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Chapter 2Resource Masters
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StudentWorksTM This CD-ROM includes the entire Student Edition along with theStudy Guide, Practice, and Enrichment masters.
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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
© 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
© 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.
© 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.
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
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
© 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
© 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
© 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
© 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.
© 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
© 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?
© 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.
© 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
© 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?
© 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
© 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
© 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
© 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
A´
BB´
C
C´
2.5� � �� ��52.5
�2.5�5
2.5�5
52.5
�21
�1�2
1�2
21
© 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°
© 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.
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
© 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
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4
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6
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4
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Chapter
2
© 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
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31
63
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31
Chapter 2 Test, Form 1B (continued)
NAME _____________________________ DATE _______________ PERIOD ________Chapter
2
© 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
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24
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1�4
30
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Chapter
2
A. � � B. � � C. � � D. � �74
�7
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54
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54
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A. � � B. � � C. � � D. � ��54
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�57
© 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
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A. �12
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2�1
14
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14
12
32
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32
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1�2
© 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
© 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
© 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
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35
1�6
54
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15
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�273
Chapter
2
2
© 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
© 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
© 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
© 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
© 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
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
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.
© 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
d°
b°c°
e°
a°
© 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
© 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
© 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
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
© 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
.. ./ /
.
99 9 987654321
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© G
lenc
oe/M
cGra
w-H
ill46
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
So
lvin
g S
yst
em
s o
f E
qu
atio
ns
in T
wo
Va
ria
ble
s
Sta
te w
het
her
eac
h s
yste
m is
con
sist
ent
and
ind
epen
den
t,co
nsi
sten
t an
d d
epen
den
t, o
r in
con
sist
ent.
1.�
x�
y�
�4
cons
iste
nt2.
2x�
5y�
8co
nsis
tent
3x�
3y�
12an
d d
epen
den
t15
y�
6x�
�24
and
dep
end
ent
Sol
ve e
ach
sys
tem
of
equ
atio
ns
by
gra
ph
ing
.
3.x
�y
�6
4.x
�y
�6
2x�
3y�
123x
�y
�6
Sol
ve e
ach
sys
tem
of
equ
atio
ns
alg
ebra
ical
ly.
5.x
�y
�4
6.3x
�4y
�10
7.4x
�3y
�15
3x�
2y�
7�
3x�
4y�
82x
�y
�5
(3, 1
)no
so
luti
on
(3,�
1)
8.4x
�5y
�11
9.2x
�3y
�19
10.
2x�
y�
63x
�2y
��
97x
�y
�9
x�
y�
6(�
1, 3
)(2
, 5)
(4, 2
)
11.
Rea
l E
sta
teA
MC
Hom
es, I
nc.
is
plan
nin
g to
bu
ild
thre
e-an
d fo
ur-
bedr
oom
hom
es i
n a
hou
sin
g de
velo
pmen
t ca
lled
Ch
estn
ut
Hil
ls. C
onsu
mer
dem
and
indi
cate
s a
nee
d fo
r th
ree
tim
es a
s m
any
fou
r-be
droo
m h
omes
as
for
thre
e-be
droo
mh
omes
. Th
e n
et p
rofi
t fr
om e
ach
th
ree-
bedr
oom
hom
e is
$16
,000
an
d fr
om e
ach
fou
r-be
droo
m h
ome,
$17
,000
. If A
MC
Hom
es m
ust
net
a t
otal
pro
fit
of $
13.4
mil
lion
from
th
is d
evel
opm
ent,
how
man
y h
omes
of
each
typ
e sh
ould
th
ey b
uil
d?20
0 th
ree-
bed
roo
m, 6
00 f
our
-bed
roo
m
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
2-1
Answers (Lesson 2-1)
© Glencoe/McGraw-Hill A3 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill47
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
2-1
Se
t Th
eo
ry a
nd
Ve
nn
Dia
gra
ms
Set
th
eory
, wh
ich
was
dev
elop
ed b
y th
e n
inet
een
th c
entu
ry G
erm
anlo
gici
an a
nd
mat
hem
atic
ian
, Geo
rg C
anto
r, b
ecam
e a
fun
dam
enta
lu
nif
yin
g pr
inci
ple
in t
he
stu
dy o
f m
ath
emat
ics
in t
he
mid
dle
of t
he
twen
tiet
h c
entu
ry. T
he
use
of
sets
per
mit
s th
e pr
ecis
e de
scri
ptio
n o
fm
ath
emat
ical
con
cept
s.
Th
e in
ters
ecti
on o
f tw
o se
ts d
eter
min
es t
he
elem
ents
com
mon
to
the
two
sets
. Th
us,
th
e in
ters
ecti
on o
f tw
o li
nes
in a
sys
tem
of
equ
atio
ns
refe
rs t
o th
e po
int
or p
oin
ts t
hat
are
com
mon
to
the
sets
of
poin
tsbe
lon
gin
g to
eac
h o
f th
e li
nes
. We
can
use
Ven
n d
iagr
ams,
wh
ich
are
nam
ed f
or t
he
Bri
tish
logi
cian
, Joh
n V
enn
, to
visu
ally
rep
rese
nt
the
inte
rsec
tion
of
two
sets
.E
xam
ple
Let
U�
the
set
of a
ll p
oin
ts i
n t
he
Car
tesi
an c
oord
inat
e p
lan
e.L
et A
�th
e se
t of
all
poi
nts
th
at s
atis
fy t
he
equ
atio
n x
�4.
Let
B�
the
set
of a
ll p
oin
ts t
hat
sat
isfy
th
e eq
uat
ion
y�
�2.
Dra
w a
Ven
n d
iagr
am t
o re
pre
sen
t U
, A, a
nd
B.
Th
e sh
aded
reg
ion
rep
rese
nts
th
ein
ters
ecti
on o
f se
ts A
and
B, w
rit-
ten
A�
B. I
n t
his
exa
mpl
e,
A�
B�
{(4,
–2)
}.
Sin
ce t
he
solu
tion
set
s of
tw
o pa
rall
el li
nes
hav
e n
o po
ints
in c
om-
mon
, th
e se
ts a
re c
alle
d d
isjo
int
sets
. In
a V
enn
dia
gram
, su
ch s
ets
are
draw
n a
s ci
rcle
s th
at d
o n
ot o
verl
ap. A
�B
��
.
Use
th
e d
iag
ram
at
the
rig
ht
to a
nsw
er t
he
follo
win
g q
ues
tion
s.
Let
A =
th
e se
t of
all
poi
nts
th
at s
atis
fy t
he
equ
atio
n f
or li
ne
p.
Let
B =
th
e se
t of
all
poi
nts
th
at s
atis
fy t
he
equ
atio
n f
or li
ne
q.
1.In
wh
ich
nu
mbe
red
regi
on d
o th
e po
ints
th
at s
atis
fy o
nly
th
eeq
uat
ion
for
lin
e p
lie?
12.
In w
hic
h n
um
bere
d re
gion
do
the
poin
ts t
hat
sat
isfy
on
ly t
he
equ
atio
n f
or li
ne
q li
e? 3
3.In
wh
ich
nu
mbe
red
regi
on d
o th
e po
ints
th
at s
atis
fy t
he
equ
atio
ns
for
nei
ther
lin
e p
nor
lin
e q
lie?
44.
If t
he
equ
atio
n o
f li
ne
pis
2x
�y
�4
and
the
equ
atio
n o
f li
ne
qis
3x�
y�
6, w
hic
h p
oin
t li
es in
reg
ion
2?
(2,0
)
Answers (Lesson 2-2)
© Glencoe/McGraw-Hill A4 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill49
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
So
lvin
g S
yst
em
s o
f E
qu
atio
ns
in T
hre
e V
ari
ab
les
Sol
ve e
ach
sys
tem
of
equ
atio
ns.
1.x
�y
�z
��
12.
x�
y�
5x
�y
�z
�3
3x�
z�
23x
�2y
�z
��
44y
�z
�8
(0, 1
, 2)
(10,
�5,
�28
)
3.3x
�5y
�z
�8
4.2x
�3y
�3z
�2
4y�
z�
1010
x�
6y�
3z�
07x
�y
�4
4x�
3y�
6z�
2
(2.2
,�11
.4,�
55.6
)��1 2�,
�2 3�,�
�1 3� �
5.2x
�y
�z
��
16.
4x�
4y�
2z�
3x
�y
�z
�1
�6x
�6y
�6z
�5
x�
2y�
z�
22x
�3y
�4z
�2
(�2,
�1,
2)
��1 33 �,�
2,�1 69 �
�
7.x
�z
�5
8.2x
�4y
�2z
�9
y�
3z�
124x
�6y
�2z
��
92x
�y
�7
x�
y�
3z�
�4
(20,
�33
, 15)
��1 2�,�3 2�,
�1 �
9.B
usi
nes
sT
he
pres
iden
t of
Spe
edy
Air
lin
es h
as d
isco
vere
d th
at h
er
com
peti
tor,
Zip
Air
lin
es, h
as p
urc
has
ed 1
3 n
ew a
irpl
ain
es f
rom
Com
mu
ter
Avi
atio
n f
or a
tot
al o
f $1
5.9
mil
lion
. Sh
e kn
ows
that
Com
mu
ter
Avi
atio
n
prod
uce
s th
ree
type
s of
pla
nes
an
d th
at t
ype
Ase
lls
for
$1.1
mil
lion
, typ
e B
se
lls
for
$1.2
mil
lion
, an
d ty
pe C
sel
ls f
or $
1.7
mil
lion
. Th
e pr
esid
ent
ofS
peed
y A
irli
nes
als
o m
anag
ed t
o fi
nd
out
that
Zip
Air
lin
es p
urc
has
ed 5
m
ore
type
Apl
anes
th
an t
ype
C p
lan
es. H
ow m
any
plan
es o
f ea
ch t
ype
did
Zip
Air
lin
es p
urc
has
e?
7 ty
pe
Ap
lane
s, 4
typ
e B
pla
nes,
and
2 t
ype
C p
lane
s
2-2
© G
lenc
oe/M
cGra
w-H
ill50
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
2-2
Gra
ph
Co
lori
ng
Th
e st
ude
nt
cou
nci
l is
sch
edu
lin
g a
voll
eyba
ll t
ourn
amen
t fo
r te
ams
from
all
fou
r cl
asse
s. T
hey
wan
t ea
ch c
lass
tea
m t
o pl
ay e
very
oth
ercl
ass
team
exa
ctly
on
ce. H
ow s
hou
ld t
hey
sch
edu
le t
he
tou
rnam
ent?
If w
e ca
ll t
he
team
s A
, B, C
, an
d D
, all
of
the
poss
ible
gam
es a
mon
g th
efo
ur
team
s ca
n b
e re
pres
ente
d as
AB
, AC
, AD
, BC
, BD
, an
d C
D.
Dra
w a
gra
ph t
o re
pres
ent
the
prob
lem
. Th
en c
olor
th
e gr
aph
.To
col
ora
grap
h m
ean
s to
col
or t
he
vert
ices
of
that
gra
ph s
o th
at n
otw
o ve
rtic
es c
onn
ecte
d by
an
edg
e h
ave
the
sam
e co
lor.
Let
th
e ve
rtic
es r
epre
sen
t th
e po
ssib
le g
ames
. Let
th
e ed
ges
repr
esen
tga
mes
th
at c
ann
ot b
e sc
hed
ule
d at
th
e sa
me
tim
e. F
or e
xam
ple,
if t
eam
Bis
pla
yin
g te
am C
, th
en t
eam
C c
ann
ot p
lay
team
D.
Ch
oose
a v
erte
x at
wh
ich
to
begi
n.
•C
olor
AB
red
. Sin
ce C
Dis
not
con
nec
ted
to A
B,
colo
r it
red
as
wel
l. A
ll o
f th
e ot
her
ver
tice
s ar
eco
nn
ecte
d to
AB
, so
do n
ot c
olor
th
em r
ed.
•C
olor
AC
blu
e. S
ince
BD
is n
ot c
onn
ecte
d to
AC
, col
or it
blu
e.•
Col
or A
D g
reen
. Sin
ce B
C is
not
con
nec
ted
toA
D, c
olor
it g
reen
.
Th
e co
lore
d gr
aph
sh
ows
that
pai
rs o
f ga
mes
can
be
sch
edu
led
as f
ollo
ws
AB
wit
h C
D; A
C w
ith
BD
; AD
wit
h B
C
Th
e ch
rom
atic
nu
mb
erof
a g
raph
is t
he
leas
t n
um
ber
of c
olor
s n
eces
sary
to
colo
r th
e gr
aph
. Th
e ch
rom
atic
nu
mbe
r of
th
e gr
aph
ab
ove
is 3
.
1.W
hat
doe
s th
e ch
rom
atic
nu
mbe
r of
th
e gr
aph
in t
he
exam
ple
abov
ere
pres
ent?
the
num
ber
of
gam
e ti
mes
sch
edul
ed
2.D
raw
an
d co
lor
a gr
aph
to
repr
esen
t th
e sa
me
type
of
tou
rnam
ent,
but
wit
h 6
tea
ms
play
ing.
See
stu
den
ts’ g
rap
hs.
3.W
hat
is t
he
chro
mat
ic n
um
ber
for
you
r gr
aph
?5
Answers (Lesson 2-3)
© Glencoe/McGraw-Hill A5 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill52
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Mo
de
ling
Re
al-
Wo
rld
Da
ta w
ith
Ma
tric
es
Fin
d t
he
valu
es o
f x
and
y f
or w
hic
h e
ach
mat
rix
equ
atio
n is
tru
e.
1.��
��
���4 3�,
�8 3� �2.
���
�� �
�1 52 �, �
9 5� �U
se m
atri
ces
A, B
, an
d C
to
fin
d e
ach
su
m, d
iffe
ren
ce, o
rp
rod
uct
.
A��
�B��
�C�
��
3.A
�B
4.A
�B
��
��
5.B
�A
6.�
2A
��
��
7.C
A8.
AB
��
��
9.A
A10
.C
B
��
��
11.
(CA
)B12
.C
(AB
)
��
��
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ere
$6 f
or e
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E__
____
____
____
____
____
____
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E__
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____
____
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2-3
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ult
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1021
2547
3652
Mov
ie 1
Mov
ie 2
Mov
ie 3
© G
lenc
oe/M
cGra
w-H
ill53
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
2-3
1.F
ind
the
elem
enta
ry m
atri
x th
at w
ill
inte
rch
ange
row
s 1
and
3 of
mat
rix
A.
��
3.F
ind
the
elem
enta
ry m
atri
x th
at w
ill
mu
ltip
ly t
he
elem
ents
of
the
firs
t ro
wof
mat
rix
Aby
–2
and
add
the
resu
lts
to t
he
corr
espo
ndi
ng
elem
ents
of
the
thir
d ro
w.
��
2.F
ind
the
elem
enta
ry m
atri
x th
at w
ill
mu
ltip
ly r
ow 2
of
mat
rix
Aby
5.
��
4.F
ind
the
elem
enta
ry m
atri
x th
at w
ill
inte
rch
ange
row
s 2
and
3 of
mat
rix
Aan
d m
ult
iply
row
1 b
y – 2
.
��
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00
0 1
0 1
0
1 0
00
5 0
0 0
1
1 0
00
1 0
– 2 0
1
0 0
10
1 0
1 0
0
5.F
ind
the
elem
enta
ry m
atri
x th
at w
ill m
ult
iply
th
e el
emen
ts o
f th
e se
con
d ro
w o
fm
atri
x A
by –
3 an
d ad
d th
e re
sult
s to
th
e co
rres
pon
din
g el
emen
ts o
f ro
w 1
.
5.�
�1
– 3 0
0 1
00
0 1
Ele
me
nta
ry M
atr
ix T
ran
sfo
rma
tio
ns
Ele
men
tary
row
tra
nsf
orm
atio
ns
can
be
mad
e by
mu
ltip
lyin
g a
mat
rix
onth
e le
ft b
y th
e ap
prop
riat
e tr
ansf
orm
atio
n m
atri
x. F
or e
xam
ple,
to
inte
rch
ange
row
s 2
and
3 in
a 3
�3
mat
rix,
mu
ltip
ly t
he
mat
rix
on t
he
left
by
the
mat
rix �
�.
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mp
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Let
C��
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ltip
ly o
n t
he
left
by�
�.
���
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ore
com
plic
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row
tra
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orm
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can
als
o be
mad
e by
a m
atri
x m
ult
ipli
er.
Exa
mp
le
Fin
d t
he
elem
enta
ry m
atri
x th
at, w
hen
mat
rix
Ais
mu
ltip
lied
b
y it
on
th
e le
ft, r
ow 1
wil
l b
e re
pla
ced
by
the
sum
of
2 ti
mes
ro
w 1
an
d r
ow 3
.
Let
A��
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d M
such
th
at M
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ry M
��
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Ch
eck
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th
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s.
2 0
10
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2a11
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a 322a
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a 33a 21
a 22
a
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a 32
a 33
a 11a 12
a 13
a 21 a 22
a 23a 31
a 32a 33
a 11a 12
a 13
a 21a 22
a 23
a 31 a 32
a 33
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3
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1
50
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0 1
0
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0
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31
15
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�21
0 0
0 0
10
1 0
Answers (Lesson 2-4)
© Glencoe/McGraw-Hill A6 Advanced Mathematical Concepts
© G
lenc
oe/M
cGra
w-H
ill55
Ad
vanc
ed M
athe
mat
ical
Con
cep
ts
Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
Mo
de
ling
Mo
tio
n w
ith
Ma
tric
es
Use
sca
lar
mu
ltip
licat
ion
to
det
erm
ine
the
coor
din
ates
of
the
vert
ices
of
each
dila
ted
fig
ure
. Th
en g
rap
h t
he
pre
-im
age
and
th
eim
age
on t
he
sam
e co
ord
inat
e g
rid
.
Use
mat
rice
s to
det
erm
ine
the
coor
din
ates
of
the
vert
ices
of
each
tran
slat
ed f
igu
re. T
hen
gra
ph
th
e p
re-i
mag
e an
d t
he
imag
e on
th
esa
me
coor
din
ate
gri
d.
Use
mat
rice
s to
det
erm
ine
the
coor
din
ates
of
the
vert
ices
of
each
refl
ecte
d f
igu
re. T
hen
gra
ph
th
e p
re-i
mag
e an
d t
he
imag
e on
th
esa
me
coor
din
ate
gri
d.
Use
mat
rice
s to
det
erm
ine
the
coor
din
ates
of
the
vert
ices
of
each
rota
ted
fig
ure
. Th
en g
rap
h t
he
pre
-im
age
and
th
e im
age
on t
he
sam
e co
ord
inat
e g
rid
.
2-4
1.tr
ian
gle
wit
h v
erti
ces
A(1
, 2),
B(2
,�1)
, an
d C
(�2,
0);
sca
le f
acto
r 2
A′(2
, 4),
B′(4
,�2)
,C
′(�4,
0)
2.qu
adri
late
ral
wit
h
vert
ices
E(�
2,�
7),
F(4
,�3)
, G(0
, 1),
an
d H
(�4,
�2)
; sca
lefa
ctor
0.5
E′(�
1,�
3.5)
F′(2
,�1.
5),
G′(0
, 0.5
), H
′(�2,
�1)
3.sq
uar
e w
ith
ver
tice
s W
(1,�
3), X
(�4,
�2)
, Y
(�3,
3),
an
d Z
(2, 2
) tr
ansl
ated
2 u
nit
s ri
ght
and
3 u
nit
s do
wn
W′(3
,�6)
,X
′(�2,
�5)
,Y
′(�1,
0),
Z′(4
,�1)
4.tr
ian
gle
wit
h v
erti
ces
J(3
, 1),
K(2
,�4)
, an
d L
(0,�
2) t
ran
slat
ed
4 u
nit
s le
ft a
nd
2 u
nit
s u
pJ′
(�1,
3),
K′(�
2,�
2),
L′(�
4, 0
)
5.�
MN
Pw
ith
ver
tice
s M
(�3,
4),
N(3
, 1),
an
d P
(�4,
�3)
ref
lect
ed
over
th
e y-
axis
M′(3
, 4),
N′(�
3, 1
),P
′(4,�
3)
6.a
rhom
bus
wit
h
vert
ices
Q(2
, 3),
R(4
,�1)
, S(�
1,�
2),
and
T(�
3, 2
) re
flec
ted
over
th
e li
ne
y�
xQ
′(3, 2
),R
′(�1,
4),
S′(�
2,�
1),T
′(2,�
3)
7.qu
adri
late
ral
CD
FG
wit
h v
erti
ces
C(�
2, 3
),D
(3, 4
), F
(3,�
1), a
nd
G(�
3,�
4) r
otat
ed 9
0°C
′(�3,
�2)
,D
′(�4,
3),
F′(1
, 3),
G′(4
,�3)
8.P
enta
gon
VW
XY
Zw
ith
ver
tice
s V
(1, 3
),W
(4, 2
), X
(3,�
2),
Y(�
1, �
4), Z
(�2,
1)
rota
ted
180°
V′(�
1,�
3),
W′(�
4,�
2),
X′(�
3, 2
), Y
′(1, 4
), Z
′(2,�
1)©
Gle
ncoe
/McG
raw
-Hill
56A
dva
nced
Mat
hem
atic
al C
once
pts
Enr
ichm
ent
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
2-4
Ma
p C
olo
rin
gE
ach
of
the
grap
hs
belo
w is
dra
wn
on
a p
lan
e w
ith
no
acci
den
tal c
ross
ings
of
its
edge
s.T
his
kin
d of
gra
ph is
cal
led
a pl
anar
gra
ph. A
plan
ar g
raph
sep
arat
es t
he
plan
e in
tore
gion
s. A
very
fam
ous
rule
cal
led
the
fou
r-co
lor
theo
rem
stat
es t
hat
eve
ry p
lan
ar g
raph
can
be
colo
red
wit
h a
t m
ost
fou
r co
lors
, wit
hou
t an
y ad
jace
nt
regi
ons
hav
ing
the
sam
eco
lor. G
rap
h A
Gra
ph
BG
rap
h C
Gra
ph
Don
e re
gion
two
regi
ons
thre
e re
gion
sth
ree
regi
ons
one
colo
rtw
o co
lors
two
colo
rsth
ree
colo
rsIf
th
ere
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.
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
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�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
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4
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0 4 5
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3 73
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Pra
ctic
eN
AM
E__
____
____
____
____
____
____
___
DAT
E__
____
____
____
_ P
ER
IOD
____
____
2-5
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
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
�0
and
y �
0u
nle
ss s
tate
d o
ther
wis
e.1.
�2y
2x
�36
infe
asib
le2.
2x�
2y
10un
bo
und
edx
�y
30
2x�
y
8ƒ(
x, y
)�3x
�3y
ƒ(x,
y)�
x�
y
Sol
ve e
ach
pro
ble
m, i
f p
ossi
ble
. If
not
pos
sib
le, s
tate
wh
eth
er t
he
pro
ble
m is
infe
asib
le, h
as a
lter
nat
e op
tim
also
luti
ons,
or
is u
nb
oun
ded
.3.
Nu
trit
ion
Adi
et is
to
incl
ude
at le
ast
140
mill
igra
ms
of
Vit
amin
Aan
d at
leas
t 14
5 m
illig
ram
s of
Vit
amin
B. T
hese
requ
irem
ents
can
be
obta
ined
fro
m t
wo
type
s of
foo
d. T
ype
Xco
ntai
ns 1
0 m
illig
ram
s of
Vit
amin
Aan
d 20
mill
igra
ms
of
Vit
amin
B p
er p
ound
. Typ
e Y
con
tain
s 30
mill
igra
ms
of
Vit
amin
Aan
d 15
mill
igra
ms
of V
itam
in B
per
pou
nd. I
f ty
pe X
foo
d co
sts
$12
per
poun
d an
d ty
pe Y
foo
d co
sts
$8 p
erpo
und
how
man
y po
unds
of
each
typ
e of
foo
d sh
ould
be
purc
hase
d to
sat
isfy
the
req
uire
men
ts a
t th
e m
inim
um c
ost?
9�2 3�
po
und
so
f Y
and
0 p
oun
ds
of
X
4.M
an
ufa
ctu
rin
gT
he C
ruis
er B
icyc
le C
ompa
ny m
akes
tw
o st
yles
of
bicy
cles
: the
Tra
vele
r, w
hich
sel
ls f
or $
200,
and
the
Tour
este
r, w
hich
sel
ls f
or $
600.
Eac
h bi
cycl
e ha
s th
e sa
me
fram
e an
d ti
res,
but
the
ass
embl
y an
d pa
inti
ng t
ime
requ
ired
for
th
e Tr
avel
er is
onl
y 1
hour
, whi
le it
is 3
hou
rs f
or t
he
Tour
iste
r. T
here
are
300
fra
mes
and
360
hou
rs o
f la
bor
avai
labl
e fo
r pr
oduc
tion
. How
man
y bi
cycl
es o
f ea
ch m
odel
shou
ld b
e pr
oduc
ed t
o m
axim
ize
reve
nue?
alte
rnat
e o
pti
mal
Pra
ctic
e
NA
ME
____
____
____
____
____
____
____
_ D
ATE
____
____
____
___
PE
RIO
D__
____
__
2-7
© 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
2. D
3. D
4. B
5. A
6. B
7. D
8. C
9. B
Page 70
10. A
11. D
12. A
13. C
14. B
15. B
Bonus: A
Chapter 2 Answer KeyForm 1A Form 1B
� �
Chapter 2 Answer Key
Page 71
1. D
2. B
3. B
4. A
5. C
6. D
7. D
8. C
9. A
Page 72
10. B
11. B
12. A
13. C
14. A
15. B
Bonus: B
Page 731.
2.
3. ���153�, �
1139��
4. (1, 6, �1)
5. � �6. impossible
7. � �8. ���1
2�, 3, ��
32��
9.
10.
11.
Page 74
12.
13.
14.
15.
16a.
16b.
Bonus:
�199
17
�181010
48�29�17
5�1
�8�6
121
Form 1C Form 2A
inconsistent
A′(�4, �3), B′(2, 5),C′(4, 7)
101
(0, 0) (4, 0)
(4, 10)
(0, 6)
�358�
��318� �
378�
�338�
��213�� � � � ��1
3�3
4�2�5
Vertices: (0, 0), (0, 6),(4, 10), (4, 0); min.: �38; max.: 12
8 master and 22 apprentice
300 A, 600 B
$1470
x � �acee
��
bbdf�
Vertices: (�3.75, 5), (3, 5), (3, �4); min.:�11.5; max.: 11
© Glencoe/McGraw-Hill A11 Advanced Mathematical Concepts
© Glencoe/McGraw-Hill A12 Advanced Mathematical Concepts
Page 75
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Page 76
12.�12�� � · � �
13.
14.
15.
16. unbounded
Bonus: x � y � 3
Page 77
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Page 78
12.
13. Vertices: (0, 0), (0, 5), (5, 0); min.: 0; max.: 25
14. Vertices: (0, 0), (0, 4), (1, 4), (3, 0); min.: �8; max.: 9
15. $150
16. $2500
(2, �3),Bonus: (�1, 1), (8, 5)
57
�23
4�5
Chapter 2 Answer KeyForm 2B Form 2C
consistent anddependent
(3, �4)
(2, 4, �3)
� �� �
� ����5
4�, �3
4��
A′(�3, 3),B ′(4, 1),C ′(�1, �2)
29
� ��41�
�81�
�16
�
��112�
�8�23
27
�655
�1
71
�18
3�18
1512
�69
�5�410
�1�7�1
consistent andindependent
no solution
(4, 1, �7)
� �� �� �(�5, �1)A′(�1, 5),B′(0, �1),C ′(�4, 3)
11
� �0�13
�
�12
�
��16
�
9�30
12
�16257
90
�18
�12�8
2
�1610
�6
49
�2
5�5
9
�12
� � � � � �510
�43
2�1
vertices: (0, 0), (�4, 0),(�4, �4); min.: �32;max.: 0
vertices: (�1, 0), (�1, 2), (1, 4), (3, 0); min.: �5;max.: 17
20 Givernybenches and 40 Kensingtonbenches
© 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.
© 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
� �1d.
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
$3500$5000$6000
CassettesVideotape
CDs
15010047
$750$1000$1000
CassettesVideotape
CDs
550400425
$2750$4000$5000
CassettesVideotape
CDs
Chapter 2 Answer KeyOpen-Ended Assessment
© Glencoe/McGraw-Hill A15 Advanced Mathematical Concepts
Mid-Chapter TestPage 80
1.
2.
3. (2, �1)
4. (�2, 1, 3)
5.
6.
7.
8. (1, �5)
9.
10.
Quiz APage 81
1.
2.
3.
4. (2, 3)
5. (�1, 4, 2)
Quiz BPage 81
1. � �
2. � �
3. � �4. (�5, �3)
5a.
5b.
5c.
Quiz CPage 82
1. 33
2. does not exist
3. ��29�, � �13
��
4.
5. min.: �8; max.: 6
Quiz DPage 82
1. $460
2. 14 gallons�15
12104
�17�11
36
�10188
6�6
�48
Chapter 2 Answer Key
consistent anddependent
� �21
24
� �� 8� 12
16� 4
012
� �12�7
�1021
�6�3
A′(�5, �2),B′(�1, �5),C′(4, �4),D′(�6, �12)
E′(�3, �4),F′(�5, 1),G′(2, �3)
consistent and independent
infinitely many solutions
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)
© Glencoe/McGraw-Hill A16 Advanced Mathematical Concepts
Page 831. C
2. A
3. D
4. A
5 D
6. C
7. E
8. C
9. A
10. D
Page 8411. E
12. C
13. D
14. E
15. D
16. A
17. C
18. A
19. 5
20. 54/6, 27/3, 9/1, or 9
Page 851. D: {�5, �2, 4};
R: {0, 1, 2, 3}; no
2. ��2x1
2�
3. y � �x � 1
4. x � 3y � 18 � 0
5.
6.
7. infinitely many
8. (3, 2)
9. (4, �1, 3)
10. � �11. �41
12. A′(5, �2), B′(7, �3),C′(�2, 6)
13.�14
�� � · � �14. max.: 6, min.:3
15. 200 lumber, 600 plywood
�51
�31
11
56
1�10
2�1
Chapter 2 Answer KeySAT/ACT Practice Cumulative Review