College of Marin Study Guide for Math103 X,Y - Pearson · College of Marin Study Guide for Math103...

College of MarinStudy Guide

for Math 103 X,Y

To AccompanyIntermediate Algebra, Tenth Edition

By Marvin L. Bittinger

Ted Broomas, Ira G. Lansing, Jeannette Woods

Revision Editor—Dan Ayer

Revised Edition

2006360755_Broomas4_TP 6/29/06 2:37 PM

Copyright © 2006, 2003 by College of Marin Department of MathematicsCopyright © 1999 by Addison Wesley, Inc.All rights reserved.

Permission in writing must be obtained from the publisher before any part of this work may be reproduced or transmitted in any formor by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system.

All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and areused herein for identification purposes only.

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

ISBN 0-536-27024-4

2006360755

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6360755_© 6/21/06 10:59 AM Page ii

1

Contents

Math 103X—Intermediate Algebra

Math 103Y—Intermediate Algebra

Lesson Page Title

1 7 Review of Elementary Algebra

2 11 Linear Equations

3 14 Inequalities

4 20 Polynomials

5 24 Factoring

6 28 Review

7 31 Fractional Expressions

8 35 Solving Fractional Equations

9 40 Radicals, Part I

10 44 Radicals, Part II

11 48 Review

Final 51 Final Exam for Math 103X

Lesson Page Title

12 59 Quadratic Equations

13 62 Equations of Quadratic Form

14 66 Graphing Linear Equations

15 69 Graphs of Quadratic Equations and Nonlinear Inequalities

16 73 More About Functions

17 80 Review

18 82 Exponential and Logarithmic Functions

19 85 Systems of Equations

6360755_FM_ppi-vi.fm Page 1 Wednesday, June 28, 2006 4:34 PM

2

20 88 Matrices

21 116 Linear Programming

22 132 Review

23 135 Comprehensive Review

Final 139 Final Exam for Math 103Y

149 Answers to Sample Tests

172 Answers to Homework Problems in the

Study Guide


Videotape Index for

Intermediate Algebra,

10th Ed. 3

Videotape Index for

Intermediate Algebra,

There are 19 videotapes that introduce the material in each section and work the follow-ing text exercises.

VHS Tape #

Run Time

9e Sect #

Section Title Presenter Exercises Used Examples Used

1 16:41 R.1 The Set of Real Numbers

C. Vance 1, 3, 5, 41, 43, 51, 53, 57, 59, 63

1, 3, 4, 7, 9, 14, 15

1 19:09 R.2 Operations with Real Numbers

C. Vance 1, 7, 15, 23, 25, 41, 53, 61, 81, 109

1, 2, 5, 6, 7, 8, 11, 12, 16, 19, 24, 38, 39, 46

1 13:22 R.3 Exponential Notation and Order of Operations

J. Penna 1, 5, 13, 23, 35, 41, 45, 75, 103

5, 13, 21

1 07:34 R. 4 Introduction to Algebraic Expressions

B. Johnson 43 1, 7

2 15:06 R.5 Equivalent Algebraic Expressions

P. Schwarzkopf 1, 5, 9, 27, 37, 51, 59

none

2 12:27 R.6 Simplifying Algebraic Expressions

J. Penna 1, 9, 17, 35, 45, 49, 57

11, 22

2 22:28 R.7 Properties of Exponents and Scientific Notation

J. Penna 1, 15, 37, 43, 77, 79, 105

7, 15, 18, 25

3 21:44 1.1 Solving Equations J. Penna 1, 7, 25, 35, 37, 43, 73

1, 3, 6

3 06:52 1.2 Formulas and Applications

C. Vance 1, 19 5, 6

3 22:30 1.3 Applications and Problem Solving

P. Schwarzkopf 23 5, 6

4 23:49 1.4 Sets, Inequalities, and Interval Notation

P. Schwarzkopf 7, 11, 15, 17, 25, 39, 59, 69

none

4 19:41 1.5 Intersections, Unions, and Compound Inequalities

C. Vance 19, 23, 39, 51 1, 4, 6, 10

4 20:45 1.6 Absolute-Value Equations and Inequalities

J. Penna 7, 9, 15, 21, 25, 33, 59, 77

3, 7, 8, 17, 18

5 24:52 2.1 Graphs of Equations P. Schwarzkopf 1, 11, 31, 47 2, 7

5 22:02 2.2 Functions and Graphs P. Schwarzkopf 11, 13 3, 5, 7, 8, 9

5 10:49 2.3 Finding Domain and Range

B. Johnson 11, 23 1, 4

6 20:50 2.4 Linear Functions: Graphs and Slope

P. Schwarzkopf 7, 15, 27 1, 8

6 18:54 2.5 More on Graphing Linear Equations

P. Schwarzkopf 5, 25, 27, 45, 53 none

6 20:38 2.6 Finding Equations of Lines; Applications

C. Vance 1, 5, 9, 21, 39 4, 6

7 16:33 3.1 Systems of Equations in Two Variables

J. Penna 1, 5, 15 2

7 17:11 3.2 Solving by Substitution

J. Penna 1, 15 2, 4

7 20:57 3.3 Solving by Elimination

J. Penna 3, 15, 17, 29 2, 3

7 18:14 3.4 Solving Applied Problems: Two Equations

B. Johnson 3, 7 6

8 14:22 3.5 Systems of Equations in Three Variables

J. Penna 1, 17, 21 none

8 16:09 3.6 Solving Applied Problems: Three Equations

C. Vance 15, 17 1

3


9th Ed. Marvin Bittinger

4

Study Guide for Math 101 X, Y

8 21:59 3.7 Systems of Inequalities in Two Variables

C. Vance 13, 15, 41, 43 1, 2

8 12:32 3.8 Business and Economics Applications

C. Vance none 1, 2

9 18:23 4.1 Introduction to Polynomials and Polynomial Functions

D. Ellenbogen 55, 67 4a, 10

9 22:18 4.2 Multiplication of Polynomials

J. Penna 27, 51 2, 4, 8, 12, 16, 23

9 16:22 4.3 Introduction to Factoring

C. Vance 27, 31, 37 1, 2, 4, 10

9 21:50 4.4 Factoring Trinomials:

x

2

�

bx

�

c

J. Penna 5, 7, 25 1, 2, 5, 6, 8

10 26:24 4.5 Factoring Trinomials:

ax

2

�

bx

�

c, a

�

1P. Schwarzkopf 11, 19 1, 5

10 17:38 4.6 Special Factoring J. Penna 5, 33, 55, 59, 67, 101

1, 7, 14

10 12:26 4.7 Factoring: A General Strategy

J. Penna 7, 27, 43 3, 5

10 20:28 4.8 Applications of Polynomial Equations and Functions

C. Vance 1, 13, 31, 35, 59 6, 7

11 11:16 5.1 Rational Expressions and Functions: Multiplying, Dividing, and Simplifying

J. Penna 7, 11, 33 11, 17

11 17:45 5.2 LCMs, LCDs, Addition, and Subtraction

J. Penna 23, 51, 63 2, 7, 14

11 18:58 5.3 Division of Polynomials

P. Schwarzkopf 5, 11, 19, 25 none

11 12:48 5.4 Complex Rational Expressions

J. Penna 7 1, 2

12 13:24 5.5 Solving Rational Equations

J. Penna 13, 27, 43 4

12 21:04 5.6 Applications and Proportions

B. Johnson 3, 7 5

12 10:58 5.7 Formulas and Applications

D. Ellenbogen none 1, 2

12 20:31 5.8 Variation and Applications

P. Schwarzkopf 11, 19, 23, 39 1, 8

13 15:28 6.1 Radical Expressions and Functions

C. Vance 1, 7, 9, 35, 43, 47, 49, 51

9, 14, 36, 43

13 15:14 6.2 Rational Numbers as Exponents

B. Johnson 1, 15, 31, 43, 55, 79

2, 7, 11, 19, 27

13 16:02 6.3 Simplifying Radical Expressions

P. Schwarzkopf 13, 19, 49, 57, 69, 85

1, 4

13 11:02 6.4 Addition, Subtraction, and More Multiplication

J. Penna 13, 23, 43, 59 7, 13

14 13:18 6.5 More on Division of Radical Expressions

C. Vance 5, 9, 17, 21, 27, 29 1

14 24:39 6.6 Solving Radical Equations

J. Penna 11, 17, 53 3, 6, 8

14 13:40 6.7 Applications Involving Powers and Roots

P. Schwarzkopf 9, 17, 21, 25 none

14 20:23 6.8 The Complex Numbers

J. Penna 7, 9, 13, 19, 25, 33, 39, 53, 55, 57, 87, 95

5

15 18:05 7.1 The Basics of Solving Quadratic Equations

J. Penna 5, 23, 43a, 57 3, 8

VHS Tape #

Run Time

9e Sect #



5

15 19:29 7.2 The Quadratic Formula

B. Johnson 5, 29 1

15 16:50 7.3 Applications Involving Quadratic Equations

C. Vance 31, 33, 41 4

15 22:05 7.4 More on Quadratic Equations

P. Schwarzkopf 5, 7, 9, 21, 23, 31, 39, 47

none

16 20:40 7.5 Graphing

f

(

x

)

�

a

(

x

�

h

)

2

�

k

P. Schwarzkopf 1, 3, 5, 13, 15, 21 2

16 19:39 7.6 Graphing

f

(

x

)

�

ax

2

�

bx

�

c

C. Vance 7 1, 4

16 14:23 7.7 Mathematical Modeling with Quadratic Functions

C. Vance 15 1, 7

16 20:54 7.8 Polynomial and Rational Inequalities

J. Penna 13, 25, 29 1, 2

17 27:29 8.1 Exponential Functions P. Schwarzkopf 7, 9, 13, 31 1, 5

17 21:39 8.2 Inverse and Composite Functions

B. Johnson 7, 25, 45 2, 11, 13

17 11:02 8.3 Logarithmic Functions C. Vance 1, 35, 39, 43, 51, 63

2, 3, 4, 5, 6, 7

18 20:33 8.4 Properties of Logarithmic Functions

J. Penna 7, 17, 27, 49 1, 3, 4, 6, 8, 21

18 07:09 8.5 Natural Logarithmic Functions

C. Vance 37 8, 10, 13, 14

18 20:25 8.6 Solving Exponential and Logarithmic Equations

C. Vance 15, 23, 51 1, 2, 5, 7

18 11:54 8.7 Mathematical Modeling with Exponential and Logarithmic Functions

C. Vance 5 1, 2, 3b

19 22:45 9.1 Parabolas and Circles B. Johnson 7, 11, 49 1, 2, 5, 6

19 24:22 9.2 Ellipses P. Schwarzkopf 17, 19 2, 3

19 17:37 9.3 Hyperbolas P. Schwarzkopf 5, 11, 15, 19 none

19 18:25 9.4 Nonlinear Systems of Equations

J. Penna 1 3, 4, 6

VHS Tape #

Run Time

9e Sect #



Videotape Index for Intermediate Algebra, 9th Ed.

7

Lesson 1: Review of Elementary Algebra

This lesson reviews some of the basic concepts from elementary algebra that you willneed to prepare for this course.

NO CALCULATORS MAY BE USED ON TESTS.

Objectives

When you have completed this lesson you will be able to:

1.

Write numbers in decimal, fractional, or percent notation.

2.

Do arithmetic using signed numbers.

3.

Simplify algebraic expressions using the properties of arithmetic.

4.

Perform basic operations with exponentials that have integer powers.

Procedures

1.

Read section R.1, pp. 2–9. Be sure to do all the margin exercises when soinstructed. The answers to these margin exercises are in the back of the text.Work one problem at a time and check each answer.

Homework:

Sect. R.1, pp. 10–12, 1–67 All Odd Problems. (Answers to theodd-numbered problems are in the back of your textbook.)

2.

Read section R.2, pp. 13–20

Note:

The rule at the bottom of page 18 can be used for subtracting any combi-nation of positive and negative numbers.

Homework:

Sect. R.2, pp. 21–24, 1–141 All Odd Problems.

3.


Homework:


4.


Homework:


5.


Homework:


6.


Homework:


7.


Many of the answers in the text are not given in simplified form. When theinstruction is to “Simplify,” the answer cannot have any negative exponents or

powers that have not been expanded. For example,

x

�

2

must be written as ,

2

3

must be expanded to be written as 8, and (3

x

)

4

must first be changed to 3

4

x

4

1

x2

-----

6360755_CH01-05_pp007-027.fm Page 7 Wednesday, June 28, 2006 3:11 PM

8


and then to 81

x

4

. You will be expected to give the answers in the correct form forthe tests, so ask a staff member to help you if this is not clear.

Examples

1.

(

�

X

2

Y

3

)(

X

�

4

Y

) = (

�

1)(

X

2

�

X

�

4

)(

Y

3

�

Y

) = (

�

1)

X

2

�

4

�

Y

3+1

=

�

X

�

2

Y

4

=

2.

3.

Recall that (

�

2)

3

= (

�

2)(

�

2)(

�

2) =

�

8 and

4.

(

�

3

x

2

y

�

3

)

4

= (

�

3)

1(4)

x

2(4)

y

�

3(4)

= (

�

3)

4

x

8

y

�

12

=

Remember (

�

3)

4

= (

�

3)(

�

3)(

�

3)(

�

3) = (9)(9) = 81, but

�

3

4

=

�

(3)

4

=

�

(3)(3)(3)(3) =

�

(9)(9) =

�

81.

Homework:


5.

Do the following as a sample test for Lesson 1:

The sample test is Lesson 1: Sample Test on the next page of this

Study Guide

.The answers are given at the end of this

Study Guide

beginning on page 149.Work the entire test before checking the answers. Be sure to time this test andallow at least this amount of time when you take the real test. Remember that

every test you take is averaged into your score for this lesson.

6.

Show your homework and sample test to a staff member and ask for LessonTest 1. If you use scratch paper, number your problems and staple your scratchpaper to your test before you turn it in.

Y4

�

X2

----------

36X7Y

2

9X3Y

8�-------------------- 36

9�-------- X

7

X3

------ Y2

Y8

-----�� 4 X7–3

Y2–8

� �� 4X4Y

6��

4X4

�

Y6

--------------� � � �

2x2y

4��( )

3�2�( )1 3�( )

x2 3�( )

y4 3�( )�

2�( ) 3�x

6�y

12� �

y12

2�( )3x

6-------------------- y

12�

8x6

------------� �

18�

-------- 18�

-------- 1�1�

--------�1�

8-------- .� �

81x8

y12

-----------


9

Lesson 1: Sample Test

1. Evaluate when x = 10 and y = 5.

2. Find the area of a triangle when the height h is 30 feet and the base b is 16 feet.Be sure to give the appropriate units.

3. Write an algebraic expression: Nine less than some number.

4. Write another inequality with the same meaning as x � �2.

Simplify.

5. a. b.

Add, subtract, multiply, or divide.

6. 3 � (�6)(2) � 8 7. 8.

Multiply. Factor.

9. 3y(6 � x) 10. �5(y � 1) 11. 12x � 22xy 12. 7x � 21 + 14y

Simplify.

13. 23 � 10[4 � (�2 + 18)3] 14.

3xy 2�-------------

7 5�6

--------

43--- 7�

6--------�

1�5

-------- 38---�

2 16( )� 2 8�( ) 53

��


10 Study Guide for Math 103 X, Y

15. 4x � [6 �3(2x � 5)] 16. 7 � 3{[5(y � 3) + 9] � 2(y + 8)}

17. Graph on a number line: 3 � x

Simplify the following.

18. �70 19. 3�3 � 3 20. 21. (23)�2

22. 23. (3x2)3(�2x5)3

24. 3(�x2)3(�2x5)3 25. (�4x�3y6)�3

x12

x2�

--------

3a2b

3�

9a2�b

5�-----------------------


11

Lesson 2: Linear Equations

In applying mathematics to solve problems it is very important that you gain ability toexpress concepts as equations and to understand the methods of solving them. In thislesson you will review methods of solving equations.


Objectives


1. Tell what an equation means and what is meant by a solution to an equation.

2. Solve equations using the addition and multiplication principles.

3. Solve certain equations containing parentheses.

4. Determine when an equation has no solution or when any real number is asolution.

5. Solve word problems by translating them to equations.

6. Solve formulas for a specified variable.

Procedures

1. Read section 1.1, pp. 76–85

Homework: Sect. 1.1, pp. 86–89, 1–99 All Odd Problems.

2. There are equations with no solution and there are equations for which any realnumber is a solution. The latter type is called an identity.

Homework: Solve the following equations. (Answers are provided at the backof this Study Guide beginning on page 172.)

1. 3(x + 5) = 25 � (10 � 3x) 2. x + 25 = 6x � 5(x � 2)

3. 8x � 4(2x � 3) = 12 4. 19 � (3y + 4) = 10 � 3y

5. 2(3m � 1) = 8m � (2m + 2)







5. Do the following as a sample test for Lesson 2:

The sample test is Lesson 2: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide beginning on page 149.Work the entire test before checking the answers. Be sure to time this test andallow at least this amount of time when you take the real test. Remember thatevery test you take is averaged into your score for this lesson.

6. Show your homework and sample test to a staff member and ask for Lesson Test 2.If you use scratch paper, number your problems and staple your scratch paper toyour test before you turn it in.


13


Solve each of these equations:

1. 7 � a = 9

2.

3. 8 = 5(n �6) + 78

4. 0.7x � 0.1 = 2.1 � 0.3x

5. 4t � (t + 10) = 4(t � 2)

6. 8 � (y � 6) = 5 � y

7. 3(5x � 4) � 5(3x � 4) = 8

For Exercises 8 and 9, translate to an equation and solve.

8. Find the dimensions of a rectangle whose length is twice its width and whoseperimeter is 54 inches.

9. Money is borrowed at 9% simple interest. After one year, $872 pays off the loan.How much was originally borrowed?

10. Solve for h.

3 x 4�( )2

--------------------- 9�

A r r h�( )2

--------------------�


14

Lesson 3: Inequalities

A comparison of any two real numbers A and B gives rise to one and only one conclu-sion. Either A and B are equal, A is less than B, or B is less than A. thus, we say that theset of real numbers is linearly ordered. In this lesson, we use the properties of the rela-tions less than and greater than to solve sentences involving them.

You will also learn a method for converting units of measure from one unit to anotherwithin a system, or converting from one system to another. This method uses a wellknown property of fractions.


Objectives

When you have completed this lesson, you will be able to:

1. Solve and graph simple inequalities in one variable.

2. Perform a given change of dimension symbols by the method of multiplying byone.

3. Write set notation for solution sets of equations and inequalities.

4. Find the union or intersection of two given sets.

5. Solve and graph compound inequalities in one variable.

6. Solve applied problems that involve inequalities.

7. Solve and graph absolute value inequalities.

Procedures

1. Read Section 1.4ab, pp. 113–114

Note: If a compound sentence uses the conjunction or, the sentence is consid-ered true if either part is true. The sentence “�3 � 8” is an abbreviated form of“�3 � 8 or �3 = 8.” To determine the truth or falseness of the sentence, weexamine each part separately. If either part is true, then the sentence is true.Since �3 � 8 is true, the sentence �3 � 8 is also true.

Homework: Set 1.4, p. 123, 1–13.

2. Read section 1.4c, pp. 115–119

Homework: Set 1.4, pp. 123–126, 15–69 All Odd Problems.

3. Read section 1.4d, pp. 119–122



Note: An easy way of remembering the meaning of intersection is to think ofthe intersection of two streets as being the portion of pavement that belongs toboth streets.


Lesson 3: Inequalities 15

On p. 129, the size of the oval shapes is immaterial. It is merely an abstract representation.

Note: If a compound sentence uses the conjunction and, the sentence is consid-ered true only if both parts are true.

Example 1

“3 � 5 and 8 2” is true because “3 � 5” is true and “8 2” is also true.

Example 2

“3 � x and x � 7” is true if x is replaced by any number between 3 and 7 (notincluding 3 or 7) because each of those numbers make both parts true. The sen-tence may be abbreviated to “3 � x � 7.” In general, the sentence “a � x � b”may be abbreviated to “a x b” provided that a � b. Similarly, “a x andx b” may be abbreviated to “a x b” provided that a b.

Note: There is no abbreviation for a compound sentence that uses the con-junction or.

Note: In the example in the middle of p. 130, if you imagine placing the secondnumber line on top of the first, then the third graph shows the points that are inboth graphs 1 and 2.

In Example 7, p. 134, the third graph is the final graph that shows the points thatbelong to either graph 1 or graph 2.

The following are additional examples of unions and intersections of sets. Inthese examples, set-builder notation is used.

In each of these problems, graph the solution set, and also indicate the solutionset by using set-builder notation.

1. {x |x � 4} � {x |x � �3}

The third graph shows all the points that belong to either graph; therefore, itis the graph of the union of the two given sets.

Using set-builder notation, we get:

{x |x � 4} � {x |x � �3} = {x |x � 4}

�3 4

4

�3 4

�3

x � 4

x � �3

x � 4



2. {x |x 5} � {x | x �7}

The third graph represents the union of the two given sets. In set builder nota-tion, we get:

{x | x 5} � {x |x �7} = {x |x �7}

3. {x |x � 5} � {x |x �7}

{x | x � 5} � {x |x 7} = {x |�7 � x � 5}

4. {x |x 4} � {x |x � �2}

{x |x 4} � {x |x � �2} = {x |x � �2 or x 4}

�7 5

5

�7 5

�7

x 5

x �7

x �7

�7 5

5

�7 5

�7

x � 5

x �7

�7 � x � 5

�2 4

4

�2 4

�2

x 4

x � �2

x � �2 or x 5


Lesson 3: Inequalities 17


5. Read section 1.6abcd, pp. 141–144

Homework: Set. 1.6, pp. 148–149, 1–69 All Odd Problems.

6. Read section 1.6e, pp. 145–147

Homework: Set 1.6, pp 150–151, 71–115 All Odd Problems.

7. Read section 1.6e, pp. 145–147 again.

Note: |5x � 3| = �7 is false for all real x since the absolute value is alwaysgreater than or equal to zero. For the same reason, |x � 5| � �2 is impossiblealso. Therefore, the answer to each of these is that the solution set is the emptyset. Whereas, |5x � 3| �2 is true for all real x, and |3x � 2| � 0 is true onlywhen 3x � 2 = 0.

Homework: Solve the following equations and inequalities. Answers are onpage 173 of this Study Guide.

1. |5x � 2| = �2 4. |6x � 18| � 0

2. |3x � 5| �3 5. |5x � 10| 0

3. |2x � 1| � �1

8. Do the following as a sample test for Lesson 3.


9. Show your homework and sample test to a staff member and ask for LessonTest 3. If you use scratch paper, number your problems and staple your scratchpaper to your test before you turn it in.


18


Solve and write set notation for the answer.

1. x + 2 � 12 2. �12x � 60 3. �4x � 3 5

4. �5x � 30 9x � 2 5. �3(2x � 5) �2(�x + 3)

6. �8(2x + 3) + 6(4 � 5x) 2(1 � 7x) �4(4 + 6x)

7. You can rent a car for either $40 per day with unlimited mileage, or $30 per daywith an extra charge of 15 cents per mile. For what number of miles, traveled inone day, would the unlimited mileage plan save you money? (Translate to an ine-quality using algebra and solve.)

8. Jean is taking a math course. There will be 5 tests, each worth 100 points. Shehas scores of 83, 85, 92, and 95. What score on the last test will give her an A inthe course if she needs an average of at least 90 on the five tests? (Translate to aninequality using algebra and solve.)


Lesson 3: Sample Test 19

9. Graph: x � �1 or x 6 10. Graph: �2 � x � 4

Solve and write set notation for the answer.

11. {2, 3} � {1, 2} 12. {�5, 3} � {3, 5}

13. 5 + 2x 1 and 3x + 2 14 14. x � 7 �5 or 2x � 5 �7

15. 2 � �3x � 7 � 5 16.

17. |x | = 9 18. |4 � 8x | = 10 19. |1 � 2x | = |4 � 3x |

20. |x | 3 21. |6 � 3x | � 0 22. |7 + 2x | �3

23. |5 � 2x | � 3 24. 25.

1 � 3 x�

2------------- � 5

1 x2---� 6 2 5x

3------� �

32---


20

Lesson 4: Polynomials

This lesson will review the vocabulary associated with polynomials. You will also learnto do arithmetic operations with polynomial expressions.


Objectives


1. Identify the terms and degrees of a polynomial.

2. Identify the coefficient and degree of a term.

3. Add, subtract, multiply, or divide polynomials.

Procedures

1. Read section 4.1abc, pp. 326–331

Note: Remember that any expression (except 0) with a 0 exponent is equal to 1.If necessary, a number like �3 can be thought of as �3 = �3 � 1 = �3y0 (or�3x0, etc.). This means that �3 remains the same no matter what values thevariable assumes, so �3 is referred to as the constant term.





Note: Don’t forget that you can use the FOIL method for multiplying:

This process can be extended when either polynomial has more than two terms(as per “Product of Two Polynomials,” just before Ex. 7 on p. 350):


( x + 2 ) ( 3 y � 7 ) = 3 x y � 7 x + 6 y � 1 4

L

I

F

O

( x + 2 ) ( 3 y – 7 + 8 z ) = 3 x y – 7 x + 8 x z + 6 y – 1 4 + 1 6 z


Lesson 4: Polynomials 21


Note: In this homework be sure to use the special product rules to multiply.Becoming familiar with these will help you in future work.


5. Read section 5.3ab, pp. 436–439

Note: Remember that any number (except 0) divided by itself is 1:

Also, remember the names of a division problem:

When dividing by polynomials, the terms of the quotient are found by multiply-ing only the first term of the divisor by the correct number. That is,

In arithmetic, 3 means 3 + where the + is understood. But in algebra,

3x means 3 times x, so the + is necessary in our answer. The answer in

mixed form is x � 8 + .

Homework: Sect 5.3, p. 442, 1–19 All Odd Problems.

77--- 1 6�

6�-------- 1 .5

.5---- 1 3x

2

3x2

-------- 1 (if x 0)��

5

14 73

70

3

quotient

dividend

remainder (may be zero)

divisor

x � 3 x2 � 5x � 8

x

x � 3 x2 � 5x � 8

x2 � 3x

�8x � 8

x

x � 3 x2 � 5x � 8

x2 � 3x

�8x � 8

x – 8

x � 3 x2 � 5x � 8

x2 � 3x

�8x � 8

to get this term, ask yourselfwhat times x (from x + 3) is x2?Then,

Multiply x and (x + 3) = x2 + 3x.

Then, subtract (x2 + 3x) from (x2 – 5x + 8). i.e., x2 – 5x + 8 – (x2 + 3x) =x2 – 5x + 8 – x2 – 3x = –8x + 8.

to get this term, ask yourselfwhat times x (from x + 3) is –8x ?

�8x � 24

32

Then, multiply –8(x + 3) = –8x – 24,and subtract –8x – 24 from –8x + 8 as(–8x + 8) – (–8x – 24) for aremainder of 32.

(Watch your signs!)

25--- 2

5---

32x

------ 3�



6. Do the following as a sample test for Lesson 4.




23


1. What is the degree of the polynomial 7x2y + 2x � 9y?

2. What is the coefficient of the third degree term of the polynomial in Exercise 1?

3. Collect like terms: �2abc + 3a2bc � 4abc

4. Add: 14x3y2 + 6x � 5 and �3x + 2x3y2 + 7y + 1

5. Subtract: (5x3 + 6x � 2) � (3x2 + 4x �1)

Multiply the following:

6. (2y2 � 1) and (y + 3)

7. (4z2 + 8)2

8. Divide:

9. Divide: (6y3 � 4y2 � 14) � (2y + 4)

10. Multiply and collect like terms: (3x + 2y � 1)(x2 � 7x + 1)

6x5

3x3

2x 1� ��

3x3

-------------------------------------------------


24

Lesson 5: Factoring

In this lesson, you will review factoring of polynomials that you learned in elementaryalgebra. Also, you will study the factorization of the sum and difference of two cubes.


Objectives


1. Factor certain polynomials.

2. Factor differences of two squares having more than two terms.

3. Factor sums and differences of two cubes.

4. Solve equations by factoring.

Procedures








Homework: Set 4.5, pp. 379–380, 1–67 All Odd Problems. Answers for 19,49, and 51 could be as follows:

19. 49.

51.


Homework: Set 4.6, pp. 380–382, 69–119 All Odd Problems. Answers for 85and 95 could be as follows:

85.

95.

1100--------- 5x 3�( )2 1

36------ 1 6z�( ) 1 6z�( )

1100--------- 2x 3y�( ) 2x 3y�( )

18--- 2a 1�( ) 4a

22a� 1�( )

11000------------ 10x 1�( ) 100x

210x� 1�( )


Lesson 5: Factoring 25


Homework: Set 4.7, pp. 386–387, 1–57 All Odd Problems. Answers for 37 and53 could be as follows:

37. 53. (3x � 1)(9x2 + 3x + 1)

7. Read section 4.8a, pp. 387–392

Note: When solving an equation by factoring and the principle of zero prod-ucts, it is essential that zero be on one side of the equation.

Example: Solve x2 + x � 6 = 6

Although the left side can be factored, there must be a zero on theright before you factor.

So, x2 + x � 12 = 0

(x + 4)(x � 3) = 0

The solutions are �4 and 3.


8. Read section 4.8b, pp. 393–396





1144--------- 3x 4y

2�( )

2 527------


26


Factor each of the following completely.

1. 6x2y � 21x3y2 + 3x2y3 2. ab + ac + db + dc

3. 9t2 � 30t + 25 4. 18x3 � 50x

5. x2 � 40x + 336 6. x2 + 8x + 16 � 100y2

7. 8x + 30x2 � 6 8. x3 � 64



9. 16a7b + 54ab7 10. 3x3 + 6x2 � 27x � 54

11. Solve: 10x2 + 6 = 16x 12. Solve: 147x = 3x3

Translate Exercises 13 and 14 into an equation, then solve:

13. A photograph is 3 cm longer 14. The shorter leg is 7 feet lessthan it is wide. Its area is 40 cm2. than the longer leg of a rightFind the dimensions of the triangle. If the hypotenuse isphotograph. one foot more than the longer

leg, find the longer leg.


28

Lesson 6: Review

We now pause to reflect upon and review the material you have learned in the first fivelessons.

When you have completed this lesson you will have firm foundation of concepts neededfor the succeeding lessons.


Procedures

1. Homework:

Test Chapter R. pp. 73–74, 1–61 All Odd Problems.

2. Homework:

Test Chapter 1, pp. 155–156, 1–45 All Odd Problems.

3. Homework:

Test Chapter 4, pp. 405–406, 1–39 All Odd Problems.

4.

Review sample test 1–5.

5.



Study Guide


Study Guide


every test you take is averaged into your score for this lesson

6.


6360755_CH06-10.pp028-047.fm Page 28 Wednesday, June 28, 2006 3:09 PM

29


1.

Remove grouping symbols and simplify:

3

a

�

[(2

�

7

a

)

�

(4

a

+ 6)]

2.

Simplify:

3.

Solve for

T:

4.

Solve and graph:

�

12

�

5

x

�

2

�

13

5.

Solve and write the solution set in set notation:

|5

x

+ 3 |

�

18

6.

Subtract and collect like terms:

(

�

4

x

2

�

6

x

+ 11

x

3

)

�

(2

x

2

+ 8

x

�

5

x

3

)

7.

Divide 9

x

4

�

x

2

�

6

x

�

2 by 3

x

�

1

8.

Factor completely: 24

x

2

+ 2

x

�

12

9.

Factor completely: 8

x

3

�

27

y

6

28b4�c

8

35b9�c

2---------------------

R 35--- T 6��


30


10.

Translate to an equation, then solve:

A hardware store drops the price of a lawn mower 15% to a sale price of $510.What was the former price?


31

Lesson 7: Fractional Expressions

The properties of rational numbers and operations within that set are now extended toalgebraic fractions. In order to solve fractional equations that arise in the solution ofcertain types of applied problems, you must first learn how to manipulate fractionalexpressions. A necessary prerequisite skill is factoring polynomial expressions.


Objectives


1.

Rename a fractional expression using multiplication by a factor of one.

2.

Simplify fractional expressions.

3.

Multiply and divide fractional expressions.

4.

Find the LCM of several algebraic expressions.

5.

Simplify complex fractional expressions.

Procedures

1.

Read Section 5.1 abcd, pp. 412–418

Homework:

Set 5.1, pp. 421–422, 1–39 All Odd Problems.

2.

Read section 5.1e, pp. 419–420

3.

Review the definition of

reciprocal

and examples 41–44 on page 19 of your text-book.

Homework:


4.

Read section 5.2a, pp. 425–426 to the end of Example 4.

Note:

It is very important that you use this method of addition of fractionsbecause it is precisely the method used in algebraic fractions. Notice that thedenominators are kept in factored form until the very last step. This makes iteasier to check if there are any common factors in the numerator and denom-inator. If there are, the fraction should then be reduced by “removing thefactor(s) of 1.”

Homework:

Set 5.2, p. 432, 1–14.

5.

Read section 5.2a, pp. 426–427

Homework:

Set 5.2, p. 432, 15–29 All Odd Problems.

6.

Read section 5.2b, pp. 427–430

Note:

In Example 6 why is 0 not a sensible replacement? In Example 7, noticethat the numerator and denominator of the sum on the first line are factored in


32


order to reduce the fraction to the lowest terms. Fractional answers should bereduced to lowest terms whenever possible.

Homework:


7.

Read section 5.2c, p. 531

Homework:


8.

Read section 5.4, pp. 444–448

Homework:


9.



Study Guide.

The answers are given at the end of this

Study Guide



10.



33


1.

Simplify:

2.

Multiply and simplify:

3.

Divide and simplify:

4.

Subtract and simplify:

5.

Find the LCM:

x

2

�

4,

x

2

+ 4

x

+ 4

6.

Add and simplify:

7.

Simplify:

y2

6y 8� �

3y2

12y�------------------------------

x 3�

x3

3x2

�--------------------- x

3

x 3�-------------�

x2

6x� 9�

x2

x� 6�------------------------------ x

22x 15��

x2

2x�---------------------------------�

1

2x2

-------- 2y 1�

2x2

-----------------�

3x 1�------------- 2

1 x�-------------�

1 3x---�

x 9x---�

--------------



8. Add and simplify:

9. Calculate and simplify:

10. Calculate and simplify:

3y

3y2

12�---------------------- 1

2y2

4y�----------------------�

x2

x2

2x 1� �------------------------------ 1

3x 3�----------------- 1

6---��

2x

x2

y2

�------------------ 1

x y�------------- 3

y x�-------------��


35

Lesson 8: Solving Fractional Equations

The solutions of applied problems in algebra oftentimes involve fractional equations.The technique used to solve these equations involves the use of the LCM.


Objectives


1. Solve fraction equations.

2. Solve applied problems involving work, motion, or variation.

3. Solve a formula containing fractional expressions for a given letter.

4. Given a description of direct or inverse variation find the variation constant andthe equation of variation.

Procedures


Note: You must check your values of x into the original equation and throw outany value of x that makes any denominator equal to zero.




3. Read the first paragraph of page 466 of section 5.5c.

Homework: Given solve for:

a. t b. d


Note: Since systems of equations have been deferred to Lesson 19, an adjust-ment is needed in the solution of the problem in Example 6. The table must befilled using only one variable. It is common practice, in this case, to fill in twocolumns from the given information, then filling in the third column using theother two entries in the same row.

The sample is redone as follows:

Example 6

Instead of using the two variables w and t, we choose to use only one. Either onewill do, but usually we choose the one asked for in the problem. So if we use w,

r dt--- ,�



two columns of the table may be filled in as follows:

Let w = speed of the wind in mph

then, 200 + w = speed of plane traveling with the wind in mph

and 200 � w = speed of plane traveling against the wind in mph

No matter which one is the unknown, it is always the speed of theplane in still air minus the speed of the wind for the speed of theplane against the wind.

Now we will fill in the remaining column using the fact that

Since the two times are the same, we write:

Time with wind = Time against wind, which is

The rest of the solution is the same as that on the top half of p. 481.

Study the following additional example.

Example 7

A car leaves a town traveling east at 40 mph. Two hours later, a truck leaves thetown in the same direction at 55 mph. How far from town will the truck catch thecar?

Distance(miles)

Rate(mph)

Time(hours)

With Wind

1062 200 � w

Against Wind

738 200 � w

Distance(miles)

Rate(mph)

Time(hours)

With Wind

1062 200 � w

Against Wind

738 200 � w

time distancerate

-------------------�

1062200 w�---------------------

738200 w�---------------------

1062200 w�--------------------- 738

200 w�---------------------�


Lesson 8: Solving Fractional Equations 37

Let d = the distance from the town in miles,

then, d = the distance of the car in miles,

and d = the distance of the truck in miles.

Now we will fill in the remaining column using the fact that

We then compare the quantities in the column last filled (in this case the twovalues in the Time column). Since the time for the car is two hours more than thetime for the truck, we write:

Time for the car = Time for the truck + 2 hrs or,

40 = 2 � 2 � 2 � 5 and 55 = 5 � 11, so the LCD = 2 � 2 � 2 � 5 � 11

so, clearing of fractions, we have 11d = 8d + 880 and or miles.

The truck will catch the car 293 miles from town.





Homework: Set 5.7, p. 486, 1–13

Distance(miles)

Rate(mph)

Time(hours)

Car d 40

Truck d 55

Distance(miles)

Rate(mph)

Time(hours)

Car d 40

Truck d 55

time distancerate

-------------------�

d40------

d55------

d40------ d

55------ 2��

d 8803

---------� 293 13---

13---



7. Read section 5.8cd, pp. 481–482

Homework: Set 5.7, p. 487, 15–27 (odds) 29–55 All Odd Problems.





39


1. Solve for x:

2. Solve for y:

3. Solve for t:

4. Solve for x:

5. Solve for t1:

6. Find the variation constant and an equation of variation where y varies directly

as x, and y = 18 when

7. Find the value of y when The description is: y varies inversely as x, and

when y = 10, x = 4.

8. Joe can paint a room in 5 hours. Cathy can paint the same room in 3 hours.Working together, how long would it take them to paint the room?

9. A fishing boat takes twice as long to go 25 miles up a river than to return. If theboat cruises at 9 mph in still water, what is the rate of the current?

10. The volume V of a gas varies inversely as the pressure p upon it. The volume of agas is 200 cubic cm under a pressure of 32 km per square cm. What will be itsvolume under a pressure of 40 km per square cm?

1x--- 1

4---�

x 2�4x

-------------�

2y 6�------------- 3

2y 3�-----------------�

2t 2�------------ t

t 2�------------� 3�

x1 x�------------- 1 1

x---��

Q Rs t1 t2�( )-----------------------�

x 54--- .�

x 25--- .�


40

Lesson 9: Radicals, Part I

In this lesson, you will review the properties of, and operations with, square roots.These will be extended to include other roots as well.


Objectives


1. Simplify expressions with integer exponents.

2. Convert to and from scientific notation.

3. Multiply and divide using scientific notation.

4. Simplify expressions involving odd and even roots.

5. Add, subtract, multiply, divide, and simplify radical expressions.

6. Identify rational and irrational numbers.

Procedures

1. Read section R.7, pp. 58–65

Note: Since scientific notation is of the form a 10b, where 1 � a � 10, toconvert to scientific notation move the decimal point in the original number to aplace where the result is a number between 1 and 10. The b is an integer, positiveor negative. For example,

4,387,200 would become 4.387200

.091 would become 9.1 (zeroes at the beginning or end may be dropped).

This new number is now a. The exponent b is how many places the decimalmoves to give back the original number; positive if to the right, negative if to theleft. So,

Thus,

Remember that while a number like 13.95 105 looks like scientific notation, itis not because 13.95 is not between 1 and 10.

4.387200 is 6 places right and b is +6

09.1 is 2 places left and b is �2

4,387,200 4.3872 106

�

.091 9.1 102�

�

13.95 105

1.395 10+1

( ) 105

�

1.395 106

�


Lesson 9: Radicals, Part I 41

Homework: Set R.7, pp. 66–69, 79–99 (odds), 107, 109, 113–117 All OddProblems.


Note: Remember that |a � b | = |a | � |b | ,but |a + b | |a | + |b | .



Note: When an index is 2, this is the square root. Also observe you need to useonly absolute value for even roots. Never use absolute values for odd roots.



Note: Examples 9–15 on pp. 517–518 do not require absolute value onlybecause the problem says assume that all expressions under radicals representnon-negative numbers. This is not usually the case! Be sure you can use absolutevalue appropriately in these problems. For example, the answers for each wouldbe the following:

9. 10. 11.

Do not use absolute values for 12, 14, or 15.



Note: The Power-Root Rule allows you to do either roots or powers first,

whichever is easier. For example, find . Here, 86 is rather complicated, notto mention its cube root.

But is 2. So, . On the other hand, for the expression

the radical part does not simplify. But squaring the radicand first gives

.


6. Read section 6.4a, p. 524



The sample test is Lesson 9: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide beginning on page 149.Work the entire test before checking the answers. Be sure to time this test andallow at least this amount of time when you take the real test. Remember thatevery test you take is averaged into your score for this lesson


x 5 3 x 2y 6x2

y 6xy

863

83 863 83( )

62

6� �

94( )2

94( )2

924 814 3

44 3� � � �


42


Simplify:

1. 2.

Write in scientific notation:

3. 4.

Simplify Exercises 5–7, assuming the letters represent any allowable real number.

5.

6. 7.

5x1�y

3

3x8�y

7-----------------

2�12 x

2[ ]a 1�

�

3x3 a�

-------------------------------

7.6 104

( ) 3.1 107�

( ) 3.0 106�

4.0 103�

---------------------------

a. 18�( )33 b. 3�( )66

16y44 9a

212a 4� �



Simplify Exercises 8–13, assuming all expressions are positive real numbers.

8. 9.

10. 11.

12. 13.

175x6

8x45 8x

45

324 3 1624� 54x43 16x3�

72x

4------------- 18x

6y

3�

25x1�y

5------------------------


44

Lesson 10: Radicals, Part II

This lesson will extend the concept of radicals to more complex expressions and toequations. New ways of expressing and using roots will also be discussed.


Objectives


1. Rationalize radical expressions.

2. Use and simplify fractional exponents.

3. Solve radical equations.

4. Perform arithmetic operations with complex numbers.

Procedures

1. Read section 6.4b, p. 525

Note: Remember that (a + b)2 a2 + b2. Also,



Homework: Set 6.5, p. 533, 1–19 All Odd Problems.








Homework: Set 6.6, pp. 542–546, 1–71 All Odd Problems. Be sure to checkyour solutions when you use the principle of powers with an even exponent.





3 3� 9 3 but 33 33�( ) 93 3.� � �


Lesson 10: Radicals, Part II 45

9. Read section 6.8cdef, pp. 555–560

Note: On p. 556, Example 13, the FOIL method of multiplication may be used:

Also, Examples 14–17 lead to the following:

i0 = 1

i1 = i

i2 = �1

i3 = �i

i4 = 1

i5 = i, etc.

It can be seen that every four times the powers of i start over again. This leads toa shortcut for simplifying powers of i: divide the exponent by 4 and use theremainder as the new exponent. For example,

i23 = i3 = �i since 23 � 4 = 5 R 3. Also,

i48 = i0 = 1 since 48 � 4 = 12 R 0.

On page 575, Example 25, the conjugate of 4i is � 4i because 4i = 0 + 4i. In gen-eral, the conjugate of bi is �bi. However, the conjugate of 4 is not � 4. Note that4 = 4 + 0i and the conjugate is 4 � 0i = 4. In general, the conjugate of a is a.





3 2i�( )23 2i�( ) 3 2i�( )�

9 6i� 6i� 4i2

��

9 12i� 4� 5 12i��


46


1. Rationalize the 2. Simplify:

denominator:

3. Simplify, leaving the 4. Use exponents to answer in exponent form: have a single radical:

Simplify Problems 5–9:

5. (9 + 3i)(2 � 5i) 6. (12 � i) � (8 + 4i)

7. i75 8.

75

8x25

-------------x

3�4

--------

x

5�12--------

2�

64x4y

26 5x23 2

2 x�( ) 3 5 x�( )



9. 10. Rationalize the denominator:

Solve Problems 11 and 12.

11. 12.

13. In a right triangle, find b if c = 10 and a = . (Simplify the answer.)

6 2i�7 3i�---------------- y x�

y 2 x�-------------------------

3x 6� 11� 2�� 2x 5� 1 x 3��

4 5


48

Lesson 11: Review

Before going on further, this lesson lets you review the material you have learned onLessons 7 through 10.

When you have completed this lesson, you will have a good grasp of fractional expres-sions and equations, and you will feel comfortable with exponents, powers, and roots.


Procedures

1. Homework:

Test: Chapter 5, pp. 493–494, 1–33 All Odd Problems.

2. Homework:

Test: Chapter 6, pp. 568–569, 1–45 All Odd Problems.

3.

Review Sample Tests 7–10.

4.



Study Guide


Study Guide

. Text references for all

Review Sample Tests

are to the left of the problem number in the answers. Besure to time this test and allow at least this amount of time when you take the realtest. Remember that

every test you take is averaged into your score for thislesson.

5.



49


1.

Simplify:

2.

Solve:

3.

A train traveling at maximum speed takes 2 hours longer to travel a distance of

120 miles than to travel a distance of 30 miles at the same maximum speed. Findthe maximum speed of the train. (Translate using algebra and then solve.)

4.

The electrical resistance of a certain wire varies directly as its length. If a wire450 ft long has a resistance of 1 ohm, what is the resistance of a wire of length100 ft? (Translate using algebra and then solve.)

5.

Simplify: (Assume

x

is any allowable real number.)

6.

Rationalize the

numerator.

7.

Simplify:

8.

Simplify:

14x 27x------�

28 4x---�

------------------------

5x

50x2

18�------------------------- 1

5x 3�-----------------�

120x 12�-----------------------�

12---

25x2

60x� 36�

3 2x 2 x�

5 x 3 2x�-------------------------------

8a3

�

x9

--------------

2�3

--------

9y87

64x57

----------------


50


9.

Solve:

10.

Simplify:

3x 1� 1� 2x 1�� 0�

8 6i�5 3i�----------------


51

Comprehensive Lesson: Final Exam for Math 103X

Congratulations, you have completed all your lesson tests for the first module Math103X of the Intermediate Algebra course. The only remaining task for you to completeis the final examination. This will demonstrate that you have mastered the material con-tained in the first eleven lessons of this

Study Guide.


Objectives

When you have completed the final exam you will be able to:

1.

Say you have mastered the material in Lessons 1–11.

2.

Receive a grade in Math 103X.

Procedures

1.

Read through your past homework assignments and redo as many problems asneeded to make sure you feel that you are comfortable with all the material.

2.

Redo your sample tests from the first eleven lessons.

3.

Do the following as a sample test for the final exam:

The sample test is Sample Final for Math 103X given on the next page of this

Study Guide.

Be sure to time this test and allow at least that amount of time whenyou take the real test. If you take this test during the designated day of finalsweek, you will be restricted to a time limit of 3 hours. During the regular classtimes you can take the entire continuous period for which the math lab is open.

4.

Show your sample test to a staff member and ask for the final for Math 103X. Ifyou use scratch paper, number your problems in sequence and staple yourscratch paper to your test before you turn it in. Be sure you are ready for this test,since

you can take it only once and it counts 35% of your course grade.

If you complete the course work for a module, but have not satisfied the atten-dance requirement, you can request an IP, in-progress grade, and complete yourattendance requirement during the following semester.

If you are on an IP grade from the previous semester and you complete thecourse work for a module, but have not satisfied the attendance requirement, youcan request an I, incomplete grade, and complete your attendance requirementduring the following semester.

If you are on an I grade from the previous semester and you complete the coursework for a module, but have not satisfied the attendance requirement, you canrequest an extension of the incomplete grade for another semester by filling out apetition with the admissions office.

Remember, you cannot receive a passing grade without the attendance hours.


52

Sample Final for Math 103X

This is a very accurate representation of the actual final, so be certain you can workthese problems easily (without a calculator).

1.

Simplify:

�

2{4

x

�

3[3(4

x

�

2)

�

(5

x

+ 2)]} + 12

2.

Simplify:

�

12

2

�

3

�

2 + (19

�

3)

3.

Simplify:

4.

Divide and write scientific notation for the answer:

5.

Solve:

2y4�x

4�

3y6x

2�----------------------

4�

2.2 107

�

3.2 103�

�------------------------

23--- 7

8--- 4x�

58---�

38---�


Lesson : Sample Final for Math 103X 53

6.

The perimeter of a rectangle is 96 feet. The length is three feet less than twice thewidth. Find the dimensions. (Translate using algebra and then solve.)

7.

Solve: for

c.

8.

Solve and graph: Use set notation. 19

�

(2

x

+ 3)

�

2(

x

+ 3) +

x

9.

Solve: 7

x

+ 4

�

�

17 or 6

x

+ 5 � �7

10. Solve: |x � 3| = 12

11. Solve: |4 � 3y| � 8

A h c d�( )2

---------------------�



12. Solve:

13. Subtract: (9x2 � 4x + 4) � (�7x2 + 4x + 4)

14. Multiply: (a � 2b)(a + 2b)(a2 + 4b2)

15. Factor: ax3 � 2bx3 � ay + 2by

16. Factor: 6x3 � 15x � x2 17. Factor: 4x2 � 16

18. Factor: 20z3 � 45zy2

19. Solve: x2 + 3x = 28

6 x�7

------------- 15



20. If 4 times the square of a number is 45 more than 8 times the number, what is thenumber? (Translate using algebra and then solve.)

21. Divide and give the answer in mixed form: (a2 � 8a � 16) � (a + 4)

22. Multiply and simplify:

23. Simplify:

24. Add. Simplify, if possible.

25. Simplify, if possible.

26. Solve for x:

x2

2x� 35�

2x3

3x2

�--------------------------------- 4x

39x�

7x 49�----------------------�

3x 15�x

-------------------- x 5�5x

-------------�

4x2

5xy�

x2

y2

�------------------------- 2xy y

2�

x2

y2

�----------------------�

2x

x2

4�---------------- 1

x 2�------------- 5

2 x�-------------��

x2--- 5

2x------� 3��



27. Solve for x:

28. An old machine does a certain job in 9 hours while a new machine can do thesame job in 5 hours. How long would it take to get the job done if both machinesworked together? (Translate using algebra and then solve.)

29. A train leaves town and travels north at 75 kph. Two hours later, a second trainleaves on a parallel track traveling north at 125 kph. How far from the startingpoint will the second train overtake the first?

30. Solve: for s.

31. Simplify:

32. Evaluate:

60x

------ 60x 5�-------------�

2x---�

R gsg s�-------------�

5x 15x------�

5 1x---�

--------------------

6�( )88



33. Simplify:

34. Simplify:

35. Simplify:

36. Simplify:

37. Rationalize the denominator:

38. Rationalize the denominator:

39. Use rational exponents to write a single radical and simplify:

40. In a right triangle, find the length of the side a if b = 12, and c = 13. Give anexact answer in simplified form.

24x4

�( )y53

8x5

27y3

-----------3

24x3 3x43�

3 5�( ) 2 5�( )

2y43

6x43

-------------

5 3 3 2�

3 2 2 3�-----------------------------

3x 2x3�



41. Divide and simplify:

42. Solve:

3 2i�2 i�

----------------

x 3� x 5� 4�� 0�


59

Lesson 12: Quadratic Equations

Many applied problems cannot be solved using only linear for first degree equations.For example, problems describing the motion of objects frequently require quadratic orsecond degree equations. In this lesson, you will review some methods of solving qua-dratic equations and learn how complex numbers, studied in Lesson 10, come into thetheory of quadratic equations.

CALCULATORS MAY BE USED ON TESTS.

Objectives


1. Solve a quadratic equation by root extraction, factoring, completing the square,and the quadratic formula.

2. Approximate solutions by square root table or calculator.

3. Solve fractional equations that are quadratic.

4. Solve applied (geometric, motion, rate) problems.

5. Describe the solutions of a quadratic equation without solving it by using thediscriminant.

Procedures


Note: Remember x2 = a has two values for x, and .

Do not multiply or divide both sides of an equation by anything that could bezero. Factor first, then use the principle of zero products.



Note: When simplifying from the quadratic formula, factor first, then line out(or cancel) like factors in numerator and denominator.


3. Read section 6.1, p. 503 Calculator Corner.

Homework: Set 6.1, p. 503 Calculator Corner, Exercises 1–8.


Note: In solving fractional equations you should first find the restrictions on thevariable. The values of the variable that make the denominators zero cannot pos-sibly be solutions. The textbook states “check possible solutions by substituting

a a�

9 6 53

--------------------- 3 3 2 5( )3

----------------------------- 3 3 2 5( )3 1

----------------------------- 3 2 5� � �11



into the original equation,” but we often forget to do this. So, it is a good idea tofirst find the restrictions.

Example Solve:

Restriction is x + 5 � 0 so x � �5.

Simplifying, we get: x2 + 5x + 2x = �10

x2 + 7x + 10 = 0

(x + 5)(x + 2) = 0

x = �5 or x = �2

The solution is only �2, since x = �5 is the restricted value.

Note: In solving the word problems in Section 7.3 you might review the stepsshown on page 99, the motion problems on pp. 466–468, and the work problemson pp. 460–463.







x 2xx 5�-------------�

10�x 5�-------------�


61


1. Solve: x(x � 7) = (2x + 1)(x � 4)

2. An artist wants to put a frame of uniform width around a 3 ft by 4 ft rectangularpainting. He wants the area of the frame to equal the area of the painting. Howwide should he make the frame?

3. Solve: 2x2 + 6x + 17 = 0 (by completing the square)

4. Solve: (Approximate solutions to the nearest tenth.)

3x2 = 9x � 5

5. Solve:

6. A boat travels 24 mi downstream and then immediately returns to its startingpoint. If the entire trip took 4.5 hours and if his speed in still water was 12 mph,find the rate of the current.

7. Determine the nature of the solutions of 9x2 + 25 = 30x without solving theequation.

8. Write a quadratic equation having solutions 3i and �3i.

9. Solve:

10. In order to finish a job as soon as possible, a contractor rents a power sprayerthat can paint a building in 9 hr less time than the sprayer he owns. Using bothsprayers, he has the building painted in 20 hours. How long would it have takenif he had used only his own sprayer?

2410 x�---------------- 1�

2410 x�----------------�

1t 3�------------ 1

t 1�------------� 1�


62

Lesson 13: Equations of Quadratic Form

Many equations that are not really quadratic equations can be put into the form of aquadratic, and as such can be solved using methods you already know.


Objectives


1. Solve more complicated literal equations and formulas.

2. Solve equations that are reducible to quadratic form.

3. Solve second degree equations of variation.

Procedures


Note: There is no single best method for solving formulas for a letter. All yourequation-solving techniques come into play when you have such a problem. Thesteps outlined in the following box are useful.

Example Solve: for p.

To solve a formula for a letter, say x:

1. Use the principle of powers and clear of fractions, as needed, to elim-inate radicals and until x does not appear in any denominator. (Insome cases you may clear of fractions first, and in others you mayuse the principle of powers first. But do these until radicals are goneand x is not in any denominator.)

2. Collect all terms with x2 in them. Do the same for terms with x.

3. If x2 does not appear, you can finish by using just the addition andmultiplication principles. Get all terms containing x on one side ofthe equation, then factor out x. Dividing on both sides will then get xalone.

4. If x2 appears but x does not appear to the first power, solve the equa-

tion for x2. Then take the square root on both sides.

5. If there are terms containing both x and x2, put the equation in stan-dard form and use the quadratic formula.

M 2Lp

------�


Lesson 13: Equations of Quadratic Form 63

Note: The exercises assume that all variables represent nonnegative numbers.Thus, when you take the square root of both sides of certain equations, only thepositive square root is taken. Without this restriction remember you will alwayshave two square roots; one positive and one negative.



Note: Be careful when using the “u substitution,” that you do not forget to solvefor the original variable. Sometimes it is more efficient to stay with the originalvariable.

Example 1 Solve x4 � 2x2 � 3 = 0

This equation is not quadratic in x. But, it is quadratic in x2 since

(x2)2 � 2(x2) � 3 = 0

Then, (x2 � 3)(x2 + 1) = 0

x2 = 3 or x2 = �1

x = or x = i

We could check these four solutions in the original equation.

Example 2 Solve y�2 � y�1 � 2 = 0

This equation is not quadratic in y, but it is quadratic in y�1.

Thus, (y�1)2 � (y�1) � 2 = 0

Then, (y�1) � 2)(y�1 + 1) = 0

y�1 = 2 or y�1 = �1

y = or y = = �1

Again, we could check these two solutions in the original equation.


3. Read section 5.8ef, pp. 483–485

Note: A variable, say y, may vary directly or inversely as the square of anothervariable, say x. Refer to Lesson 8 for direct and inverse variation.

Note: In this lesson, a third type of variation problem is introduced, joint variation.


M( )2 2Lp

------⎝ ⎠⎛ ⎞ 2

�

M2 2L

p------�

M2p 2L�

p 2L

M2

-------�

Principle of powers (squaring)

Clearing of fractions

Multiplying by 1

M2

-------

3

12--- 1

1�--------


65


1. Solve: 5x4 � x2 � 4 = 0

2. Solve:

3. Solve:

4. Solve: 6x�2 � 5x�1 � 6 = 0

5. Solve: (x � 3)2 + 3(x � 3) = 4

6. Solve a2 + b2 + c2 = d2 for b.

7. Solve for a:

8. Solve for r: A = �r2 + �rs

9. Suppose Q varies jointly as m and the square of n, and inversely as p. If Q = �4when m = 6, n = 2, and p = 12, find an equation of variation representing Q.

10. The weight w of an object on or above the surface of the earth varies inversely asthe square of the distance d between the object and the center of the earth. If aperson weighs 100 lb on the surface of the earth, how much would that personweigh (to the nearest pound) 400 mi above the earth's surface? (Assume theradius of the earth is 4000 mi.)

y

23---

y

13---

� 6� 0�

m 7 m� 12� 0�

(Assume all variables represent nonnegative numbers.)

q a

a2

b2

�

-----------------------� (Assume all variables represent nonnegative numbers.)

(Assume all variables represent nonnegative numbers.)


66

Lesson 14: Graphing Linear Equations

There is a saying that “a picture is worth a thousand words.” In this lesson, you willlearn to draw graphs that are pictures of equations containing two variables. Graphs areoften used to clarify the relationship between two variables.


Objectives


1. Find ordered pairs of numbers that are solutions of a given equation.

2. Determine the slope and y-intercepts of an equation of the form y = mx + b andsketch the graph.

3. Determine the x- and y-intercept of a linear equation and sketch its graph.

4. Find the slope of a line containing two given points.

5. Find an equation of a line given its slope and a point on the line.

6. Find an equation of a line given two points on a line.

7. Find an equation of a line containing a given point and parallel to a given line.

8. Find an equation of a line containing a given point and perpendicular to a givenline.

9. Find the distance between two given points.

10. Find the midpoint of a line segment.

11. Find a linear equation that fits the data collected from a given situation.

12. Given the equation of a circle, find its center and radius, and graph it.

13. Write an equation of a circle given its center and radius.

Procedures


Note: If one of the coordinates of a point is zero, then the point is not in anyquadrant. The point is on one of the axes. For example, the point (3,0) is on thex-axis.





Note: Another way to determine whether or not an equation is linear is to lookat each term of the equation. A liner equation must contain only one variable ineach term and its power must be one.


Lesson 14: Graphing Linear Equations 67







Homework: Set 2.6, p. 229, 1–20



8. Read section 2.6e, pp. 223–228


9. Read section 9. 1bc, pp. 762–763








68


1. Find the slope and y-intercept of x + 3y = 6.

2. Find the x- and y-intercepts of 2y = 10 � 5x and graph the line.

3. Graph the line with a y-intercept of �1 and slope .

4. Graph: 4x � 3y = 12

5. Write an equation of a circle having center at (�3, 5) and radius 2.

6. Find an equation of the line containing the point (1, �6) with slope �3.

7. Find the distance between (5, �6) and (1, 4), and find the midpoint of the segm-ent with these as endpoints.

8. Find y if the slope of the line containing (2, y) and (�4, 1) is �1.

9. Find the center and radius of the circle and sketch the graph.

x2 + y2 + 4x � 10y + 26 = 0

10. Find an equation of the line containing the point (6, �2) and perpendicular to theline 3x + y = 4. Write the equation in the form y = mx + b.

45---


69

Lesson 15:Graphs of Quadratic Equations and

Nonlinear Inequalities

This lesson will present nonlinear inequalities, and quadratic equations will be graphedin detail.


Objectives


1. Graph a quadratic equation and determine its principal parts.

2. Solve maximum/minimum problems.

3. Graph quadratic and other polynomial inequalities.

4. Graph rational inequalities.

Procedures








Note: The standard form for a parabola is y = a(x � h)2 + k. In completing thesquare, be careful of the effect “a” has on other terms. As in Example 3 on p. 647, (25/4) � (25/4) is added inside the parentheses, but when the �(25/4)comes outside the parentheses it is really (�2) � (�25.4), since a = �2. Considerf(x) = 3x2 � 6x + 1. Completing the square by adding and subtracting 1 gives

When the �1 comes out, it is really 3 � (�1) = �3, so


f(x) = 3(x2 – 2x + 1 – 1) + 1perfectsquare

moveout

f(x) = 3(x – 1)2 – 2–3 + 1






Homework: Set 7.8, p. 658, 1–18







71


1. Use the vertex and at least two 2. Explain how to change the pointsother points to graph y = �2x2. from Problem 1 to graph

y = �2(x � 1)2 + 3.

3. For y = �3(x + 6)2 + 8, without 4. For y = 3x2 � 2x � 5, findgraphing find the following: the x-intercepts.

a. vertex

b. line of symmetry

c. maximum value

d. minimum value

Use completing the square to change each of the following to standard form.

5. y = x2 � 3x + 5 6. y = �2x2 � 2x + 3



7. What is the minimum product of 8. What are the dimensions of thetwo numbers whose difference is largest rectangular pen that can20. Explain why it is a minimum. be enclosed by 120 feet of fencing.(Use algebra.) (Use algebra.)

Solve and give all answers in set notation.

9. (2x + 5)(x � 2) � 0 10. 5(x � 2)(x + 1)(x � 3) � 0

11. 12. x2 � x � 2 13.2x 5�x 4�

----------------- � 1 x2

4x� 12�

2x2

x� 3�--------------------------------- � 0


73

Lesson 16: More About Functions

One of the most important aspects of science and social science is defining relationsbetween various types of phenomena. Functions, which are a certain type of relation,can be used to predict outcomes given pertinent data.

In this lesson, you will be introduced to inverse functions. You will also learn additionalgraphing techniques.


Objectives


1.

Determine whether a given correspondence is a function.

2.

Evaluate a function for a specific input.

3.

Recognize the graph of a function.

4.

Find the domain of a function.

5.

Find function values and combinations of function values.

6.

Write a formula for the inverse of a given function.

7.

Graph translations of functions.

Procedures

1.


Note:

In terms of domain and range, Example 1 can be illustrated as follows:

Homework:


2.


Homework:


3.

Read Appendix D, pp 823–824

Homework:

Set D, p. 825, 1–31 All Odd Problems.

4.


Homework:


•

••

•

••

RangeDomain

Not a function

Domain Range

A function

Ex. h: Ex. p:


74


5.

Study the following examples:

Note:

In the middle of page 620, there is a rule for parabolas being translated tothe left or right. There is a similar rule for general functions as follows:

For example, the graph of

y

=

f

(

x

�

3) is found by moving all the points of

y

=

f

(

x

) three units to the right since

h

= 3.

Also, the graph of

y

=

f

(

x

+ 3) is found by moving all the points of

y

=

f

(

x

) threeunits to the left since

h

=

�

3.

Example 1

Sketch the graph of

y

= |

x

�

3| by translating the graph of

y

= |

x

| .

The graphs of

y

=

f

(

x

) = |

x

| and

y

=

f

(

x

�

3) = |

x

�

3| are shown on the sameaxes. In comparing

y

= |

x

�

3| with

y

= |

x

| , we note that

x

has been replaced by

x

�

3, so

h

= 3. Therefore, the graph is moved three units to the right. This isadding 3 to the

x

-coordinates.

Note:

In the middle of page 621, there is a rule for parabolas being translatedup or down. There is a similar rule for general functions as follows:

For example, the graph of

y

=

f

(

x

) + 3 is found by moving all the points of

y

=

f

(

x

) three units up.

The graph of

y

=

f(x

�

h

) looks just like the graph of

y

=

f(x

), except it is translated to the left or right. If

h

�

0, then the function is translated

h

units to the right. If

h

�


h

units to the left.

The graph of

y

=

f

(

x

) +

k

looks just like the graph of

y

=

f

(

x

), except it is translated up or down. If

k

�


k

units up. If

k

�

0, then the function is translated |

k

| units down.

y = |x – 3|y = |x|


Lesson 16: More About Functions 75

Also, the graph of

y

=

f

(

x

)

�

3 is found by moving all the points of

y

=

f

(

x

) threeunits down.

Example 2

Sketch the graph of

y

=

f

(

x

) = |

x

|

�

3 by translating the graph of

y

=

f

(

x

) = |

x

| .In comparing

f

(

x

) + (

�

3) = |

x

|

�

3 with

f

(

x

) = |

x

| , we note that

f

(

x

) has beenreplaced by

f

(

x

)

�

3, so

k

=

�

3. Therefore, the graph is moved down three units.This is subtracting 3 from the y-coordinates. Both graph are shown below.

Example 3 The graph of y = f (x) is shown below. Sketch the graph of y = f (x � 4) � 2.

In comparing y = f (x � 4) � 2 with y = f (x), we have h = 4 and k = �2. There-fore, the graph is moved 4 units to the right, and 2 units down. This is adding 4 tothe x-coordinates and subtracting 2 from the y’s.

Example 4 Sketch the graph of the inverse of the y = f (x) given in Example 3above.

y = |x|

y = |x| – 3

y = f(x)

y = f(x – 4) – 2



Three points from y = f (x) are (�2, 1), (2, 5), and (4, 0). These would become (1,�2), (5, 2), and (0, 4) in the inverse of y = f (x). Plot (1, �2), and (5, 2) and con-nect them. Then plot (0, 4) and connect it with (5, 2).





77


In Problems 1–3, use the relation f = {(2, �3), (5, 6), (7, �3)}.

1. a. If f a function? 2. a. Find the domain of f.

b. Is f one to one? b. Find the range of f.

3. a. Find the inverse of f.

b. Is the inverse a function?

In Problems 4–7, let f (x) = 5x � 3 and g(x) = 2x � x2. Find:

4. g(�2) 5. (f � g)(a)

6. (f � g)(x) 7.

In Problems 8–9, find the domain.

8. 9.

10. Find if

g a h�( ) g a( )�h

-----------------------------------------

f x( ) x 2�

x2

16�-------------------� g x( ) 9 6x��

f1�

x( ) f x( ) 3x 1�2x

-----------------�



11. Graph f (x) = |x + 2| by translating

the graph of f (x) = |x | . Show both

graphs on one set of axes.

In Problems 12–15, use the graph given below as y = f (x).

12. Graph y = f (x + 3). 13. Graph y = f (x) + 2.



14. Graph y = f (x + 1) � 3. 15. Graph the inverse of f (x).


80

Lesson 17: Review

This lesson consists of a set of problems illustrating the basic concepts you havelearned in Lessons 12–16.


Procedures

1. Do Summary and Review: Chapter 7, pp. 660–662, 1–40 All Odd Problems.

2. Do Test: Chapter 7, pp. 663–664, 1–23 All Odd Problems.

3. Do Test: Chapter 2, pp. 236–239, 1–35 All Odd Problems.

4. Do Test: Chapter 8, pp. 750–751, 1, 5, 7, 9.

5. Do Test: Chapter 9, pp. 801–802, 1–9.

6. Do Exercise Set 8.2, p. 693, 5–12 (odds).

7. Review sample tests 12–16.


The sample test is Lesson 17: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide beginning on page 149.Work the entire test before checking the answers. Be sure to time this test andallow at least this amount of time when you take the real test. Remember thatevery test you take is averaged into your source for this lesson.



81


1. Solve: 3x(x + 1) = 6 + 7x(x + 2)

2. A car travels 10 mph faster than a truck. The car travels 300 miles in one hourless time that the truck. Find the rate of each vehicle. (Use algebra.)

3. Solve for x, assume x � 0:

4. Solve: 6x4 � 17x2 = �5

5. For A(�1, �1) and B(2, �7), find the following:

a. distance from A to B b. midpoint from A to B

6. Find the equation of the line containing (�3, �2) and (�1, �5).

7. Graph: y = �x2 + 2x + 2

8. Find if

9. If f (x) = �3x2 � x + 4 and g(x) = 2x � 3, find:

a. f (�4) b. (f � g)(x)

10. Solve and write in set notation: 3x2 � x � 10

z x

x2 y2

�

----------------------�

f1�

x( ) f x( ) 3x 5�2x 7�----------------- .�


82

Lesson 18: Exponential and Logarithmic Functions

This lesson will present in detail two very important functions. Their graphs, propertiesand uses will be explored.


Objectives


1. Graph exponential and logarithmic functions.

2. Convert between exponential and logarithmic equations.

3. Use the product, quotient, and power properties of logarithms to simplifyexpressions.

4. Use a calculator or a table to find certain logarithmic values.

5. Solve certain applications of these functions.

Procedures






Note: Remember that previously to find an inverse you interchanged x and y,and solved for y. With the inverse of exponentials, solving for y causes problems:

How can y be isolated?

This is what logarithms accomplish and why we say that logarithms are expo-nents. In general, the rule is:

logax = y and ay = x

are interchangeable equations. In words, we have: the logarithm, y, is the expo-nent that must be placed on the base, a, to give the number, x.

Consider: log10 100 = 2, 102 = 100




y ax

�x a

y�

after the interchange

⎧ ⎪ ⎨ ⎪ ⎩


Lesson 18: Exponential and Logarithmic Functions 83

5. Read section 8.4, pp. 709–712 (You may omit the proofs).

Note: Do not make up your own (incorrect) properties. The following have nofurther simplifications and are not shortcuts:

loga(M + N) � logaM + logaN

logaM logaN � loga(M N)

(logaM)P � P(logaM)

loga(M � N) � logaM � logaN





Note: It is absolutely necessary to check all the values obtained for X into theoriginal equation in order to get the final solutions. Read Example 6 on p. 727very carefully!






10. Show your homework and sample test to a staff member and ask for Lesson Test 18.If you use scratch paper, number your problems and staple your scratch paper to yourtest before you turn it in.

logaM

logaN--------------- loga

MN-----�


84


1. Graph y = 5x.

2. Convert to a log equation: M5 = 161

3. Convert to an exponential equation: log10 5.33 = 0.7267

4. Express in terms of logs of x, y, or z: log xy

5. Express as a single simplified logarithm:

log x + log y � log (x + 5)

6. Solve:

7. Solve: log2 x + log2 (x � 2) = 3

8. Solve: log (x + 3) � log x = 1

9. How long will it take ten dollars to double at 10% compounded annually?

10. The average walking speed R of people living in a city of population P, in thou-sands, is given by R = 1.98 log P + 0.05, where R is in feet per second. The peo-ple living in a particular city have an average walking speed of 2.5 feet persecond. Find the population.

z

12---

2x

2

24�

32�


85

Lesson 19: Systems of Equations

When you have completed this lesson you will be able to solve systems of linearequations in two variables by using several methods. We will now review those meth-ods, introduce new ones, and extend our theory to include linear systems in three vari-ables. In addition, we will learn how to solve some nonlinear systems of equationswhere one or more equations are of the second degree.


Objectives


1. Determine whether an ordered pair is a solution of a system of two equations.

2. Solve systems of two linear equations by graphing.

3. Solve systems on two linear equations by the substitution and the additionmethods.

4. Determine whether an ordered triple is a solution of a system of three equations.

5. Solve systems of three equations in three variables.

6. Solve applied problems by translating to a system of two or three equations.

7. Determine whether a system of linear equations is consistent, inconsistent, ordependent.

8. Solve systems of two equations where one or both equations are second degree.

Procedures


Note: Do not confuse a set with an ordered pair.

Example 1: (2, 3) is an ordered pair of numbers.

{2, 3} is a set of two numbers, not ordered.

Example 2: The system has one solution, (2, 3); but the

quadratic equation x2 + 5x + 6 = 0 has two solutions, {2, 3}.



Note: The substitution method is usually best when one variable has a coeffi-cient of 1. However, when in doubt, use the addition method.

Homework: Set 3.2, pp. 257–29 All Odd Problems.


x y� 5�

2x y� 1�⎩⎨⎧







Note: Solving 3-by-3 systems graphically is extremely difficult because thegraph of each equation is a plane, and seeing where they intersect (if they do) isnot an easy job. We will solve 3-by-3 systems by the addition method. In Lesson20, we will look at a more regimented version of the addition method.




7. Appendix B, pp. 813–816, is optional material. You will not be tested on it.

Note: Determinants are useful in advanced mathematics. Cramer’s Rule is asystematic method for solving systems of equations employing determinants,and is useful for the systems not larger than 3-by-3. For larger systems, themethod of matrices in Lesson 20 is better.

Homework: (Optional) Appendix B, p. 817, 1–31 All Odd Problems.


Note: When solving systems with one first degree equation and one seconddegree equation, use the substitution method.

9. Homework: Set 9.4, pp. 793–796, 1–57 All Odd Problems.





87


1. Solve by graphing. Use your own graph paper.

x � 2y = 3

y � 2x = 3

2. Solve by the substitution method: 3x � 3y = 7

y = 1 � 4x

3. Solve by the addition method: 7x + 15y = 12

3x + 10y = 8

4. Solve the system: x + y � z = �1

2x + 3y + 2z = 8

2x � 3y � 4z = 2

5. Determine whether this system of equations is consistent or inconsistent.

x + y = 4

2x � y = 3

6. Determine whether this system of equations is dependent or independent.

9x � 4y = 12

4.5x � 6 = 2y

7. Solve the system: x + y = 1

x2 � 4y = 8

8. Solve the system: x2 � y2 = 4

4x2 � y2 = 16

9. A grower bought two brands of frozen juice. She paid $4.70 per case for onebrand and $5.20 per case for the other brand. If the total cost was $271 for 55cases, how many cases of each brand of juice did she buy? (Use 2 equations in 2variables to solve.)

10. A total of $105 is to be divided among three men. The first is to receive $10 morethan the second, and the first and second together receive twice as much as thethird. How much does each man receive?


88

Lesson 20: Matrices

Arithmetic is the study of numbers. Geometry is the study of shapes. Algebra is thestudy of equations. When you study matrices, you are dealing with the subject of linearalgebra. In this lesson our study of matrices is limited to their role in systems of linearequations.


Objectives


1. Solve a system of linear equations using matrices.

2. Perform matrix addition and subtraction.

3. Perform scalar multiplication of a number and a matrix.

4. Obtain the product of two matrices.

5. Write a matrix equation equivalent to a system of linear equations.

6. Calculate the inverse of a square matrix, if it exists, using the Gauss-Jordanreduction method.

7. Use matrix inverses to solve systems of linear equations.

Procedures

1. Read Appendix C, pp. 844–847

Our goal is to put zeros below the main diagonal as shown on theright. The rows do not have to be exchanged unless there is a zeroon the diagonal. The zeros in the first column should be foundbefore the zero in the second column. The zero in the second col-umn can not be found using row 1, since the zero in column 1would be lost.

Example Solve the system:

Write the matrix in which the first column is the coeffi-cients of x, the second is of y, the third of z, and the fourth isthe column of constants.

We need zeros below the main diagonal. It is possible thatwe do not have a 1 in the first column, and the numbers arenot multiples of each other. In that case, we would need to adjust both rows. Toget a zero in the second row first column, we need to multiply row 1 by 3, row 2by 2, and then add the new rows 1 and 2 together.

x

0

0

x

x

0

x

x

x

x

x

x

||||

–2x 3y� 2z� 1�

–3x 2y 2z� � 11��

–4x 2y 3z�� 16�

2

–3

4

–3

2

1

2

1

–3

1

–11

16

||||


Lesson 20: Matrices 89

If you write out a code beside each row, itwill be easier for you and others to checkyour work. 3 R1 → R1 means multiply theold row 1 by 3 and this becomes your newrow 1.

Now add row 1 to row 2. The new row 1was only needed to get the 0 in row 2, souse the old row 1.

Multiply row 1 by �2.

Now add row 1 to row 3.

Write the old row 1 again.

Since row 3 has all multiples of 7, it will be easier to work with, if we multiply

by .

Exchange row 2 and row 3.

Multiply row 2 by 5.

Add row 2 to row 3.

Use the old row 2.

Recover your equations.

6

–6

4

–9

4

1

6

2

–3

3

–22

16

||||

3 R1 → R1

2 R2 → R2

2

0

4

–3

–5

1

2

8

–3

1

–19

16

||||

R1 + R2 → R2

2

0

4

–3

–5

1

2

8

–3

1

–19

16

||||

–2 R1 → R1

2

0

0

–3

–5

7

2

8

–7

1

–19

14

||||R1 + R3 → R3

17---

2

0

0

–3

–5

1

2

8

–1

1

–19

2

|||| R3 → R3

17

2

0

0

–3

1

–5

2

–1

8

1

2

–19

|||| R2 → R3

R3 → R2

2

0

0

–3

5

–5

2

–5

8

1

10

–19

||||

5 R2 → R2

2

0

0

–3

1

0

2

–1

3

1

2

–9

|||| R2 + R3 → R3

2x 3y� 2z� 1�

y z� 2�

3z 9��



Solve for z, y and x in that order, working from the bottom equation upwards.

If you can handle the arithmetic, it is possible to skip some of the above steps.We can multiply a row by a number to add to another row and still leave one ofthe rows unchanged. We can multiply a row by a number and exchange it withanother row in one step. The above system is redone using these shortcuts. If youcannot follow everything, do the arithmetic along the side.

At this point, recover the equations and back-substitute as above.

Homework: Appendix C, p. 822, 1–17 All Odd Problems.

2. Read from this Study Guide, Section 20.2, Operations on Matrices, pp. 95–100.

Multiplying matrices is tricky. Each entry of the product matrix is formed by“multiplying” a row of the first matrix with a column of the second matrix.

Example Find if and

Homework: Exercise Set 20.2 from this Study Guide, p. 100, 1–11 (odds), 17,21, 23, 25, 27. (Answers are on p. 174.)

z 9 3� 3��

y 3�( )� 2 y; 1��

2x 3 1�( )� 2 3�( )� 1�

2x 3 6�� 1�

2x 4 x; 2� �

Solution: 2 1 3�,�,( )

2

–3

4

–3

2

1

2

1

–3

1

–11

16

||||

The system in matrix form.

2x 3y� 2z� 1�

�3x 2y 2z� � 11��

rx 2y 3z�� 16�

2

0

0

–3

–5

7

2

8

–7

1

–19

14

||||

Row 1 is unchanged.

3 R1 + 2 R2 → R2

–2 R1 + R3 → R3

2

0

0

–3

1

–5

2

–1

8

1

2

–19

||||

Row 1 is unchanged.

R3 → R2

R2 → R3

17

2

0

0

–3

1

0

2

–1

3

1

2

–9

||||

Row 1 is unchanged.

Row 2 is unchanged.

5 R2 + R3 → R3

A B A 1 2

3 4� B 5 6

7 8�

1 2

3 4

5 6

7 8 1 5 2 7� 1 6 2 8�

3 5 4 7� 3 6 4 8�

19 22

43 50� �



3. Read from this Study Guide, Section 20.3. Inverses of Matrices, on pp. 103–107.We will only cover the Gauss-Jordan Reduction Method.

Our goal is to put zeros above and below the diagonal as shown onthe right. The rows do not have to be exchanged unless there is azero on the diagonal. The zeros in the first column should be foundbefore the zeros in the second column, then the zeros in the thirdcolumn should be found last. The order the zeros are found for agiven column does not matter, but the order given below can be used in all cases.

If no zeros occur on the diagonal, find the zeros as shown onthe right. The rows should be exchanged if a zero occurs onthe diagonal, and then continue to find the zeros as shown.

Find A�1 when .

Put the matrix on the left and a2 � 2 identity matrix on theright.

Example Find A�1 for the following A.

Put the matrix on theleft and a 3 � 3identity matrix onthe right to find A�1.

x

0

0

0

x

0

0

0

x

||||

x

1st

2nd

3rd

x

4th

5th

6th

x

||||

A 1 2

2 5�

1

2

2

5

1

0

0

1 1

0

2

1

1

–2

0

1

1

0

0

1

5

–2

–2

1

5

–2

–2

1

Row 1 is unchanged.

–2 R1 + R2 → R2

–2 R2 + R1 → R1

Row 2 is unchanged.

So, we have A–1 =

2

–3

4

–3

2

1

2

1

–3

||||

1

0

0

0

1

0

0

0

1

2

0

0

–3

–5

7

2

8

–7

||||

1

3

–2

0

2

0

0

0

1

10

0

0

0

–5

0

–14

8

21

||||

–4

3

11

–6

2

14

0

0

5

30

0

0

0

–105

0

0

0

21

||||

10

–25

11

10

–70

14

10

–40

5

1

0

0

0

1

0

0

0

1

||||

13

1121

521

132323

13821521

Row 1 is unchanged.

3 R1 + 2 R2 → R2

–2 R1 + R3 → R3

5 R1 – 3 R2 → R1

Row 2 is unchanged.

7 R2 + 5 R3 → R3

3 R1 + 2 R3 → R1

21 R2 – 8 R3 → R2

Row 3 is unchanged.

R1 → R1

R2 → R2

R3 → R3

130

–1105

121



Homework: From this Study Guide, Exercise Set 20.4, p. 175, 1–13 (odds).(Answers are on p. 175.)

4. Read from this Study Guide, Section 20.4, Solving Systems Using Inverses onpages 108–109.

Example 1 Solve x + 2y = 3 (by using the inverse matrix)

2x + 5y = �1

To use the inverse of the coefficient matrix method to solve the above system,first write the system using matrices.

A X � B

Now multiply A�1 on the left of the matrix of constants, B, to find the solutionmatrix, X. Use the A�1 from Procedure 3, Example 1.

A–1 B � X

So the solution is (x, y) = (17, �7).

Example 2 To use the inverse of the coefficient matrix method to solve the fol-lowing system, first write the system using matrices.

A X � B

Now multiply A�1 on the left of the matrix of constants, B, to find the solutionmatrix, X. Use A�1 from Procedure 3, Example 2.

A–1 B � X

So, the solution is (x, y, z) = (2, �1, �3).

Homework: From this Study Guide, Exercise Set 20.4, p. 110, 15–18.(Answers on p. 175.)


13

1121

521

132323

13821521

So, we have A–1 =

1 2

2 5

x

y 3

1��

5 2�

2� 1

3

1� 5 3( ) 2� 1�( )�

2 3( ) 1 1�( )��

17

7��

We have the following using matrices.

2x 3y� 2z� 1�

3x� 2y 2z� � 11��

4x 2y 3z�� 16�

2 3� 2

3� 2 1

4 1 3�

x

y

z

1

11�

16

�

13--- 1

3--- 1

3---

521------ 2

3--- 8

21------

1121------ 2

3--- 5

21------

1

11�

16

13--- 1 + 1

3--- 11�( ) + 1

3--- 16

521------ 1 + 2

3--- 11�( ) + 8

21------ 16

1121------ 1 + 2

3--- 11�( ) + 5

21------ 16

2

1�

3�

� �



The sample test is Lesson 20: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide. Be sure to time this test andallow at least this amount of time when you take the real test. Remember thatevery test you take is averaged into your score for this lesson.



94

Section 20.2: Operations on Matrices

A matrix of m rows and n columns is called a matrix with dimensions m � n (read “mby n”).

A matrix with the same number of rows as columns is called a square matrix.

Matrix Addition

To add matrices, we add the corresponding members. In order to do this the matricesmust have the same dimensions.

Example 1 Add.

Example 2 Add.

Addition of matrices is both commutative and associative.

Zero Matrices

A matrix having zeros for all of its entries is called a zero matrix and is often denotedby 0. When a zero matrix is added to another matrix of the same dimensions, that samematrix is obtained. Thus, a zero matrix is an additive identity.

Example 3 Add.

Inverses and Subtraction

To subtract matrices, we subtract the corresponding members. Of course, the matricesmust have the same dimensions for that to be possible.

Example 4 Subtract.

5� 0

4 12---

6 3�

2 3�

�5 6� 0 3�

4 2� 12--- 3�

�

1 3�

6 3 12---

�

1 3 2

1� 5 4

6 0 1

1� 2� 1

1 2� 2

3� 1 0

�0 1 3

0 3 6

3 1 1

�

2 1� 3

1 0 1�

0 0 0

0 0 0� 2 1� 3

1 0 1��


Section 20.2: Operations on Matrices 95

The additive inverse of a matrix can be obtained by replacing each member by itsadditive inverse. Of course, when two matrices that are inverses of each other areadded, a zero matrix is obtained.

Example 5 Let A represent

and �A represent

.

Show that A + (�A) = 0.

A + (–A) = 0.

With numbers we can subtract by adding an inverse. The same is true of matri-ces. If we denote matrices by A and B, this fact can be stated as follows:

Example 6 Subtract directly and also by adding an inverse.

a.

A – B

b.

A + (–B)

For matrices A and B of the same dimensions,

A � B = A + (�B).

1 2

2� 0

3� 1�

1 1�

1 3

2 3

�0 3

3� 3�

5� 4�

�

1 0 2

3 1� 5

1� 0 2�

3� 1 5�

1 0 2

3 1� 5

1� 0 2�

3� 1 5�� 0 0 0

0 0 0.�

3 1�

2� 4

2 1

3 2�� 1 2�

5� 6�

3 1�

2� 4

2� 1�

3� 2� 1 2�

5� 6�



Multiplying Matrices and Numbers

We define a product of a matrix and a number, and call that product a scalar product.

Examples

Let .

7. Find 3A.

.

8. Find (�1)A.

.

Products of Matrices

We do not multiply two matrices by multiplying their corresponding members. Themotivation for defining matrix products comes from systems of equations.

Let us begin by considering one equation:

We will write the coefficients on the left side in a 1 � 3 matrix (a row matrix) and thevariables in a 3 � 1 matrix (a column matrix). The 4 on the right is written in a 1 � 1matrix:

We can return to our original equation by multiplying the members of the row matrix bythose of the column matrix, and adding:

We define multiplication accordingly. In this special case, we have a row matrix A and acolumn matrix B. Their product AB is a 1 � 1 matrix, having the single member 4 (alsocalled 3x + 2y � 2z).

The (scalar) product of a number k and a matrix A is the matrix, denoted kA, obtained by multiplying each number in A by the number k.

A 3� 0

4 5�

3A 3 3� 0

4 5

9� 0

12 15� �

1�( )A 1 3� 0

4 5� 3 0

4� 5��

3x 2y 2z�� 4.�

3 2 2�[ ]x

y

z

4[ ].�

3 2 2�[ ]x

y

z

[3x � 2y � 2z].�



Example 9 Find the product of these matrices.

Let us continue by considering a system of equations:

3x � 2y � 2z = 4,

2x � 2y � 5z = 3,

�x � 2y � 4z = 7.

Consider the following matrices:

A X B

If we multiply the first row of A by the (only) column of X, as we did above, weget 3x + 2y � 2z. If we multiply the second row of A by the column in X, in thesame way we get the following:

Note that the first members are multiplied, the second members are multiplied,the third members are multiplied, and the results are added, to get the singlenumber 2x � y + 5z. What do we get when we multiply the third row of A by thecolumn in X?

We define the product AX to be the column matrix

Now consider this matrix equation:

.

Equality for matrices is the same as for numbers, that is, a sentence such as a = bsays that a and b are two names for the same thing. Thus if the above matrixequation is true, the “two” matrices are really the same one. This means that 3x +2y � 2z is 4, 2x � y + 5z is 3, and �x + y + 4z is 7, or that

3 2 1�[ ]1

2�

3

3 1 2 2�( )� 1�( ) 3�[ ] 4�[ ]� �

3 2 2�

2 1� 5

1� 1 4

x

y

z

4

3

7

.

3 2 2�

2 1� 5

1� 1 4

x

y

z

2x y� 5z��

3 2 2�

2 1� 5

1� 1 4

x

y

z

x� y 4z� ��

3x + 2y – 2z

2x – y + 5z

x� + y + 4z

.

3x + 2y – 2z

2x – y + 5z

x� + y + 4z

4

3

7

�



3x � 2y � 2z = 4,

2x � 2y � 5z = 3,

and

�x � y � 4z = 7.

Thus the matrix equation AX = B is equivalent to the original system of equations.

Example 10 Write a matrix equation equivalent to this system of equations:

4x � 2y � 2z = 3,

9x � z = 5,

4x � 5y � 2z = 1,

4x � 2y � 2z = 0.

We write the coefficients on the left in a matrix. We multiply that matrix by thecolumn matrix containing the variables, and set the product equal to the columnmatrix containing the constants on the right:

Example 11 Multiply.

In all the examples so far, the second matrix had only one column. If the secondmatrix has more than one column, we treat it in the same way when multiplyingthat we treated the single column. The product matrix will have as many col-umns as the second matrix.

Example 12 Multiply (compare with Example 11).

Example 13 Multiply.

4 2 1�

9 0 1

4 5 2�

1 1 1

x

y

z

3

5

1

0

.�

3 1 1�

1 2 2

1� 0 5

4 1 2

1

2

1

–3 1 1� 2 1� 1

–1 1 2� 2 2� 1

–1 1 0� 2 5� 1

–4 1 1� 2 2� 1

4

7

4

8

� �

� 0 2

0 1

�

3 1 1�

1 2 2

1� 0 5

4 1 2

1 0

2 1

1 3

3 � 1

1 � 1

1 0 0� 1

4 � 1

4

7

4

8

4

74

8

�2

815

7

0 1 (�1)3

2 3

5 32 3

�

�

�

�

�

The rows of Amultiplied by

the secondcolumn of B

Same as inExample 11

A B



If matrix A has n columns and matrix B has n rows, then we can compute theproduct AB, regardless of the other dimensions. The product will have as manyrows as A and as many columns as B.

3 1 1�

2 0 3

1 4 6

3 1� 9

2 5 1

3 1 1 3 1 2�� 3 4 1 1�( ) 1 5�� 3 6 1 9 1 1��

2 1 0 3 3 2� � 2 4 0 1�( ) 3 5�� 2 6 0 9 3 1� ��

4 6 26

8 23 15�


100

Exercise Set 20.2

For Exercises 1–16, let

Find.

1. A + B 2. B + A 3. E + O 4. 2A

5. 3F 6. (�1)D 7. 3F + 2A 8. A � B

9. B � A 10. AB 11. BA 12. OF

13. CD 14. EF 15. AI 16. IA

In Exercises 17–20, let

, , , and

.

Find.

17. AB 18. BA 19. CI 20. IC

Multiply.

21. 22.

23. 24.

A 1 2

4 3,� B 3� 5

2 1�,� C 1 1�

1� 1,� D 1 1

1 1�

E 1 3

2 6,� F 3 3

1� 1�,� O 0 0

0 0, and� I 1 0

0 1.�

A1 0 2�

0 1� 3

3 2 4

� B1� 2� 5

1 0 1�

2 3� 1

� C2� 9 6

3� 3 4

2 2� 1

�

I1 0 0

0 1 0

0 0 1

�

3� 24

2�2� 0 4

8

6�12---

5� 1 2

1 3

1� 0

4 2�

3� 2

0 1

4� 5

4

2


Lesson 20: Exercise Set 20.2 101

Write a matrix equation equivalent to each of the following systems of equations.

25. 3x � 2y � 4z = 17, 26. 3x � 2y � 5z = 9,

2x � y � 5z = 13 4x � 3y � 2z = 10

27. x � y � 2z � 4w = 12, 28. 2x � 4y � 5z � 12w = 2,

2x � y � z � w = 0, 4x � y � 12z � w = 5,

x � 4y � 3z � w = 1, �x � 4y � 2w = 13,

3x � 5y � 7z � 2w = 9 2x � 10y � z = 5

Compute.

29.

30.

3.61 2.14� 16.7

4.33� 7.03 12.9

5.82 6.95� 2.34

3.05 0.402 1.34�

1.84 1.13� 0.024

2.83� 2.04 8.81

1.23� 4.51 17.4�

61.2 8.81� 0.123

14.14 6.92 14.4�

4.24 16.1 41.3

0.146 6.06� 18.9�

8.43� 1.12 0.0245


102

Review 20.2

1. Solve using matrices.

3x � 2y � z = 4

3x � 2y � z = 5

4x � 5y � z = �1

Let .

Find each of the following

2. A + B 3. �3A 4. �A 5. AB 6. A � B

A1 1� 0

2 3 2�

2� 0 1

,� B1� 0 6

1 2� 0

0 1 3�

�


103

Section 20.3: Inverses of Matrices

Objectives

You should be able to:

1. Calculate the inverse of a square matrix, if it exists, using the cofactormethod.

2. Calculate the inverse of a square matrix, if it exists, using the Gauss-Jordanreduction method.

3. Use matrix inverses to solve systems of n equations in n variables.

We use the symbol I to represent matrices of the type

Note that these are square matrices with 1’s extending from the upper left to the lowerright along the main diagonal, and 0’s elsewhere.

The following property can be shown easily for the 2 � 2 and 3 � 3 cases (see Exercise 19 in Exercise Set 20.4).

Do Exercise 1.

1. Let

and

.

a. Find AI.

b. Find IA.

c. Compare AI and IA.

Theorem

For any n � n matrices A and I, AI = IA = A (I is a multiplicative identity).

1 0

0 1,

1 0 0

0 1 0

0 0 1

.

A a b

c d�

I 1 0

0 1�



We usually say simply that I is an identity matrix.

Suppose a matrix A has a multiplicative inverse, or simply inverse, A�1. Then A�1 is amatrix for which

A A�1 = A�1 A = I.

For example, for the matrix

we have

We can check that A A�1 = I as follows:

We leave it to the student to verify that A�1 A = I.

Do Exercise 2.

2. Let

and

a. Find AA�1.

b. Find A�1A.

c. Compare AA�1 and A�1A.

A 5 3

3 2,�

A1� 2 3�

3� 5.�

A A1�

5 3

3 2

2 3�

3� 5

1 0

0 1I.� � �

A3 1 0

1 1� 2

1 1 1

�

A1� 1

8---

3 1 2�

1� 3� 6

2� 2 4

�


Section 20.3: Inverses of Matrices 105

Calculating Matrix Inverses

In this section, we consider a way of calculating the inverse of a square matrix, if itexists. We shall see later that such inverses exist only when the determinant of thematrix is nonzero.

The Gauss-Jordan Reduction Method

Suppose we want to find the inverse of the matrix A:

.

Matrices and Determinants

First we form a new (augmented) matrix consisting, on the left, of the matrix A and, onthe right, of the corresponding identity matrix I:

We now proceed in a manner very much like that described earlier. We attempt to trans-form A to an identity matrix, but whatever operations we perform, we do on the entireaugmented matrix. When we finish we will get a matrix like the following:

The matrix on the right,

will be A�1.

Example 1 Find A�1.

A2 1� 1

1 2� 3

4 1 2

�

2 1� 1

1 2� 3

4 1 2

1 0 0

0 1 0

0 0 1

.

⎧ ⎪ ⎨ ⎪ ⎩ ⎧ ⎪ ⎨ ⎪ ⎩

The matrix A The identity matrix I

1 0 0 a b c

0 1 0 d e f

0 0 1 g h i

.

a b c

d e f

g h i

,

A2 1� 1

1 2� 3

4 1 2

�



a. We find the augmented matrix consisting of A and I:

.

b. We interchange the first and second rows so that the elements of the first col-umn are multiples of the top number on the main diagonal:

.

c. Next we obtain 0’s in the rest of the first column. We multiply the first rowby �2 and add it to the second row. Then we multiply the first row by �4and add it to the third row:

.

d. Next we move down the main diagonal to the number 3. We note that thenumber below it, 9, is a multiple of 3. We multiply the second row by �3and add it to the third:

.

e. Now we move down the main diagonal to the number 5. We check to see ifeach number above 5 in the third column is a multiple of 5. Since this is notthe case, we multiply the first row by �5:

.

f. Now we work back up. We add the third row to the second. We also multiplythe third row by 3 and add it to the first:

.

2 1� 1 1 0 0

1 2� 3 0 1 0

4 1 2 0 0 1

1 2� 3 0 1 0

2 1� 1 1 0 0

4 1 2 0 0 1

1 2� 3 0 1 0

0 3 5� 1 2� 0

0 9 10� 0 4� 1

1 2� 3 0 1 0

0 3 5� 1 2� 0

0 0 5 3� 2 1

5� 10 15� 0 5� 0

0 3 5� 1 2� 0

0 0 5 3� 2 1

5� 10 0 9� 1 3

0 3 0 2� 0 1

0 0 5 3� 2 1


Section 20.3: Inverses of Matrices 107

g. We move back to the number 3 on the main diagonal. We multiply the firstrow by �3, so the element on the top of the second column is a multipleof 3:

.

h. We multiply the second row by 10 and add it to the first:

.

i. Finally, we get all 1’s on the main diagonal. We multiply the first row by ,the second by , and the third by :

We now have the matrix I on the left. Thus

With either method the student can check by doing the multiplication A�1A orAA�1. If we cannot obtain the identity matrix on the left using the Gauss-Jordanreduction method, as would be the case when a system has no solution or infi-nitely many solutions, then A�1 does not exist.

Do Exercises 3 and 4.

Find A�1. Use the Gauss-Jordan reduction method.

3. 4.

Do Exercise 5.

5. Consider the system

4x1 � 2x2 = �1,

x1 � 5x2 = 1.

15 30� 0 27 3� 9�

0 3 0 2� 0 1

0 0 5 3� 2 1

15 0 0 7 3� 1

0 3 0 2� 0 1

0 0 5 3� 2 1

115------

13--- 1

5---

1 0 0 715------ 1

5---� 1

15------

0 1 0 23---� 0 1

3---

0 0 1 35---� 2

5--- 1

5---

.

A1�

715------ 1

5---� 1

15------

23---� 0 1

3---

35---� 2

5--- 1

5---

.�

A1 0 1

2 1 0

1 1� 1

� A 3 5

2 4�

a. Write a matrix equation equivalent to the system.

b. Find the coefficient matrix A.

c. Find A�1.

d. Use the inverse of the coefficient matrix to solve the system.


108

Section 20.4: Solving Systems Using Inverses

We can use matrix inverses to solve certain kinds of systems. Consider the system

3x1 � 5x2 = �1,

x1 � 2x2 = 4.

We write a matrix equation equivalent to this system:

Now we let

and

Then we have

A X = B.

To solve this equation, we first find A�1 by the Gauss-Jordan Reduction Method.

3 5

1 2�

x1

x2

1�

4.�

3 5

1 2�A,�

x1

x2

X,� 1�

4B.�

A1�

211------ 5

11------

111------ 3�

11--------

�


Section 20.4: Solving Systems Using Inverses 109

We solve the matrix equation A X = B:

A�1 A X = A�1 B Multiplying by A�1

I X = A�1 B Since A A�1 = I

X = A�1 B Since I is an identity

The solution to the system of equations is and

x1

x2

211------ 5

11------

111------ 3�

11--------

1�

4

1811------

13�11

-----------

� �

x11811------� x2

13�11

----------- .�


110

Exercise Set 20.4

Find A�1, if it exists. Check your answers by calculating AA�1 and A�1A.

1. 2. 3. 4.

5. 6. 7. 8.

9. 10. 11.

12. 13. 14.

For Exercises 15 and 16, write a matrix equation equivalent to the system. Find the inverse of the coefficient matrix. Use the inverse of the coefficient matrix to solve each system.

15. 7x � 2y = �3, 16. 5x1 � 3x2 = �2,

9x � 3y = 4 4x1 � 3x2 = 1

17. x1 � x3 = 1, 18. �2x � 2y � 3z = �1,

2x1 � x2 = 3, �2x � 3y � 4z = 2,

x1 � x2 � x3 = 4 �3x � 5y � 6z = 4

19. Let

and

Show that AI = IA = A.

In each of the following, state the conditions under which A�1 exists. Then find aformula for A�1.

20. 21.

A 3 2

5 3� A 3 5

1 2� A 11 3

7 2� A 8 5

5 3�

A 4 3�

1 2� A 0 1�

1 0� A

3 1 0

1 1 1

1 1� 2

� A = 1 0 1

2 1 0

1 1� 1

A1 1� 2

0 1 3

2 1 2�

� A1 1� 2

0 1 2

1 3� 4�

� A

2� 3� 4 1

0 1 1 0

0 4 6� 1

2� 2� 5 1

�

A2� 5 3

4 1� 3

7 2� 5

� A

1 2 3 4

0 1 3 5�

0 0 1 2�

0 0 0 1�

� A

2� 3� 4 1

0 1 1 0

0 4 6� 1

2� 2� 5 1

�

[obj. 2]

Aa b c

d e f

g h i

� I1 0 0

0 1 0

0 0 1

�

A x[ ]� A x 0

0 y�


Lesson 20: Exercise Set 20.4 111

22. 23.

24. Consider

a11x � a12y = c1,

a21x � a22y = c2.

Use the Gauss-Jordan method and the equivalent matrix equation AX = C toprove Cramer’s rule.#

A0 0 x

0 y 0

z 0 0

� A

x 1 1 1

0 y 0 0

0 0 z 0

0 0 0 w

�


112

Lesson 20: Review

Let

, , and

[obj. 1] Find each of the following, if possible.

1. A � B 2. �3A 3. �A 4. AB

5. B � C 6. A � B 7. A�1 8. B�1

[obj. 2] Find A�1 if it exists.

9. 10. 11.

[obj. 2] 12. Write a matrix equation equivalent to the following system of equations.

3x � 2y � 4z = 13,

3x � 5y � 3z = 7,

2x � 3y � 7z = �8

[obj. 1] Evaluate.

13. 14.

15. 16.

[obj. 1] Without expanding, show that

A1 1� 0

2 3 2�

2� 0 1

� B1� 0 6

1 2� 0

0 1 3�

� C 2� 0

1 3�

a 2� 0

1 3� A

0 0 3

0 2� 0

4 0 0

� A

1 0 0 0

0 4 5� 0

0 2 2 0

0 0 0 1

�

4� 3

3 7

1 1� 2

1� 2 0

1� 3 1

0 a b

a� 0 c

b� c� 0

4 7� 6 7

0 3� 9 8�

0 0 2� 6

0 0 0 5

17.5a 5b 5c

3a 3b 3c

d e f

0.�


Lesson 20: Review 113

[obj. 2] Factor.

18. 19. 20.

21. On the basis of Exercise 16, conjecture and prove a theorem regardingdeterminants.

1 a bc

1 b ac

1 c ab

1 x2

x3

1 y2

y3

1 z2

z3

1 a a2

a3

1 b b2

b3

1 c c2

c3

1 d d2

d3


114


Solve using matrices for Problems 1 and 2.

1. 2x � 3y = 9 2. 2x � 2y � 2z = 35x � 2y = 13 2x � 6y � 3z = 4

3x � 2y � z = 0

For Problems 3–6, let , , and .

3. 2A � 3B 4. AB

A 2 1�

4 3�� B 4 1

3 2�� C

1 1 1

1 0 1�

0 2 1�

�



5. B�1 6. C�1

7. Find

8. Write a matrix equation equivalent to the system in Problem 10 below.

Use the inverse of the coefficient matrix to solve the systems in Problems 9 and 10.

9. 4x � y = �7 10. 2x � 2y � z = �23x � 2y = �19 2x � 2y � z = �2

2x � 2y � z = �5

2 3 1�

0 4 2

3 4 1

0 1� 2

2 0 3�


116

Lesson 21: Linear Programming

Solving systems of linear inequalities is a prerequisite skill for linear programming.Liner programming is a powerful mathematical method used to help determine the best(or optimal) utilization of limited resources for a desired goal. The goal may be to max-imize profit or production, minimize costs or risks, etc.

The term

programming

is not used in the sense of computer programming, but as a syn-onym for planning. However, a great many computer programs are devoted to this typeof problem because several hundred variables and many inequalities may be involved.


Objectives

When you have finished this lesson you will be able to:

1.

Determine whether an ordered pair is a solution of a linear inequality in two vari-ables.

2.

Graph a linear inequality in two variables.

3.

Graph systems of inequalities and find vertices, if they exit.

4.

Maximize or minimize linear functions subject to constraints given by a systemof linear inequalities.

5.

Solve applied problems involving a linear function and a system of constraints(Linear Programming).

Procedures

1.

Read section 3.7ab, pp. 298–302

Homework:


2.

Read sections 3.7c, pp. 302–308

Homework:


3.

Read pp. 119–121 of this

Study Guide.

Homework:

Exercise Set 21.3 on p. 122 of this

Study Guide;

17, 23, 25, 27.(Answers are on p. 176.)

4.

Read pp. 123–126 of this

Study Guide.

Homework:


Study Guide;

1, 3, 5–8.(Answers are on p. 177.)

5.

Many of the more complicated problems can be simplified by using a table toorganize the information. Consider the following example:

Example 1

An office manager needs to purchase new filing cabinets. Heknows that Ace cabinets cost $10 each, require 6 square feet of


Lesson 21: Linear Programming 117

floor space, hold 8 cubic feet of files. On the other hand, eachExcello cabinet costs $20, requires 8 square feet of floor space, andholds 12 cubic feet. His budget permits him to spend no more than$140 on files, while the office has room for no more than 72 squarefeet of cabinets. The manager desires the greatest storage capacitywithin the limitations imposed by funds and space. How many ofeach type cabinet should he buy?

Use letters to label what you are asked to find. If each Ace costs$10, then

x

Ace’s costs $10

x

. Continue for all the given informa-tion. The objective equation is determined by what must be maxi-mized. The other information forms inequalities for the constraints.

The storage capacity

S

(

x

,

y

)

�

8

x

�

12

y

is to be a maximum sub-ject to the following constraints:

10

x

�

20

y

�

140 for the total cost and 6

x

�

8

y

�

72 for thetotal space required.

Also, the logical constraints

x

�

0, and

y

�

0 are needed since thenumber of cabinets cannot be negative.

We proceed as before at this point to graph the region of feasible solutions. Thevertices are substituted into the storage capacity formula to find the maximum.

S

(0, 0)

�

8(0)

�

12(0)

�

0

�

0

�

0

S

(0, 7)

�

8(0)

�

12(7)

�

0

�

84

�

84

S

(12, 0)

�

8(12)

�

12(0)

�

96

�

0

�

96

S

(8, 3)

�

8(8)

�

12(3)

�

64

�

36

�

100

Since

x

�

8 and

y

�

3 give a maximum of 100 cu. ft. of storage space from theobjective function, the manager should buy 8 Ace cabinets and 3 Excello cabi-nets.

Homework:


Study Guide;

9–11, and theProblems 13–16 below. (Answers are on pp. 177–179.)

13.

A company produces two types of parts that require time on a drill press and on alathe. Part A requires 5 minutes on the drill press and 10 minutes on the lathe.

Numberto buy of

Cost(Dollars)

Floor Space(Square feet)

Storage Capacity

(Cubic feet)

Acecabinets

x

10

x

6

x

8

x

Excellocabinets

y

20

y

8

y

12

y

Total

�

140

�

72 = S(

x, y

)


118


Part B requires 6 minutes on the drill press and 4 minutes on the lathe. The timeper day available on the drill press is 1 hour and 40 minutes, and the time avail-able on the lathe is 2 hours. To meet demand, at least 5 parts of each type shouldbe produced each day. If the profit on part A is $3 and on part B is $2, how manyparts per day of each type should be made for maximum profit?

14.

A candy company has 100 pounds of nuts and 125 of raisins to be sold as twodifferent mixtures. One mix will contain half nuts and half raisins and will sellfor $1.00 per pound. The other mix will contain 1/3 nuts and 2/3 raisins and willsell for $.80 per pound. How many pounds of each mix should the company pre-pare for maximum revenue?

15.

Fisher makes two amplifiers: the A model on which their profit is $100, andthe B model on which their profit is $150. The A model requires 3 hours onthe assembly line, while the B model requires 4 hours. It takes 1 hour tofinish the cabinet for the A model and 2 hours for the B model. There are4800 man-hours available on the assembly line and 2100 man-hours avail-able in the finishing department. How many of each model should the com-pany produce to maximize profit? Find the maximum profit.

16.

A company has two plants where three models of machines are produced: stan-dard, superior, and luxury. There are orders for 600 standard, 340 superior, and200 luxury models. Plant A can produce 10 standard, 8 superior, and 3 luxurymodels per day, whereas plant B can produce 20 standard, 9 superior, and8 luxury models per day. It costs $500 per day to operate plant A and $800 perday to operate plant B. How many days should each plant be used to meetdemand and minimize costs?

Procedures

(continued)

6.


The sample test is Lesson 21: Sample Test on page 129 of this

Study Guide.

Theanswers are given at the end of this

Study Guide.

Be sure to time this test andallow at least this amount of time when you take the real test. Remember that


7.



119

Section 21.3: Systems of Inequalities

Systems of Inequalities

To get a picture of the solution set of a system or conjunction of inequalities, we willgraph the inequalities separately and find their intersection. A system of linear inequali-ties may have a graph that is a polygon and its interior. In certain applications it isimportant to find the vertices.

Example 1

Graph this system in inequalities. Find the coordinates of any vertices formed.

Do Exercise 1.

2x y 2,��

4x 3y� 12,�

12--- x 2,� �

y 0�

1 2 3 4 512

(1, 0)

12( , 1)

6

5

4

3

2

1

2x + y = 2

4x + 3y = 12x = x = 21

2

(2, 0)

(2, )43

( , )12

103

y

x2


120


1.

Graph 1

�

y

�

2 .

We find the vertex ( , 1) by solving the system

2

x

�

y

�

2,

x

�

.

We find the vertex (1, 0) by solving the system

2

x

�

y

�

2,

y

�

0.

We find the vertex (2, 0) from the system

x

�

2,

y

� 0.

The vertices (2, ) and ( , ) were found by solving, respectively, the systems

x � 2,and

x � ,4x � 3y � 12 4x � 3y � 12.


2. Graph. If a polygon is formed, find the vertices.

12---

y

x

5

4

3

2

1

–1

–2

–3

–4

–5

0 1 2 3 4 55 4 3 2 1– – – – –

12---

12---

43--- 1

2--- 10

3------

12---

x y� 1, �

y x� 2�

y

x

5

4

3

2

1

–1

–2

–3

–4

–5

0 1 2 3 4 55 4 3 2 1– – – – –


Section 21.3: Systems of Inequalities 121

3. Graph. If a polygon is formed, find the vertices.

5x 6y� 30,�

0 y 3,� �

0 x 4� �

y

x

5

4

3

2

1

–1

–2

–3

–4

–5

0 1 2 3 4 55 4 3 2 1– – – – –


122

Exercise Set 21.3

Graph.

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

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

9. 3x � 2 � 5x � y 10. 2x � 6y � 8 � 4y 11. x �4 12. y � 5

13. 0 � x 5 14. �4 y �1 15. y � |x | 16. y |x |

Graph these systems. Find the coordinates of any vertices formed.

17. x � y � 1, 18. x � y � 3, 19. y � 2x 1, 20. y � x 0,x � y � 2 x � y � 4 y � 2x 3 y � x 2

21. 2y � x � 2, 22. x � 3y � 9, 23. y � 2x � 1, 24. x � y � 2,

y � 3x � �1 3x � 2y � 5 y � �2x � 1, x � 2y � 8,

x � 2 y � 4

25. x � 2y � 12, 26. 8x � 5y � 40, 27. 3x � 4y � 12, 28. y � 2x � 3,2x � y � 12, x � 2y � 8, 5x � 6y � 30, y � 2x � 5,

x � 0, x � 0, 1 � x � 3 6 � y � 8y � 0 y � 0

Graph.

29. y � x2 � 2, 30. x � 2y2 � 1, 31. y x � 1, 32. y � �x2 � 5,

y � 2 � x2 x y y � x2 y x2 � 1

Graph.

33. |x | � |y | � 1 34. |x � y | � 1 35. |x � y | 0 36. |x | |y |

[obj. 1]

12---

[obj. 2]

12---


123

Section 21.4: Linear Programming

Objective

You should be able to:

1. Solve linear programming problems.

Suppose you are taking a test in which different items are worth different numbers ofpoints. Presumably those with higher point values take more time. How many items ofeach kind should you do, in order to make the best score? A kind of mathematics calledlinear programming (developed during World War II) may provide an answer. Let usconsider an example.

Example 1

You are taking a test in which items of type A are worth 10 points and items of type Bare worth 15 points. It takes three minutes for each item of type A and six minutes foreach items of type B. The total time allowed is 60 minutes and you are not allowed toanswer more than 16 questions. Assuming that all of your answers are correct, howmany items of each type should you answer to get the best score?

Let x � the number of items of type A, and y � the number of items of type B. The totalscore T is a function of the two variables x and y:

T � 10x � 15y.

This function has a domain that is a set of ordered pairs of numbers (x, y). This domainis determined by the following conditions, called constraints:

Total number of questions allowed, not more than 16 x � y � 16,

Time, not more than 60 min 3x � 6y � 60,

Numbers of items answered will not be negative x � 0,

y � 0.

We now graph the domain of the function T. This is the graph of the system of inequali-ties above. We will determine the vertices, if any are formed.

(0, 10)

(12, 4)

(16, 0)(0, 0) 20 x

y

16



The domain consists of a convex polygon* and its interior. Under this condition, ourlinear function does have a maximum value and a minimum value. Moreover, the maxi-mum and minimum values occur at the vertices of the polygon. All we need do to findthese values is substitute the coordinates of the vertices in T � 10x � 15y.

From the table we see that the minimum value is 0 and the maximum is 180. To get thismaximum you would answer 12 items of type A and 4 items of type B.

We now state the main theorem needed to perform linear programming.

Example 2

Find the maximum and minimum values of F � 9x � 40y subject to the constraints

y � x � 1,y � x � 3,2 � x � 5.

We graph the system of inequalities, determine the vertices, and find the function valuesfor those ordered pairs.

* A convex polygon, roughly speaking, is one that has no indentation, unlike this one:

Vertices(x, y)

ScoreT = 10x + 15y

(0, 0) 0

(16, 0) 160

(12, 4) 180

(0, 10) 150

Theorem

If a linear function F � ax � by � c is defined on a domain described by a system of linear inequalities (constraints), then any maximum or minimum value of F will occur at a vertex. If the domain consists of a convex polygon and its interior, then both maximum and minimum val-ues of F actually exist.


Section 21.4: Linear Programming 125

The maximum value of F is 365 when x � 5 and y � 8. The minimum value of Fis 138 when x � 2 and y � 3.


1. Find the maximum and minimum values of

F � 34x � 6y

subject to

x � y � 6,x � y � 1,1 � x � 3.

2. Find the maximum and minimum values of

G � 3x � 5y � 27

subject to

x � 2y � 8,0 � y � 3,0 � x � 6.

Example 3

A company manufacturers motorcycles and bicycles. To stay in business it must pro-duce at least 10 motorcycles each month, but it does not have facilities to produce morethan 60 motorcycles. It also does not have facilities to produce more than 120 bicycles.The total production of motorcycles and bicycles cannot exceed 160. The profit on amotorcycle is $134 and on a bicycle is $20. Find the number of each that should bemanufactured to maximize profit.

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7

(2, 3)

(2, 5)

(5, 6)

(5, 8)

y

x

Vertices

(x, y)

(2, 3)

(2, 5)

(5, 6)

(5, 8)

Score

F = 9x + 40y

138

218

285

365



Let x � the number of motorcycles to be produced, and y � the number of bicycles tobe produced. The profit P is given by

P � $134x � $20y,

subject to the constraints

10 � x � 60,0 � y � 120,x � y � 160.

We graph the system of inequalities, determine the vertices, and find the function valuesfor those ordered pairs.

Thus the company will make a maximum profit of $10,040 by producing 60 motorcy-cles and 100 bicycles.

Do Exercise 3.

3. A college snack bar cooks and sells hamburgers and hot dogs during the lunchhour. To stay in business it must sell at least 10 hamburgers but cannot cookmore than 40. It must also sell at least 30 hot dogs but cannot cook more than 70.It cannot cook more than 90 sandwiches altogether. The profit on a hamburger is$.33 and on a hot dog it is $.21. How many of each kind of sandwich should theysell to make the maximum profit? What is the maximum profit?

160

120

80

40

16040

(10, 0) (60, 0)

(60, 100)

(40, 120)

(10, 120)

x

y Vertices

(x, y)

(10, 0)

(60, 0)

(60, 100)

(40, 120)

(10, 120)

Profit

P = $134x + $20y

0,$1340

0,$8040

$10,040

,0$7760

$13,740


127

Exercise Set 21.4

In Exercises 1–4, maximize and also minimize (find maximum and minimumvalues of the function, and the values of x and y where they occur).

1. F � 4x � 28y, 2. G � 14x � 16y, 3. P � 16x � 2y � 40,subject to subject to subject to5x � 3y � 34, 3x � 2y � 12, 6x � 8y � 48,3x � 5y � 30, 7x � 5y � 29, 0 � y � 4,x � 0, x � 0, 0 � x � 7.y � 0. y � 0.

4. Q � 24x � 3y � 52, subject to5x � 4y � 20,0 � y � 4,0 � x � 3.

Solve Exercises 5–12.

5. You are about to take a test that contains questions of type A worth 4 points andquestions of type B worth 7 points. You must do at least 5 questions of type A buttime restricts doing more than 10. You must do at least 3 questions of type B buttime restricts doing more than 10. In total, you can do no more than 18 questions.How many of each type of question must you do to maximize your score? Whatis this maximum score?

6. You are about to take a test that contains questions of type A worth 10 points andquestions of type B worth 25 points. You must do at least 3 questions of type Abut time restricts doing more than 12. You must do at least 4 questions of type Bbut time restricts doing more than 15. In total you can do no more than20 questions. How many of each type of question must you do to maximize yourscore? What is this maximum score?

7. A man is planning to invest up to $22,000 in bank X or bank Y or both. He wantsto invest at least $2000 but no more than $14,000 in bank X. Bank Y does notinsure more than a $15,000 investment so he will invest no more than that inbank Y. The interest in bank X is 6% and in bank Y it is 6 % and this will besimple interest for one year. How much should he invest in each bank to maxi-mize his income? What is the maximum income?

8. A woman is planning to invest up to $40,000 in corporate or municipal bonds orboth. The least she is allowed to invest in corporate bonds is $6000 and she doesnot want to invest more than $22,000 in corporate bonds. She also does not wantto invest more than $30,000 in municipal bonds. The interest on corporate bondsis 8% and on municipal bonds it is 7 %. This is simple interest for one year.How much should she invest in each type of bond to maximize her income?What is the maximum income?

[obj. 1]

12---

12---



9. It takes a tailoring firm 2 hr of cutting and 4 hr of sewing to make a knit suit. Tomake a worsted suit it takes 4 hr of cutting and 2 hr of sewing. At most, 20 hr per day are available for cutting and, at most, 16 hr per day are available for sewing.The profit on a knit suit is $34 and on a worsted suit is $31. How many of eachkind of suit should be made to maximize profit? What is the maximum profit?

10. A pipe tobacco company has 3000 lb of English tobacco, 2000 lb of Virginiatobacco, and 500 lb of Latakia tobacco. To make one batch of SMELLO tobaccoit takes 12 lb of English tobacco and 4 lb of Latakia. To make one batch ofROPPO tobacco it takes 8 lb of English and 8 lb of Virginia tobacco. The profit is$10.56 per batch for SMELLO and $6.40 for ROPPO. How many batches ofeach kind of tobacco should be made to yield maximum profit? What is the max-imum profit? Hint: Organize the information in a table.

11. An airline with two types of airplanes, P-1 and P-2, has contracted with a tourgroup to provide accommodations for a minimum of each of 2000 first-class,1500 tourist, and 2400 economy-class passengers. Airplane P-1 costs $12,000per mile to operate and can accommodate 40 first-class, 40 tourist, and 120economy-class passengers, whereas airplane P-2 costs $10,000 per mile to oper-ate and can accommodate 80 first-class, 30 tourist, and 40 economy-class pas-sengers. How many of each type of airplane should be used to minimize theoperating cost?

12. A new airplane, the P-3, becomes available, having an operating cost of $15,000per mile and accommodating 40 first-class, 80 tourist, and 80 economy-classpassengers. If airplane P-1 of Exercise 11 were replaced by airplane P-3, howmany of P-2 and P-3 would be needed to minimize the operating cost?


129


Graph each inequality in Problems 1–3 on separate axes.

1. �3 � y 2 2. x � �2

3. 2x � 3y 12 4. Graph the system and label the vertices:2x � y � 4, �x � y 7, and y � � 2.

5. Given the graph of a system of constraints, minimize F(x, y) � �3x � y.

x

B (0, )

y

52

C (1, 2)

D (2, 0)A (0, 0)

2

1

1 2 3



6. Maximize F(x, y) � 2x � y given the following constraints:

0 � x � 6

0 � y � 8

4x � 4y � 42

Use the following for Problems 7 and 8.

Randy Manufacturing Company produces two types of toy cars, a sedan anda convertible. Each model has to be processed by a grinder and a finisher.The sedan requires 2 hours on the grinder and 2 hours on the finisher, whilethe convertible requires 2 hours on the grinder and 4 hours on the finisher.The company has two grinders and three finishers, each of whom works atmost 40 hours per week. Each sedan brings a profit of $3.00 and each con-vertible brings a profit of $4.00. Assuming every car will be sold, how manyof each should be made to maximize profit?

7. Organize the information in a table, give the system of constraints, and theobjective function.

8. Sketch the feasibility region.

a. Label the vertices.

b. Evaluate the objective function at each vertex.

c. Answer the question.




A hospital dietician must plan a day’s menu using some combination ofchicken and tuna as a primary source of protein. Each ounce of chicken pro-vides 30 grams of protein, 180 calories, and costs $0.70, while each ounce oftuna provides 20 grams of protein, 140 calories, and costs $0.30. Each personmust receive at least 25 grams of protein from the combination, and the costmust not exceed $0.45 per person. How many ounces of chicken and tunashould be given if the number of calories is to be made as small as possible.

9. Organize the information in a table, give the system of constraints, and theobjective function.

10. a. Sketch the feasibility region.

b. Label the vertices.

c. Evaluate the objective function at each vertex.

d. Answer the question.


132

Lesson 22: Review

This lesson consists of a set of problems illustrating the concepts you have learned inLessons 18 through 21.


Procedures

1. Do test: Chapter 8: pp. 750–752, 1–35.

2. Do test Chapter 3: pp. 323–324, 1–17.

3. Do test: Chapter 9: pp. 801–802, 1–21.

4. From the Study Guide, do page 103; 1, 2, 3, 5. (Answers are on p. 174.)

5. From the Study Guide, do page 112; 9, 10. (Answers are on p. 175.)

6. From the Study Guide, do page 100; 27, 28, 29. (Answers are on p. 174.)

7. Review Sample Tests 18–21.


The sample test is Lesson 22: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide. Text references for allReview Sample Tests are to the left of the problem number in the answers. Be sureto time this test and allow at least this amount of time when you take the real test.Remember that every test you take is averaged into your score for this lesson.



133


1. Express as a single logarithm:

logbw � 2 logbx � logby � logbz

2. Solve for x: 7(82�x) � 28

3. Solve for x: log2(x � 5) � log2(x � 2) � 3

4. Suppose $8000 is invested at 15% interest compounded annually and yields$14,000. For how many years was it invested?

5. Solve the system: x � y � 1

x2 � 4y � 8

6. Solve using matrices: 2x � y � 2z � 3x � 6y � 3z � 4

3x � 2y � z � 0

7. If , , and , find AB � C.

8. Solve using the inverse of the coefficient matrix:

2x � y � 3�x � 3y � 5

9. Maximize P(x, y) � 3x � y subject to: 3x � 2y � 12x � 2y � 8x � 0, y � 0

13---

A 1 1�

0 2� B 2 1 0

1� 3 1� C 1� 5 1

2 0 3�



10. Jim wants to buy filing cabinets. He knows that Ace cabinets cost $10 each,require 6 square feet of floor space, hold 8 cubic feet of files. On the other hand,each Excello cabinet costs $20, requires 8 square feet of floor space, and holds12 cubic feet. He needs at least 100 cubic feet of storage capacity, while theoffice has room for no more than 72 square feet of cabinets. How many of eachtype cabinet should he buy to minimize cost?


135

Lesson 23: Comprehensive Review

This lesson is only required if you are making up an I (incomplete grade). If not, youmay go straight to the final exam given in the next lesson. Since it may have been a longtime since you first began Math 103X, this will help you review for the final exam.


Procedures

1. Do Final Examination, pp. 829–834, 1–87.


The sample test is Lesson 23: Sample Test on the next page of this Study Guide.The answers are given at the end of this Study Guide. Text references for allReview Sample Tests are to the left of the problem number in the answers. Be sureto time this test and allow at least this amount of time when you take the real test.Remember that every test you take is averaged into your score for this lesson.



136


1. Solve for y: |4y � 3| 9

2. Simplify: (4x2y�7)(�2x�3y8)2

3. Factor: a2 � 2ab � b2 � 1

4. Factor: 125p3 � 8

5. Simplify:

6. Solve for x:

7. Simplify:

8. In a computer center, a card-sorter operator is given the job of alphabetizing agiven quantity of data cards. The operator knows that an older sorter can do thejob by itself in 6 hours. With the help of a newer machine, the job is completed in2 hours. How long would it take the new machine to do the job alone?

9. Solve:

10. Graph: 2x � 3y � 8 � �1

8x1/3

y1/4�

y1/12�

------------------------2

x 2 x2

4�x

----------------��

a8b

4

16c12

-------------4

1 3x 1�3� 4�



11. Graph: y � �3x2 � 6x � 1

12. Find an equation of a line containing the points (2, �1/3) and (�4, 5). Write theresulting equation in the form y � mx � b.

13. Solve for x: x4 � 34x2 � 225 � 0

14. If f(x) � (x � 1)2 � 1, what is f(a � 1)?

15. The accompanying graph is of y � f(x). What is the graph of y � f(x � 1) � 1?

16. If what is

17. Write in exponential form: logkH � 4

18. Write as logarithms of A or B only: log A2Bt

y

x

f x( )12---x 3�

7---------------- ,� f

1�x( )?



19. Solve for x:

20. If p is directly proportional to the square of q, find p when q � 2, if p � 5 whenq � 4.

21. Solve: x � 2y � 2x � y � z � 1x � 2z � 1

22. Find the inverse of .

23. Simplify:

24. Graph the system: y �x � 6�7 � x � �5y � �7

25. In an ecology drive, a town sets a goal of processing at least 40 tons of paper,30 tons of aluminum cans, and 50 tons of glass bottles each month. Eachpickup of garbage made in a commercial district has 3 tons of paper, 1 ton ofaluminum, 1 ton of glass, and costs $50, while each pickup in a residential dis-trict has 1 ton of paper, 1 ton of aluminum, 5 tons of glass, and costs $25. Howmany pickups of each kind, commercial and residential, should be made inorder to minimize the cost of the project?

210

2

2--------log xlog��

2 1

3 2

2� 3

1 1

1 2

2 1

1 0

5 4��


139

Comprehensive Lesson: Final Exam for Math 103Y

Congratulations, you have completed all your lesson tests for the second module Math103Y of the Intermediate Algebra course. The only remaining task for you to completeis the final examination. This will demonstrate that you have mastered the material con-tained in the last eleven lessons of this Study Guide.


Objectives

When you have completed the final exam, you will be able to:

1. Say you have mastered the material in Lessons 11–22.

2. Receive a grade in Math 103Y.

Procedures

1. Read through your past homework assignments, and redo as many problems asneeded to make sure you feel that you are comfortable with all the material.

2. Redo your sample tests from the last eleven lessons.

3. Do the following as a sample test for the final exam:

The sample test is Sample Final for Math 103Y given on the next page of thisStudy Guide. Be sure to time this test and allow at least that amount of time whenyou take the real test. If you take this test during the designated day of finalsweek, you will be restricted to a time limit of 3 hours.

During the regular class times you can take the entire continuous period forwhich the math lab is open.

4. Show your sample test to a staff member and ask for the final for Math 103Y. Ifyou use scratch paper, number your problems in sequence and staple yourscratch paper to your test before you turn it in. Be sure you are ready for this test,since you can take it only once and it counts 35% of your course grade.

If you complete the course work for a module, but have not satisfied the atten-dance requirement, you can request an IP, in-progress grade, and complete yourattendance requirement during the following semester.

If you are on an IP grade from the previous semester and you complete thecourse work for a module, but have not satisfied the attendance requirement, youcan request an I, incomplete grade, and complete your attendance requirementduring the following semester.

Remember, you cannot receive a passing grade without the attendance hours.



If you are on an I grade from the previous semester and you complete the coursework for a module, but have not satisfied the attendance requirement, you canrequest an extension of the incomplete grade for another semester by filling out apetition with the Admissions Office.


141

Sample Final for Math 103Y

This is a very accurate representation of the actual exam, so be certain that you canwork these problems easily.

1. Graph using the slope and y-intercept.

y � �3x � 2

2. Find the slope of the line containing the points (13, 7) and (10, �4).

3. Find an equation of the line containing the point (�4, �5) and is perpendicularto the line 2y � 3x � 8.

4. Find the center and radius of

x2 � y2 � 6x � 8y � 39 � 0 and graph it.

5. Solve by addition or substitution.

5x � 3y � 24

3x � 5y � 28



6. Solve:

x2 � y2 � 100

2x2 � 3y2 � �120

7. Solution A is 16% acid and solution B is 9% acid. How many ounces of eachshould be mixed to make 350 ounces of 12% acid? (Translate to a system ofequations and then solve.)

8. Solve by completing the square: 3x2 � 4x � 3 � 0


Sample Final for Math 103Y 143

9. A boat travels 24 miles downstream and then immediately returns to its startingpoint. If the entire trip took 4.5 hours, and if his speed in still water was 12 mph,find the rate of the current. (Translate using algebra and then solve.)

10. Solve:

11. Solve: 5x4 � x2 � 4 � 0

12. Solve for a:

1x 3�------------- 1

x 1�-------------� 1�

q a

a2

b2

�

-----------------------�



13. The volume, V, of a given mass of a gas varies directly as the temperature, T, and

inversely as the pressure, P. If the volume is 231 cm3 at a temperature of 42� and

a pressure of 20 kg/cm2, what is the volume when the temperature is 30� and the

pressure is 15 kg/cm2?

14. If f(x) � 2(x � 3) � x2, find f(�4).

15. Use completing the square to change y � �2x2 � 2x � 3 to a standard form.

16. What are the numbers whose difference is 20 and whose product is a minimum?(Translate using algebra and then solve.)



17. If f(x) � 4x � 5x2 � 7, find f (x � h).

18. If

19. If the graph is y � f(x), graph y � f(x � 2) � 1.

20. Write exponential form for logc 6 � 5.

21. Write in terms of logarithms of x, y, z, and w only:

22. Solve for x:

23. How long would it take an initial investment of $3000 to double, if interest is10% compounded annually?

f x( ) x 1�2

------------- , find f–1

x( ).�

y

x

x3y

2

w z5------------log

2x

2

24�

� 32�



24. Solve using matrices:

x � 2y � 3z � 92x � y � 2z � �83x � y � 4z � 3

25. Use the inverse of the coefficient matrix to solve:

7x � 2y � �39x � 3 y � 4

26. Simplify:

27. Graph: 2x � 3y 12

1 2

2 1

1 0

5 4� 2� 3

1 1�



28. Graph this system and label the vertices:

2x � y � �4�x � y �7

y � �2


Randy Manufacturing Company produces two types of toy cars: a sedan and aconvertible. Each model has to be processed by a grinder and a finisher. Eachsedan requires 2 hours on the grinder and 2 hours on the finisher. While eachconvertible requires 2 hours on the grinder and 4 hours on the finisher. The com-pany has two grinders and three finishers, each of whom work at most 40 hoursper week. Each sedan brings a profit of $3.00 and each convertible brings a profitof $4.00. Assuming every car will be sold, how many of each should be made tomaximize profit?

29. Organize the information in a table, give the system of constraints, and the objec-tive function.



30. a. Sketch the feasibility region.

b. Label the vertices.

c. Evaluate the objective function at each vertex.

d. Answer the question.

31. Solve x2

4x� 12�

2x2

x� 3�--------------------------------- � 0.


149

Answers to Sample Tests

Sample Test 1

1.

10

2.

240 ft

2

3.

x

�

9

4.

�

2

�

x

5a.

7

5b.

6.

7

7. 8. 9. 10.

11. 12. 13.

448

14.

�

173

15. 16.

73

�

9

y

17.

18.

�

1

19. 20.

x

14

21. 22. 23.

24.

24

x

21

25.

Sample Test 2

1.

–2

6.

no solution

2.

10

7.

any real number

3.

�

8

8.

18 in. by 9 in.

4.

2.2

9.

$800.00

5.

�

2

10.

56---

8�7

-------- 740------ 18y 3xy� 5y� 5�

2x 6 11y�( ) 7 x 3� 2y�( )

10x 21�–1 0 1 2 3 4 5

19--- 1

64------ a

4�

3b8

---------- 216x21

�

x9

�

64y18

-------------

2A r2

�r

------------------- or 2Ar

------- r�

6360755_ANS_pp149-180.fm Page 149 Wednesday, June 28, 2006 4:36 PM

150


Sample Test 3

1. 2. 3.

4. 5. 6.

7.

Anything over 66 miles.

8.

Any score from 95 up to and including 100.

9.

10. 11.

{1, 2, 3}

12.

{3}

13.

14. 15. 16.

17.

{

�

9, 9}

18. 19.

{1, 3}

20.

21.

{2}

22.

All real numbers

23.

24. 25.

Sample Test 4

1.

3rd

6.

2.

7

7.

3. 8.

4. 9.

5. 10.

x x 10�{ } x x 5��{ } x x 2��{ }

x x 2��{ } x x 218

------�⎩ ⎭⎨ ⎬⎧ ⎫

x x 74---�

⎩ ⎭⎨ ⎬⎧ ⎫

23---

–1 6

–2 4 x x 4�{ }

x x 1��{ } x 4 x 3��{ } x 7 x 1� ��{ }

3�4

-------- , 74---

⎩ ⎭⎨ ⎬⎧ ⎫

x x 3� or x 3��{ }

x 1 x 4� �{ }

x x 10 or x 14��{ } x 310------ x 21

10------� �

⎩ ⎭⎨ ⎬⎧ ⎫

2y3

6y2

y� 3��

16z4

64z2

64� �

6abc� 3a2bc� 2x

21�

2

3x2

-------- 1

3x3

--------� �

16x3y

23x 7y 4�� 3y

28y� 16 78�

2y 4�-----------------� �

5x3

3x2

� 2x 1�� 3x3

2x2y 14xy� 22x

210x 2y 1��


Answers to Sample Tests 151

Sample Test 5

1. 8.

2. 9.

3. 10.

4. 11.

5. 12.

6. 13. 8 cm by 5 cm

7. 14. 12 feet

Sample Test 6

(R.6) 1. 14a + 4 (4.1) 6.

(R.7) 2. (5.3) 7.

(1.2) 3. (4.5) 8.

(1.5) 4.

(1.6) 5. (4.6) 9.

(1.3) 10. $600.00

3x2y y

27xy� 2�( ) x 4�( ) x

24x 16� �( )

a d�( ) b c�( ) 2ab 2a2

3b2

�( ) 4a4

6a2b

29b

4��( )

3t 5�( )23 x 3�( ) x 3�( ) x 2�( )

2x 3x 5�( ) 3x 5�( ) x 1 or x 35---� �

x 28�( ) x 12�( ) x 7, or x� 0, or x 7� � �

x 10y 4� �( ) x 10y� 4�( )

2 3x 1�( ) 5x 3�( )

16x3

6x2

14x��

4b5c

6

5-------------- 3x

3x

22�

4�3x 1�-----------------� �

T 5R 30�3

---------------------� 2 4x 3�( ) 3x 2�( )

2 x 3�� –2 3

x x 215

------ or x 3��⎩ ⎭⎨ ⎬⎧ ⎫

2x 3y2

�( ) 4x2

6xy2

9y4

� �( )



Sample Test 7

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.

Sample Test 8

1. 6.

2. 7. y = 100

3. no solution 8. 1 hours

4. 9. 3 miles per hour

5. 10. 160 cm3

y 2�3y

------------- 1x 1�-------------

xx 3�------------- 1

x 3�-------------

xx 5�------------- 2y

2y 2��

2y y 2�( ) y 2�( )-------------------------------------------

1 y�

x2

-------------- 5x2

1�

6 x 1�( )2------------------------

x 2�( )2x 2�( ) 2�

x y�------------- or 2

y x�-------------

x 3� k 725

------ , y 725

------x� �

y 24��

78---

x 12---��

t1

R QSt2�

QS-----------------------�



Sample Test 9

1. 2. 3. 2.356 10�2 4. 7.5 10�4

5a. �18 5b. 3 6. 7. 8.

9. 10. 11. 12.

13.

Sample Test 10

1. 2. 3. 4.

5. 33 � 39i 6. 4 � 5i 7. �i 8.

9. 10. 11.

12. Only since does not check. 13.

9y8

25x14

------------- 4x3a–1

�

2 y 3a 2� 5x3

7

2x 2x35 11 24 3x 2�( ) 2x3 3 2x

3x3

2x

5y4

-------------------

28x35

2x---------------- x

73---

2x

23---

y

13---

200x46

6 7 x 5x��

24 2i�29

------------------- or 2429------ 2

29------i�

y 3 xy 2x� �y 4x�

------------------------------------- x 25�

x 3,� x 7� b 2 5�



Sample Test 11

(5.4) 1. (6.5) 6.

(5.5) 2. x = 1 (6.2) 7.

(Study Guide, page 36, Example 7)

3. 36 mph (6.5) 8.

(5.8) 4. (6.6) 9.

(6.1) 5. (6.8) 10.

Final: There are text references in front of each problem on this final.103X

For example, [1.5c] means Chapter 1, Section 5c of your textbook.

[R.3c] 1. [R.3c] 2. �80 [R.7b] 3.

[R.7c] 4. 6.875 109 [1.1d] 5. [1.3a] 6. 17 ft by 31 ft

[1.2a] 7. [1.4c] 8.

[1.5b] 9. [1.6c] 10. {�9, 15}

[1.6e] 11. [1.6e] 12.

7x 1�14

----------------- 2

3 2 4�---------------------

x6

4a2

--------

y 18x2y7

2x----------------------

29--- ohm x 1 or x 5� �

5x 6�29 3i�

17------------------- or 29

17------ 3

17------� i

34x 36�81y

40

16x24

-------------

532------

c 2A hd�h

---------------------� x x 2�{ }

x x 3��{ }

2

y y 4�3

-------- or y 4��⎩ ⎭⎨ ⎬⎧ ⎫

x –99 x 111� �{ }



[4.1d] 13. [4.2d] 14.

[4.3b] 15. [4.5b] 16.

[4.7a] 17. [4.7a] 18.

[4.8a] 19. �7, 4 [4.8b] 20. [5.3b] 21.

[5.1d] 22. [5.1e] 23. 15 [5.2b] 24.

[5.2c] 25. [5.5a] 26. �1, �5 [5.5a] 27. �145

[5.6a] 28. 3 hrs [5.6c] 29. 375 km [5.7a] 30.

[5.4a] 31. [6.1d] 32. 6 [6.2d] 33.

[6.3b] 34. [6.4a] 35.

[6.4b] 36. [6.5a] 37.

[6.5b] 38. [6.2d] 39.

[6.7a] 40. a = 5 [6.8e] 41. [6.6b] 42. x = 4

16x2

8x� a4

16b4

�

x3

y�( ) a 2b�( ) x 3x 5�( ) 2x 3�( )

4 x 2�( ) x 2�( ) 5z 2z 3y�( ) 2z 3y�( )

5�2

-------- , 92--- a 12�

32a 4�--------------�

2x 3�( ) x 5�( )7x

---------------------------------------- 4x y�x y�

----------------

42 x�-------------

314------ s Rg

g R�--------------�

5x 1�5

----------------- 2xy 3xy23�

2x x23

3y---------------- 2 x�( ) 3x3

1 5�y 9x

2y3

3x2

--------------------

3 6 4�2

--------------------- 108x56

8 i�5

------------



Sample Test 12

1. 2 or �2 6.

2. 7. one real solutionx = 0.71 ft to nearest hundredth

3. 8.

4. 2.3 or 0.7 9.

5. �50 or 2 10.

Sample Test 13

1. 6.

2. y = 27, �8 7.

3. m = 9, 16 8.

4. 9.

5. 10.

2412 x�---------------- 24

12 x�----------------�

92--- , 4mph�

2x 4�( ) 2x 3�( ) 24�

3� 5i2

---------------------- x2

9� 0�

1� 2

20x

------ 20x 9�-------------� 1, 45 hr�

x 1, 2i 55

-----------------� b d2

a2

c2

��

a bq

q2

1�

--------------------�

r �s� �2s

24�A��

2�--------------------------------------------------------�

x 23--- , 3

2---�� Q 2mn

2�

p------------------�

x 1, 4�� w 83 lb.�



Sample Test 14

1. Slope:

y-intercept: (0, 2)

2. x-intercept: (2, 0)

y-intercept: (0, 5)

3. 4.

5. 8. �5

6. 9. Center: (�2, 5), Radius:

7. 10.

y

x

13---�

y

x

y

x

x 3�( )2y 5�( )2

� 4�

y 3x� 3�� 3

2 29, 3 1�,( ) y 13--- x 4��



Sample Test 15

1. 2. Add 1 to the x’s and add 3 to the y’s.

1

3a. (�6, 8) 3b. x = �6 3c. y = 8 3d. none

4. 5. 6.

7. Minimum product is �100. It is a minimum since the coefficient of the squareterm is positive

8. 30 ft by 30 ft 9.

10. 11.

12. 13.

1 2�,�( ), V 0 0,( ), 1 2�,( )

–1 1 2� 3�,�( ) 0 1,( )�

0 1 0 3�,�( ) 1 3,( )�

1 1 2� 3�,�( ) 2 1,( )�

1� 0,( ) 53--- 0,⎝ ⎠

⎛ ⎞, y x 32---�⎝ ⎠

⎛ ⎞ 2 114

------�� y –2 x 12---�⎝ ⎠

⎛ ⎞ 2 72---��

x x 5�2

-------- or x 2��⎩ ⎭⎨ ⎬⎧ ⎫

x x 1 or 2 x 3� ��{ } x 9� x 4��{ }

x x 1 or x 2��{ } x x 2 or 1 x 32--- or x 6��

⎩ ⎭⎨ ⎬⎧ ⎫



Sample Test 16

1a. Yes 1b. No 2a. {2, 5, 7} 2b.{�3, 6}

3a. {(�3, 2), (6, 5), (�3, 7)} 3b. No

4. �8 5.

6. 7.

8. 9.

10.

12. 13.

14. 15. The inverse of

11.a2

3a 3��

10x 5x2

3�� 2 2a� h�

all � x; 4�{ } x x 32---�

⎩ ⎭⎨ ⎬⎧ ⎫

f1�

x( ) 13 2x�-----------------�

y f x 3�( )� y f x( ) 2��

y f x 1�( ) 3�� f x( )



Sample Test 17

(7.1) or (7.2) 1.

(7.3) 2. Truck is 50 mph and the car is 60 mph.

(7.3) 3.

(7.4) 4.

(9.1) 5. a. b.

(2.5) 6. (7.6) 7.

2�34---�,

x yz

1 z2

�

------------------- or x yz 1 z2

�

1 z2

�-------------------------� �

x 33

------------ or x 102

---------------� �

3 5 12--- , 4�⎝ ⎠

⎛ ⎞

y 3�2

--------x 132

------��

or 3x 2y� 13��

y

x



(8.2) 8. (2.2) 9. a. �40

(Appendix D) 9. b. (7.8) 10.

Sample Test 18

1. 2.

3.

4.

5.

6.

7. Only x = 4, since x = �2does not check.

8. 9. 7.27 years 10. 17,273

f1�

x( ) 5 7x�3 2x�-----------------�

12x2

� 34x 20�� x x 5�3

-------- or x 2��⎩ ⎭⎨ ⎬⎧ ⎫

y

x

y = 5x

logM161 5�

100.7267

5.33�

xlog ylog 12--- zlog� �

x yx 5�-------------log

x 3� 3,�

x 13---�



Sample Test 19

1. 6. dependent

2. 7.

3. 8.

4. 9. 30 cases at $4.7025 cases at $5.20

5. consistent 10. $40 for the first man$30 for the second$35 for the third

Sample Test 20

1. 2.

3� 3�,( )

23--- , 5

3---�⎝ ⎠

⎛ ⎞ 6� 7,( ) 2 1�,( ),

0 45---,⎝ ⎠

⎛ ⎞ 2 0,( ) 2� 0,( ),

4 2 3,�,( )

2 3� 9

5 2 13

R2 R1→R1 2 R2 � R2→R3 3 R2 � R3→

20 R2 11 R3 � R3→

2 1 2 3

1 6 3 4

3 2� 1 0

1 6 3 4

0 11� 4� 5�

0 20� 8� 12�

1 6 3 4

0 11� 4� 5�

0 0 8 32

5 R1 2 R2 � R2→ 2 3� 9

0 19� 19

2x 3y� 9, 2x 3 1�( )� 9� �

19y� 19 y, 19 19�( )� 1��

Answer is 3 1�,( ).

x 6y 3z� � 4 x 6 1�( ) 3 4( )� �, 4� �

11y� 4z� 5�� 11y� 4 4( )�, 5 y;� 1��

8z 32 z, 32 8� 4� � �

Answer is 2 1 4,�,�( )



3.

4.

5.

2 2 2 1�( )

2 4 2 3�( )

3� 4 3� 1

3� 3 3� 2�( ) � 4 12� 2� 3�

8 9� 6� 6��

8� 5�

1� 0�

2 1�

4 3�

4 1

3 2� 2 4 1 3 � 2 1 1 2�( )�

4 4 3 3 � 4 1 3 2�( ) �

5 4

7 10� �

4 1 1 0

3 2� 0 1

3 R1 4 R2 � R2→4 1 1 0

0 11 3 4�

–11 R1 R2� R1→ 44� 0 8� 4�

0 11 3 4�

1�44-------- R1 R1→

111------ R2 R2→

1 0 211------ 1

11------

0 1 311------ 4�

11--------

So, we have B1�

211------ 1

11------

311------ 4�

11--------

�

6. 1 1 1 1 0 0

1 0 1� 0 1 0

0 2 1� 0 0 1

R2 R1→R1 1� R2 R2→

1 0 1� 0 1 0

0 1 2 1 1� 0

0 2 1� 0 0 1



7.

8.

9. Using from Problem 5, we have the following:

Answer is (�3, 5).

6. (continued)

–2 R2 R3� R3→

1 0 1� 0 1 0

0 1 2 1 1� 0

0 0 5� 2� 2 1

5� R1 R3� R1→5 R2 2 R3 � R2→

5� 0 0 2� 3� 1

0 5 0 1 1� 2

0 0 5� 2� 2 1

1�5

-------- R1 R1→

15--- R2 R2→

1�5

-------- R3 R3→

1 0 0 25--- 3

5--- 1�

5--------

0 1 0 15--- 1�

5-------- 2

5---

0 0 1 25--- 2�

5-------- 1�

5--------

So, we have C1�

25--- 3

5--- 1�

5--------

15--- 1�

5-------- 2

5---

25--- 2�

5-------- 1�

5--------

�

2 3 3 0 1� 2 � � 2 4 3 1�( ) 1� 0 � � 2 1 3 2 1� 3�( ) � �

0 3 4 0 2 2 � � 0 4 4 1�( ) 2 0 � � 0 1 4 2 2 3�( ) � �

4 5 11

4 4� 2�

1 1 1

1 0 1�

0 2 1�

x

y

z

2

2

5�

�

B1�

211------ 1

11------

311------ 4�

11--------

7�

19�

2

11------ 7�( ) 1

11------ 19�( ) �

311------ 7�( ) 4�

11-------- 19�( ) �

3�

5

� �



10. Using C�1 from problem 6, we have the following:

Sample Test 21

1. 2.

3. 4.

5. The minimum value is �6; it occurs at the point D (2, 0).

6. The maximum value is 16 ; it occurs at the point .

25--- 3

5--- 1�

5--------

15--- 1�

5-------- 2

5---

25--- 2�

5-------- 1�

5--------

2

2

5�

25--- 2( ) 3

5--- 2( ) 1�

5-------- 5�( ) � �

15--- 2( ) 1�

5-------- 2( ) 2

5--- 5�( ) � �

25--- 2( ) 2�

5-------- 2( ) 1�

5-------- 5�( ) � �

3

2�

1

� � Answer is(3, �2, 1).

y

x x

y

y

x

y

x

(–1, 6)

y = –2

2x + y = 4

–x +

y =

7

(–9, –2) (3, –2)

12--- 6 9

2---,( )



7.

Objective function in Constraints are

8. Twenty of each should be made; the maximum profit is $140.

9.

Objective function in Constraints are

10. The dietician should use 0.3 oz of chicken and 0.8 oz of tuna. The minimumnumber of calories is 166.

Number tobe made of

Grinder(Hours)

Finisher(Hours)

Profit(Dollars)

Sedan x 2x 2x 3x

Convertible y 2y 4y 4y

Total � 2(40) � 3(40) � P(x, y)

P x y,( ) 3x 4y.�� 2x 2y� 80,�

2x 4y� 120�

x 0�

y 0�

Amount inounces of

Protein(Grams) Calories

Cost(Dollars)

Chicken x 30x 180x 0.70x

Tuna y 20y 140y 0.30y

Total � 25 = C(x, y) � 0.45

C x y,( ) 180x 140y��

30x 20y� 25,�

0.70x 0.30y� 0.45�

x 0�

y 0�



Sample Test 22

(8.4) 1. (SG 21.4, pp. 124–127)

(8.6) 2.

(8.6) 3. x = 3

(8.7) 4. Approx. 4.004 years

(3.5) 5. (�6, 7), (2, �1)

(Appendix C) 6. (�2, �1, 4)

(SG 20.2, pp. 95–100) 7.

(SG 20.3, pp. 104–108) 8.

(SG 21.4, pp. 124–127) 10.

Objective function is Constraints are

C(x, y) = 10x + 20y

Answer is 8 Ace cabinets

and 3 Excello cabinets.

log bwx

2

y z3---------

y

x

6

4

4 (2, 3)

8

x 43---�

9. P(4, 0) = 12 is the maximum.

2 3 0

0 6 5

145

------ 135

------,

Numberto buy of

Cost(Dollars)

Floor Space(Square feet)

StorageCapacity

(Cubic feet)

Acecabinets

x 10x 6x 8x

Excellocabinets

y 20y 8y 12y

Total = C(x, y) � 72 � 100

6x 18y� 72,�

8x 12y� 100,�

x 0�

and y 0�



Sample Test 23

(1.6) 1. y � �3 or y � (5.5) 6. x = 2

(R.7) 2. (6.1) 7.

(4.5) 3. (a + b + 1)(a + b � 1) (5.6) 8. 3 hours

(4.5) 4. (5p � 2)(25p2 + 10p + 4) (6.6) 9.

(6.2) 5.

(2.5) 10. (7.6) 11.

32---

16y9

x4

----------- a2

b

2 c3

-----------

x 283

------�

64x

23---

y

13---

-----------

y

x

3

92

–

y

x(2, –1)(0, –1)

(1, 2)



(2.5) 12. (SG, p. 76) 15.

(7.4) 13. x = 5, 3

(2.3) 14. a2 + 1

(8.2) 16. f�1(x) = 14x � 6 (SG 21.3, pp. 129–130) 24.

(8.3) 17. k4 = H

(8.4) 18. 2 log A + t log B

(8.6) 19. x = 2

(5.8) 20.

(3.5) 21.

(SG 20.3, pp. 104–108) 22.

(SG 20.2, pp. 95–100) 23.

(SG 21.4, pp. 124–127) 25. residential: 25commercial: 5

y 89--- x�

139

------��

y

x

(2, 3)

(1, 1)

(0, 3)

x = –7 y = –x – 6

y = –7

B (–5, –1)

x = –5

(–7, 1)A

C(–5, –7)(–7, –7)

D

K 516------ , p 5

4---� �

12--- , 3

4--- , 1�

4--------

2 1�

3� 2

9 11

8 5



Final 103Y: There are text references in front of each problem on this final.

For example, [3.3d] means Chapter 3, Section 3d of your textbook.

[2.5b] 1. [2.4b] 2. [2.5c] 3.

[9.1d] 4.

[3.2 & 3.3] 5. (6, 2) [9.4a] 6. (�6, �8), (�6, 8), (6, �8), (6, 8)

[3.4a] 7. 150 oz of A, 200 oz of B [7.1b] 8.

[3.4b, 5.6c, 7.3a] 9. 4 mph [5.5a] 10.

[7.4b] 11. [7.3b] 12.

[5.8e] 13. 220 cm3 [2.2b] 14. 2

[7.6a] 15. [7.7a] 16. �10 and 10

[2.2b] 17. 4x + 4h � 5x2 � 10xh � 5h2 + 7

y

x

(1 ,�1)

(0 ,�2)

C 3� 4,( )r 8�

113

------ y 23---x 7

3---��

y

x

(�3 , 4)

–2 133

------------------------

–1 2

1, 2i 55

----------------- a bq

q2

1�

--------------------�

y –2 x 12---�⎝ ⎠

⎛ ⎞ 2 72---��



[8.2c] 18. f�1(x) = 2x + 1 [Study Guide Lesson 16, Example 3, p. 75]

[8.3b] 20. c5 = 6

[8.4d] 21. 3 log x + 2 log y

� log w � log z

[8.6a] 22. �3, 3

[8.7b] 23. 7.27 years

[Study Guide Lesson 20.1] 24. (�1, 2, �2)

[Study Guide Lesson 20.4] 25.

[Study Guide Lesson 20.2] 26.

[3.7b] 27. See the answer to 3 of Sample Test 21.

[3.7c, Study Guide 21.3] 28. See the answer to 4 of Sample Test 21.

[Study Guide Lesson 21.4, 5] 29. See the answer to 7 of Sample Test 21.

[Study Guide Lesson 21.4] 30. See the answer to 8 of Sample Test 21.

[7.8b] 31. {x | x � �2 or �1 � x � or x � 6}

19.

y

x

(�4 ,�2)

(4 ,�5)

( 0 ,�1)

15---

–139------ , 55

39------⎝ ⎠

⎛ ⎞

9 11

8 5

32---



Answers to Homework Problems in the Study Guide

Lesson 2, page 11.

1. 2.

Replacing x with any No matter what number we try for x,number gives a true equation. we get a false equation. Thus, theThus, all real numbers are equation has no solution.solutions.

3. 4.

Replacing x with any real No matter what number we try for y,number gives a true equation. we get a false equation. Thus, theThus, all real numbers are equation has no solution.solutions.

5.

Replacing m with any realnumber gives a true equation.Thus, all real numbers aresolutions.

3 x 5�( ) 25 10 3x�( )��

3x 15� 25 10� 3x��

3x 15� 15 3x��

0 0�

x 25� 6x 5 x 2�( )��

x 25� 6x 5x� 10��

x 25� x 10��

25 10�

8x 4 2x 3�( )� 12�

8x 8x� 12� 12�

12 12�

19 3y 4�( )� 10 3y��

19 3y� 4� 10 3y��

15 3y� 10 3y��

0 5��

2 3m 1�( ) 8m 2m 2�( )��

6m 2� 8m 2m� 2��

6m 2� 6m 2��

0 0�


Answers to Homework Problems in the Study Guide 173

Lesson 3, page 17.

a. {x |x � 6}

{x |x � 0}

{x |x � 0}

b. {x |x � 10}

{x |x � �2}

{x |�2 � x � 10}

c. {x |x � �10}

{x |x � �3}

{x |x � �3}

d. {x |x � �7}

{x |x � 11}

{x |x � �7 or x � 11}

1. No solution 2. All real x 3. No solution

4. x = 3 5. x � 2 or x � 2

(all real x except x = 2)

0 6

0

0

10

�2 10

�2

�10 �3

�3

�10 �3

�10

�7 11

11

�7 11

�7



Lesson 20

Exercise 20.2, page 100.

1. 3. 5. 7. 9.

11. 13. 15. 17.

19. 21. 23. 25.

27. 29.

Margin Exercises, Section 20.3, pages 103, 104, 107.

1. a. A; b. A; c. both equal A

2. a. ; b. I; c. both equal I

3. 4. 5. a.

b. ; c. ; d.

2� 7

6 2

1 3

2 6

9 9

3� 3�

11 13

5 3

4� 3

2� 4�

17 9

2� 1

0 0

0 0

1 2

4 3

5� 4 3

5 9� 4

7 18� 17

2� 9 6

3� 3 4

2 2� 1

16� 2 19�3 2� 4

2 1 5�

x

y

17

13�

1 1� 2 4�

2 1� 1� 1

1 4 3� 1�

3 5 7� 2

x

y

z

w

12

0

1

9

�40.19� 37.94 142.24

36.78� 16.63 119.62

1.659� 14.97 12.65

1 0 0

0 1 0

0 0 1

I�

A1�

12---�

12--- 1

2---

1 0 1�

32--- 1

2---�

12---�

� A1�

2 52---�

1�32---

� 4 2�

1 5

x1

x2

1�

1;�

A 4 2�

1 5� A

1� 122------ 5 2

1� 4� x1

322------ , x2�

522------� �



Exercise Set 20.4, pages 110–111.

1. 3. 5. 7.

9. 11. does not exist. 13.

15. 16. 17. 18.

Lesson 21; pages 116–118.

Section 21.3 (margin exercises)

1. 2. 3. Vertices: (0, 0), (4, 0),

(4, ), (0, 3), ( 3)

3� 2

5 3�

2 3�

7� 11

211------ 3

11------

1�11-------- 4

11------

38--- 1�

4-------- 1

8---

1�8

-------- 34--- 3�

8--------

1�4

-------- 13--- 1

4---

13--- 0 1

3---

2�5

-------- 25--- 1

5---

215------ 1

5--- 1�

15--------

A1�

1 2� 3 8

0 1 3� 1

0 0 1 2�

0 0 0 1�

1�39-------- 55

39------,⎝ ⎠

⎛ ⎞ 117------ 13�

17-----------,⎝ ⎠

⎛ ⎞ 3 3� 2�, ,( ) 124 14 51�, ,( )

1

y

x

1 � y � 212

1

y

x1

–1 1

y

x

6

5 6

53--- 12

5------,



Exercise Set 21.3, page 122.

1. 3. 5.

7. 9. 11.

13. 15. 17.

19. 21. 23.

25. 27.

1

y

x

y < x

1

y

x

y + x � 0

1

y

x

3x – 2y � 6

1

y

x

2x + 3y � 6

1

y

x

3x – 2y � 5x + y

1

y

x–4

x < –4

1

y

x

0 � x � –5 12

y

x(0, 0)

y � |x|

y

x32

12( , – )

y

x

y

x

45

75( , )

y

x(0, 1)

(2, 5)

(2, –3)

y

x

(4, 4)(0, 6)

(0, 0) (6, 0)

y

x

256(1, )

94(1, )

52(3, )

34(3, )



Margin Exercises, Section 21.4, pages 125–126.

1. Maximum 120, when x = 3, y = 3; minimum 34, when x = 1, y = 0.


3. Maximum $23.70, by selling 50 hot dogs and 40 hamburgers.

Exercise Set 21.4, page 127.



5. Maximum 102, 8 of A, 10 of B 6. Maximum 425, 5 of A, 15 of B

7. $7000 bank X, $15,000 bank Y, maximum income $1395

8. $22,000 in corporate bonds, $18,000 in municipal bonds, maximum income$3110

9.


P(x, y) = 34x + 31y

Answer is 2 knit suits and 4 worsted

suits for a profit of $192.

10.


P(x, y) = 10.56x + 6.40y

Answer is 125 batches of SMELLO and 187.5 batches of ROPPO for a profit of $2520.

Number tobe made of

Cutting(Hours)

Sewing(Hours)

Profit(Dollars)

Knit suits x 2x 4x 34x

Worsted suits y 4y 2y 31y

Total � 20 � 16 = P(x, y)

2x 4y� 20,�

4x 2y� 16�

x 0�

y 0�

Number ofbatches of

EnglishPounds

Virginia(Pounds)

Latakia(Pounds)

Profit(Dollars)

SMELLO x 12x 4x 10.56x

ROPPO y 8y 8y 6.40y

Total � 3000 � 2000 � 500 = P(x, y)

12x 8y� 3000,�

8y 2000�

4x 500�

x 0�



11.


C(x, y) = 12,000x + 10,000y

Answer is 30 P-1 airplanes and 10 P-2

airplanes for a minimum of $460,000.

Procedure 5, pages 117–118.

13.

Objective function is P(x, y) = 3x + 2y. Constraints are

Answer is 8 parts of A and 10 parts of B.

14.


R(x, y) = 1.00x + 0.80y

Answer is 150 pounds of the one mix

and 75 pounds of the other mix.

Number tobe used of

First-classPassengers

TouristPassengers

EconomyPassengers

Cost(Dollars)

P-1 planes x 40x 40x 120x 12,000x

P-2 planes y 80y 30y 40y 10,000y

Total � 2000 � 1500 � 2400 = C(x, y)

40x 80y� 2000,�

40x 30y� 1500�

120x 40y� 2400�

x 0�

y 0�

Number tobe made of

Drill Press(Minutes)

Lathe(Minutes)

Profit(Dollars)

Part A x 5x 10x 3x

Part B y 6y 4y 2y

Total � 100 � 120 = P(x, y)

5x 6y� 100,�

10x 4y� 120�

x 5�

y 5�

Number ofpounds from

Nuts(Pounds)

Raisins(Pounds)

Revenue(Dollars)

One mix x x x 1.00x

Other mix y y y 0.80y

Total � 100 � 125 = R(x, y)

12--- 1

2---

13--- 2

3---

12---x 1

3---y� 100,�

12---x 2

3---y� 125�

x 0�

y 0�



15.


P(x, y) = 100x + 150y

Answer is 600 of the A model and 750

of the B model for a profit of $172,500.

16.


C(x, y) = 500x + 800y

Answer is 20 days for plant A

and 20 days for plant B.

Number tobe made of

AssemblyLine (Hours)

Finishing(Hours)

Profit(Dollars)

A model x 3x 1x 100x

B model y 4y 2y 150y

Total � 4800 � 2100 = P(x, y)

3x 4y� 4800,�

x 2y� 2100�

x 0�

y 0�

Number ofdays for

Standard(Models)

Superior(Models)

Luxury(Models)

Cost(Dollars)

Plant A x 10x 8x 3x 500x

Plant B y 20y 9y 8y 800y

Total � 600 � 340 � 200 = C(x, y)

10x 20y� 600,�

8x 29y� 340�

3x 28y� 200�

x 0�

y 0�



Lesson 22

Review 20.2, page 102.

1.

2. 3. 4.

5. 6.

Lesson 20

Review, Pages 112–113.

1. 2. 3. 4.

5. Not possible 6. 7. 8.

9. 10. 11.

12. 13. �31 14. �1 15. 0 16. 120

17.

18. 19. (y � x)(z � x)(z � y)(yz + xz + yx)

20. (b � a)(c � a)(d � a)(c � b)(d � b)(d � c)

21. If a matrix has all 0’s below the main diagonal, then its determinant is the prod-uct of the elements on the main diagonal. Proof: Expand about the first column.

32--- , 13

14------ , 33

14------⎝ ⎠

⎛ ⎞

0 1� 6

3 1 2�

2� 1 2�

3� 3 0

6� 9� 660 3�

1� 1 0

2� 3� 2

2 0 1�

2� 2 6

1 8� 18

2 1 15�

2 1� 6�

1 5 2�

2� 1� 4

0 1� 6

3 1 2�

2� 1 2�

3� 3 0

6� 9� 6

6 0 3�

1� 1 0

2� 3� 2

2 0 1�

2� 2 6

1 8� 18

2 1 15�

2 1� 6�

1 5 2�

2� 1� 4

3 1 2

2 1 2

6 2 5

B1� does not exist.

12---� 0

16--- 1

3---

0 0 14---

0 12---� 0

13--- 0 0

1 0 0 0

0 19--- 5

18------ 0

0 19---�

29--- 0

0 0 0 1

3 2� 4

1 5 3�

2 3� 7

x

y

z

13

7

8�

�

5a 5b 5c

3a 3b 3c

d e f

5 3( )a b c

a b c

d e f

0, since the first two rows are the same.� �

b a�( ) c b�( ) c a�( )


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