0,

0)�

0

Let

P a

nd

Q b

e ve

rtic

es o

f a

clos

ed c

onve

x p

olyg

on, a

nd

let

W li

e on

PQ__

. Let

f(x

, y) =

ax

+ b

y.

1.If

f(Q

)�f(

P),

wh

at is

tru

e of

f?

of f

(W)?

It p

rod

uces

alt

erna

te o

pti

mal

solu

tio

ns a

ll al

ong

P �Q �

;f(W

)�f(

P).

2.If

f(Q

)�f(

P),

fin

d an

equ

atio

n o

f th

e li

ne

con

tain

ing

Pan

d Q

.S

amp

le a

nsw

er: a

x+

by

=f(

P)

2-7

© G

lenc

oe/M

cGra

w-H

ill64

Ad

vanc

ed M

athe

mat

ical

Con

cep

ts

Lin

ea

r P

rog

ram

min

g

Gra

ph

eac

h s

yste

m o

f in

equ

alit

ies.

In

a p

rob

lem

ask

ing

you

to f

ind

th

e m

axim

um

val

ue

of ƒ

(x, y

), st

ate

wh

eth

er t

he

situ

atio

n is

infe

asib

le, h

as a

lter

nat

e op

tim

al s

olu

tion

s, o

r is

un

bou

nd

ed. I

n e

ach

sys

tem

, ass

um

e th

at x

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© Glencoe/McGraw-Hill A10 Advanced Mathematical Concepts

Page 67

1. A

2. B

3. D

4. C

5. B

6. B

7. A

8. D

9. D

Page 68

10. C

11. A

12. B

13. D

14. C

15. C

Bonus: C

Page 69

1. C

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3. D

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6. B

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Page 70

10. A

11. D

12. A

13. C

14. B

15. B

Bonus: A

Chapter 2 Answer KeyForm 1A Form 1B

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� �

Chapter 2 Answer Key

Page 71

1. D

2. B

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Page 72

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© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts

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© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts

Page 75

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12.

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15. $150

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Chapter 2 Answer KeyForm 2B Form 2C

consistent anddependent

(3, �4)

(2, 4, �3)

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© Glencoe/McGraw-Hill A13 Advanced Mathematical Concepts

Chapter 2 Answer KeyCHAPTER 2 SCORING RUBRIC

Level Specific Criteria

3 Superior • Shows thorough understanding of the concepts addition,subtraction, and multiplication of matrices, sclar multiplicationof a matrix, and finding the maximum value of a function for apolygonal convex set.

• Uses appropriate strategies to solve problem.• Computations are correct.• Written explanations are exemplary.• Word problem concerning linear inequalities is appropriate and makes sense.

• Goes beyond requirements of some of all problems.

2 Satisfactory, • Shows thorough understanding of the concepts addition,with Minor subtraction, and multiplication of matrices, scalar multiplicationFlaws of a matrix, and finding the maximum value of a function for a

polygonal convex set.• Uses appropriate strategies to solve problem.• Computations are mostly correct.• Written explanations are effective.• Word problem concerning the product of matrices is appropriate and mades sense.

• Satisfies most requirements of problems.

1 Nearly • Shows understanding of the concepts addition, subtraction,Satisfactory, and multiplication of matrices, scalar multiplication of awith Serious matrix, and finding the maximum multiplication value of aFlaws function for a polygonal convex set.

• May not use appropriate strategies to solve problem.• Computations are mostly correct.• Written explanations are mostly correct.• Word problem concerning the product of matrices is mostly appropriate and sensible.

• Satisfies most requirements of problems.

0 Unsatisfactory • Shows little or no understanding of the concepts addition,subtraction, and multiplication of matrices, scalar multiplication of a matrix, and finding the maximum value of a function for a polygonal convex set.

• May not use appropriate strategies to solve problem.• Computations are incorrect.• Written explanations are not satisfactory.• Word problem concerning the product of matrices is not appropriate or sensible.

• Does not satisfy requirements of problems.

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© Glencoe/McGraw-Hill A14 Advanced Mathematical Concepts

Page 791a. A � B:

Total UnitsSales Sold

� �A � B gives the total sales and number of units sold for both stores. For example, the two stores sold 550,000 cassettes. The combinedcassette sales were $2,750,000

1b. B � A:Total UnitsSales Sold

� �B � A give the difference between total sales and number of units sold for the two years. For example, 100 more albums were sold in 1994 than in 1993.

1c. 2B:Total UnitsSales Sold

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Total UnitsSales Sold

CB: [$20,000 2200]Mrs. Carl wishes to sell twice asmany cassettes, three times asmany videotapes, and three times asmany CDs in 1995 as in 1994.What isthe total value and the total numberof units of her 1995 goal?

1e. CB � BC because BC is undefined.

2a. Let x � numbers of ExecutivesLet y � numbers of Suburbans.x � y � 10 total number of housesx � 6 number of Executives

Profit � $30,000x � $25,000y. Check(0, 10), (0, 0), (6, 0,) and (6, 4).$30,000(0) � $25,000(10) � $250,000$30,000(6) � $25,000(0) � $180,000$30,000(6) � $25,000(4) � $280,000He should build 6 Executives and 4 Suburbans. This answer makessense because he would be buildingthe maximum number of Executives, which are the most profitable.

2b. He should build 8 Executives and 2 Suburbans because he makes agreater profit on the Executives.

2c. He should increase the number ofSuburbans because he is already atthe limit on Executives.

700500500

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CassettesVideotape

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Chapter 2 Answer KeyOpen-Ended Assessment

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© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts

Mid-Chapter TestPage 80

1.

2.

3. (2, �1)

4. (�2, 1, 3)

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Chapter 2 Answer Key

consistent anddependent

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E′(�3, �4),F′(�5, 1),G′(2, �3)

consistent and independent

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A′(�1, 4), B′(3, �5), C ′(�6, �2)

A′(1, 4), B′(�3, �5), C ′(6, �2)

A′(7, �4), B′(�2, �8), C ′(1, 1)

Vertices: (0, 0), (0, 2), (0, 4), (4, 4)

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© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts

Page 831. C

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