UNIT 2

SYSTEMS OF EQUATIONS AND INEQUALITIES

S T U D E N T T E X T AND HOMEWORK HELPER Randall I. Charles • Allan E. Bellman • Basia Hall William G. Handlin, Sr. • Dan Kennedy Stuart J. Murphy • Grant Wiggins

Boston, Massachusetts • Chandler, Arizona • Glenview, Illinois • Hoboken, New Jersey

Copyright © 2016 Pearson Education, Inc., or its affiliates. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions, write to Rights Management & Contracts, Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030. Pearson, Prentice Hall, Pearson Prentice Hall, and MathXL are trademarks, in the U.S. and/or other countries, of Pearson Education, Inc., or its affiliates. Apple, the Apple logo, and iTunes are trademarks of Apple Inc., registered in the U.S. and other countries. Google Play is a trademark of Google Inc.

ISBN-13: 978-0-13-330072-7 ISBN-10: 0-13-330072-2 1 2 3 4 5 6 7 8 9 10 V0YJ 20 19 18 17 16 15 14

unit 2

SYSTEMS OF EQUATIONS AND INEQUALITIES

TOPIC 3

Systems of Linear Equations

TOPIC 3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3-1 Solving Systems Using Tables and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3-2 Solving Systems Algebraically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3-3 Systems of Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3-4 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Technology Lab  Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Activity Lab  Graphs in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3-5 Systems In Three Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3-6 Solving Systems Using Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

TEXAS ESSENTIAL KNOWLEDGE AND SKILLS (TEKS) FOCUS

xiv Contents

(3)(A)

Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

(3)(B)

Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

(3)(E)

Formulate systems of at least two linear inequalities in two variables.

(3)(F)

Solve systems of two or more linear inequalities in two variables.

(3)(G)

Determine possible solutions in the solution set of systems of two or more linear inequalities in two variables.

(8)(B)

Use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data.

unit 2

SYSTEMS OF EQUATIONS AND INEQUALITIES

TOPIC 4

Matrices

TOPIC 4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4-1 Adding and Subtracting Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Technology Lab  Working With Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4-2 Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4-3 Determinants and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4-4 Systems and Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

TEXAS ESSENTIAL KNOWLEDGE AND SKILLS (TEKS) FOCUS (3)(B)

Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

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Topic 3

Systems of Linear Equations

TOPIC OVERVIEW

VOCABULARY

3-1 Solving Systems Using Tables

English/Spanish Vocabulary Audio Online: English Spanish constraint, p. 88 restriccion dependent system, p. 70 sistema dependiente equivalent systems, p. 77 sistemas equivalentes feasible region, p. 88 región factible inconsistent system, p. 70 sistema incompatible independent system, p. 70 sistema independiente linear programming, p. 88 programación lineal linear system, p. 70 sistema lineal matrix, p. 102 matriz objective function, p. 88 función objetiva row operation, p. 102 operación de fila system of equations, p. 70 sistema de ecuaciones

and Graphs 3-2 Solving Systems Algebraically 3-3 Systems of Inequalities 3-4 Linear Programming 3-5 Systems in Three Variables 3-6 Solving Systems Using Matrices

DIGITAL

APPS

PRINT and eBook

Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you.

Your Digital Resources PearsonTEXAS.com

Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book.

68

Topic 3  Systems of Linear Equations

3--Act Math

The Piggy Bank The first coins were made of different metals, such as gold, silver, or copper. The value of each coin was directly related to the value of the metal from which it was made. As the value of certain metals has increased, the metal a coin is made of has sometimes become worth more than the fixed value of the coin itself. This has led to decisions to use less valuable metals for the coins. This 3-Act Math task poses an interesting problem about the value of a bunch of coins.

Scan page to see a video for this 3-Act Math Task.

If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support.

Learning Animations You can also access all of the stepped-out learning animations that you studied in class.

Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding.

Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities.

Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device.

Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support.

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3-1  Solving Systems Using Tables and Graphs TEKS FOCUS

VOCABULARY

Foundational to TEKS (3)(A) Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic. TEKS (1)(D)  Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

• Consistent system – A system of linear equations is consistent if it has at least one solution.

• Dependent system – A system of equations that has an infinite set of solutions is a dependent system.

• Inconsistent system – A system of equations that has no solution is an inconsistent system.

• Independent system – A system

Additional TEKS (1)(A), (8)(B)

of linear equations that has a unique solution is an independent system.

• Linear system – A linear system is a set of two or more linear equations that use the same variables.

• Solution of a system – A solution of a system is a set of values for the variables that makes all the equations true.

• System of equations – A system of equations is a set of two or more equations using the same variables.

• Implication – a conclusion that follows from previously stated ideas or reasoning without being explicitly stated

• Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

ESSENTIAL UNDERSTANDING To solve a system of equations, find a set of values that replace the variables in the equations and make each equation true.



Concept Summary  Graphical Solutions of Linear Systems Intersecting Lines y

Coinciding Lines y x

x O

O

one solution consistent independent

70

Parallel Lines y

infinitely many solutions consistent dependent

Lesson 3-1  Solving Systems Using Tables and Graphs

O

no solution inconsistent

x

Problem 1

TEKS Process Standard (1)(D)

Using a Graph or Table to Solve a System What is the solution of the system? b How can you use a graph to find the solution of a system? Find the point where the two lines intersect.

∙3x ∙ 2y ∙ 8 x ∙ 2y ∙ ∙8

Method 1 Graph the equations. The point of intersection appears to be ( -4, -2).

3x  2y  8 4 2

Check by substituting the values into both equations.

(4, 2)



-3x + 2y = 8

x + 2y = -8

   -3( -4) + 2( -2) = 8    ✔

y O x 2

x  2y  8

-4 + 2( -2) = -8   ✔

Both equations are true, so ( -4, -2) is the solution of the system. Method 2 Use a table. Write the equations in slope-intercept form.

-3x + 2y = 8

x + 2y = -8

    2y = 3x + 8   2y = -x - 8 3

    y1 = 2x + 4   y2 = - 12x - 4 Enter the equations in the Y∙ screen as Y1 and Y2. View the table. Adjust the x-values until you see y1 = y2 .

X

Y1

–5 –4 –3 –2 –1 0 1

X = -4

–3.5 –2 –.5 1 2.5 4 5.5

Y2 –1.5 –2 –2.5 –3 –3.5 –4 –4.5

When x = -4, both y1 and y2 equal -2. So, ( -4, -2) is the solution of the system.

Problem 2 Using a Table to Solve a Problem Biology  The diagrams show the birth lengths and growth rates of two species of shark. If the growth rates stay the same, at what age would a Spiny Dogfish and a Greenland shark be the same length? Step 1 Define the variables and write the equation for the length of each shark.

How can you use slope-intercept form to write each equation? Use the growth rate for m and the length at birth for b.



Let x = age in years.



Let y = length in centimeters.



Length of Greenland:



y1 = 0.75x + 37



Length of Spiny Dogfish: y2 = 1.5x + 22

STEM

GREENLAND SHARK

Growth rate: 0.75 cm/yr Birth length: 37 cm

SPINY DOGFISH SHARK

Growth rate:1.5 cm/yr Birth length: 22 cm continued on next page ▶

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Problem 2

continued

Step 2 Use the table to solve the problem.

Shark Length in cm

List x-values until the corresponding y-values match.

Age

Greenland

Spiny Dogfish

x

y1  0.75x  37

y 2  1.5x  22

The sharks will be the same length when they are 20 years old.

15

48.25

44.5

16

49

46







20

52

52

Problem 3

TEKS Process Standard (1)(A)

Using Linear Regression Population ​The table shows the populations of the New York City and Los Angeles metropolitan regions from the census reports for 1950 through 2000. Populations of New York City and Los Angeles Metropolitan Regions (1950–2000) New York City Los Angeles

1950

1960

1970

1980

1990

2000

12,911,994

14,759,429

16,178,700

16,121,297

18,087,251

21,199,865

4,367,911

6,742,696

7,032,075

11,497,568

14,531,529

16,373,645

SOURCE: U.S. Census Bureau

Assuming these linear trends continue, when will the populations of these regions be equal? What will that population be?

Population data for two regions

The point in time when their populations will be the same

• Use a calculator to find linear regression models. • Plot the models. • Find the point of intersection.

Enter all the numbers as millions, rounded to the nearest hundred thousand. For example, enter 12,911,994 as 12.9. Step 1

Enter the data into lists on your calculator. L1: number of years since 1950 L2: New York City populations L3: Los Angeles populations

Step 2 Use LinReg(ax ∙ b) to find lines of best fit. Use L1 and L2 for New York City. Use L1 and L3 for Los Angeles.

L1 0 10 20 30 40 50 -----L1(1) = 0

L2 12.9 14.8 16.2 16.1 18.1 21.2 ------

L3 4.4 6.7 7.0 11.5 14.5 16.4 ------

1

continued on next page ▶

72

Lesson 3-1  Solving Systems Using Tables and Graphs

Problem 3

What does x represent? The x-value is the number of years since the zero year.

continued

Step 3 Graph the linear regression lines. Use the Intersect feature. The x-value of the point of intersection is about 87, which represents the year 2037. The data suggest that the populations of the New York City and Los Angeles metropolitan regions will each be about 25.6 million in 2037.

Problem 4 Classifying a System Without Graphing

What should you compare to classify the system? Compare the slopes and y-intercepts of each line.

Without graphing, is the system independent, dependent, or inconsistent? 4y − 2x = 6 b 8y = 4x − 12 Rewrite each equation in slope-intercept form. Compare slopes and y-intercepts.

4y - 2x = 6

8y = 4x - 12

3 y = 12x + 2

3

y = 12x - 2 3

3

NLINE

HO

ME

RK

O

     m = 12 ; y-intercept is 2 m = 12 ; y-intercept is - 2 The slopes are equal and the y-intercepts are different. The lines are different but parallel. The system is inconsistent.

WO

PRACTICE and APPLICATION EXERCISES

For additional support when completing your homework, go to PearsonTEXAS.com.

Scan page for a Virtual Nerd™ tutorial video.

Use Multiple Representations to Communicate Mathematical Ideas (1)(D) Solve each system by graphing or using a table. Check your answers. y=x-2 y = -x + 3 1. b 2. b 3 y = -2x + 7 y = 2x - 2 2x + 4y = 12 3. b x+ y= 2

4. b

x = -3 y=5

2x - 2y = 4 5. b y-x=6

6. b

3x + y = 5 x- y=7

7. You and a friend are both reading a book. You read 2 pages each minute and have already read 55 pages. Your friend reads 3 pages each minute and has already read 35 pages. Graph and solve a system of equations to find when the two of you will have read the same number of pages. Since the number of pages you have read depends on how long you have been reading, let x represent the number of minutes it takes to read y pages.

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Write and solve a system of equations for each situation. Check your answers. 8. A store sells small notebooks for $8 and large notebooks for $10. If you buy 6 notebooks and spend $56, how many of each size notebook did you buy? 9. A shop has bags of peanuts for sale. If you buy 5 bags and spend $17, how many of each size bag did you buy?

10. Apply Mathematics (1)(A)  You can choose between two tennis courts at two university campuses to learn how to play tennis. One campus charges $25 per hour. The other campus charges $20 per hour plus a one-time registration fee of $10. a. Write a system of equations to represent the cost c for h hours of court use at each campus. b. Find the number of hours for which the costs are the same. c. If you want to practice for a total of 10 hours, which university campus should you choose? Explain. Without graphing, classify each system as independent, dependent, or inconsistent.

74

11. b

7x - y = 6 -7x + y = -6

12. b

-3x + y = 4 x - 13y = 1

13. b

4x + 8y = 12 x + 2y = -3

14. b

y = 2x - 1 y = -2x + 5

15. b

x=6 y = -2

16. b

2y = 5x + 6 -10x + 4y = 8

Lesson 3-1  Solving Systems Using Tables and Graphs

Select Tools to Solve Problems (1)(C)  Find linear models for each set of data. In what year will the two quantities be equal? U.S. Life Expectancy at Birth (1970–2000)

17. Year Men (years)

1970

1975

1980

1985

1990

1995

2000

67.1

68.8

70.0

71.1

71.8

72.5

74.3

Women (years)

74.7

76.6

77.4

78.2

78.8

78.9

79.7

SOURCE: U.S. Census Bureau

Annual U.S. Consumption of Vegetables

18. Year Broccoli (lb/person) Cucumbers (lb/person)

1980

1985

1990

1995

1998

1999

2000

1.5

2.6

3.4

4.3

5.1

6.5

6.1

3.9

4.4

4.7

5.6

6.5

6.8

6.4

SOURCE: U.S. Census Bureau

19. Use Representations to Communicate Mathematical Ideas (1)(E)  Your friend used a graphing calculator to solve a system of linear equations, shown below. After using the TABLE feature, your friend says that the system has no solution. Explain what your friend did wrong. What is the solution of the system? 2x + y = 6 y = 6 − 2x

3x + 2y = 8 8 − 3x y= 2

X 4 3 2 1 0 1 2

X=2

Y1 14 12 10 8 6 4 2

Y2 10 8.5 7 5.5 4 2.5 1

20. Explain Mathematical Ideas (1)(G)  Is it possible for an inconsistent linear system to contain two lines with the same y-intercept? Explain. 21. Connect Mathematical Ideas (1)(F)  Summarize the possible relationships for the y-intercepts, slopes, and number of solutions in a system of two linear equations in two variables. Write a second equation for each system so that the system will have the indicated number of solutions. 22. infinite number of solutions x y b 4+3=1

__________ ?

23. no solutions 5x + 2y = 10 b ______?_______

24. Write a system of linear equations with the solution set 5(x, y) 0 y = 5x + 26.

25. Analyze Mathematical Relationships (1)(F)  What relationship exists between the equations in a dependent system?

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26. Apply Mathematics (1)(A)  Research shows that in a certain market only 2000 widgets can be sold at $8 each, but if the price is reduced to $3, then 10,000 can be sold. a. Let p represent price and n represent the number of widgets. Identify the independent and dependent variables. b. Write a linear equation that relates price and the quantity demanded. This type of equation is called a demand equation. c. A shop can make 2000 widgets for $5 each and 20,000 widgets for $2 each. Use this information to write a linear equation that relates price and the quantity supplied. This type of equation is called a supply equation. d. Find the equilibrium point where supply is equal to demand. Explain the meaning of the coordinates of this point within the context of the exercise.

TEXAS Test Practice

27. Which graph shows the solution of the following system?  b y

A.

x 2

O

x

O

2

2

2

y

B.

y

C.

2

4x + y = 1 x + 4y = -11

D.

2

y 2

x 2

O

2

O

1

x

28. Which is the equation of a line that is perpendicular to the line in the graph? F. y = -3x + 2 G. y = 13x + 5

H. y = - 13x - 4 J. y = 3x - 1

y x 2

O 2

29. Amy ordered prints of a total of 6 photographs in two different sizes, 5 * 7 and 4 * 6, from an online site. She paid $7.50 for her order. The cost of a 5 * 7 print is $1.75 and the cost of a 4 * 6 print is $.25. Explain how to solve a system of equations using tables to find the number of 4 * 6 prints Amy ordered.

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Lesson 3-1  Solving Systems Using Tables and Graphs

2

3-2  Solving Systems Algebraically TEKS FOCUS

VOCABULARY

• Equivalent systems – Equivalent

Foundational to TEKS (3)(A)  Formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the second quadratic.

systems are systems that have the same solution(s).

• Justify – explain with logical reasoning.

TEKS (1)(G)  Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

You can justify a mathematical argument.

• Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim

Additional TEKS (1)(A)

ESSENTIAL UNDERSTANDING You can solve a system of equations by writing equivalent systems until the value of one variable is clear. Then substitute to find the value(s) of the other variable(s).

Problem 1 Solving by Substitution What is the solution of the system of equations? e

Which variable should you solve for first? In the second equation, the coefficient of y is 1. It is the easiest variable to isolate.

3x + 4y = 12 2x + y = 10

Step 1

Step 2

Solve one equation for one of the variables.

Substitute the expression for y in the other equation. Solve for x.

2x + y = 10 y = -2x + 10

3x + 4y = 12 3x + 4(-2x + 10) = 12 3x - 8x + 40 = 12 x = 5.6

Step 3 Substitute the value for x into one of the original equations. Solve for y. 2x + y = 10 2(5.6) + y = 10 11.2 + y = 10 y = -1.2 The solution is (5.6, -1.2).

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Problem 2

TEKS Process Standard (1)(A)

Using Substitution to Solve a Problem Music  A music store offers piano lessons at a discount for customers buying new pianos. The costs for lessons and a one-time fee for materials (including music books, CDs, software, etc.) are shown in the advertisement. What is the cost of each lesson and the one-time fee for materials?

# cost of one lesson 12 # cost of one lesson Relate

 6

Define Let  c

+ one-time fee = $300 + one-time fee = $480

= the cost of one lesson.

f = the one-time fee. Let  Write

6 •  12

# #

c

+

f

= 300

c

+

f

= 480

6c + f = 300

Choose one equation. Solve for f in terms of c.

f = 300 - 6c

Which equation should you use to find f ? Use the equation with numbers that are easier to work with.



Substitute the expression for f into the other 12c + (300 - 6c) = 480  equation, 12c + f = 480. Solve for c. c = 30 Substitute the value of c into one of the 6(30) + f = 300 equations. Solve for f. f = 120

Check Substitute c = 30 and f = 120 in the original equations. 6c + f = 300    12c + f = 480 6(30) + 120 ≟ 300 12(30) + 120 ≟ 480 180 + 120 ≟ 300    360 + 120 ≟ 480 300 = 300 ✔      480 = 480 ✔ The cost of each lesson is $30. The one-time fee for materials is $120.

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Lesson 3-2  Solving Systems Algebraically

Problem 3 Solving by Elimination

How can you use the Addition Property of Equality? Since - 4x + 3y is equal to 16, you can add the same value to each side of 4x + 2y = 9.

What is the solution of the system of equations? e 4x + 2y = 9

4x + 2y = 9 −4x + 3y = 16

One equation has 4x and the other has 4x. Add to eliminate the variable x.

-4x + 3y = 16 5y = 25 y = 5

Solve for y.

4x + 2y = 9

Choose one of the original equations.

4x + 2(5) = 9

Substitute for y.

4x = -1

Solve for x.

x = - 14

(

)

The solution is - 14, 5 .

Problem 4 Solving an Equivalent System What is the solution of the system of equations?

By multiplying ① by 3 and ② by -2, the x-terms become opposites, and you can eliminate them. Add ③ and ④. Solve for y.

Now that you know the value of y, use either equation to find x.

① 2x + 7y = 4 e ② 3x + 5y = −5

① 2x + 7y = 4   ② 3x + 5y = −5

③     6x + 21y = 12 ④  −6x − 10y = 10



 11y = 22



y=2

① 2x + 7(2) = 4 2x + 14 = 4 2x = −10 x = −5 The solution is (−5, 2).

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Problem 5

TEKS Process Standard (1)(G)

Solving Systems Without Unique Solutions What are the solutions of the following systems? Explain.

NLINE

HO

ME

RK

O

How are the two equations in this system related? Multiplying both sides of the first equation by - 1 results in the second equation.

WO

-3x + y = -5 4x - 6y = 6 B e 3x y = 5 -4x + 6y = 10     0 = 0     0 = 16 A e

Elimination gives an equation that is always true. The two equations in the system represent the same line. This is a dependent system with infinitely many solutions.

Elimination gives an equation that is always false. The two equations in the system represent parallel lines. This is an inconsistent system. It has no solutions.

PRACTICE and APPLICATION EXERCISES

Scan page for a Virtual Nerd™ tutorial video.

Solve each system by substitution. Check your answers. For additional support when completing your homework, go to PearsonTEXAS.com.

4x + 2y = 7 1. e y = 5x

4p + 2q = 8 4. e q = 2p + 1

2. e

5. e

3c + 2d = 2 d=4



x + 3y = 7 2x - 4y = 24

3. e 6. e

x + 12y = 68 x = 8y - 12 x + 6y = 2 5x + 4y = 36

7. Apply Mathematics (1)(A)  A student has some $1 bills and $5 bills in his wallet. He has a total of 15 bills that are worth $47. How many of each type of bill does he have? 8. A student took 60 minutes to answer a combination of 20 multiple-choice and extended-response questions. She took 2 minutes to answer each multiple-choice question and 6 minutes to answer each extended-response question. a. Write a system of equations to model the relationship between the number of multiple-choice questions m and the number of extended-response questions r. b. How many of each type of question were on the test? 9. Apply Mathematics (1)(A)  A youth group with 26 members is going skiing. Each of the five chaperones will drive a van or sedan. The vans can seat seven people, and the sedans can seat five people. Assuming there are no empty seats, how many of each type of vehicle could transport all 31 people to the ski area in one trip? Solve each system by elimination. 10. e 13. e 80

x + y = 12 x-y = 2 4x + 2y = 4 6x + 2y = 8

11. e 14. e

Lesson 3-2  Solving Systems Algebraically

x + 2y = 10 x + y = 6 2w + 5y = -24 3w - 5y = 14

12. e 15. e

3a + 4b = 9 -3a - 2b = -3 3u + 3v = 15 -2u + 3v = -5

For each system, choose the method of solving that seems easier to use. Explain why you made each choice. Solve each system. 16. e

3x - y = 5 y = 4x + 2

17. e

2x - 3y = 4 2x - 5y = -6

Solve each system by elimination.

19. e 22. e 25. e

4x - 6y = -26 -2x + 3y = 13 2x - 3y = 6 6x - 9y = 9 2x - 3y = -1 3x + 4y = 8

20. e 23. e 26. e

9a - 3d = 3 -3a + d = -1 20x + 5y = 120 10x + 7.5y = 80 3x + 2y = 10 6x + 4y = 15

18. e 21. e 24. e 27. e

6x - 3y = 3 5x - 5y = 10

2a + 3b = 12 5a - b = 13 6x - 2y = 11 -9x + 3y = 16 3m + 4n = -13 5m + 6n = -19

28. Apply Mathematics (1)(A)  Suppose you have a part-time job delivering packages. Your employer pays you a flat rate of $9.50 per hour. You discover that a competitor is hiring (see ad below). How many deliveries would the competitor’s employees have to make in four hours to earn the same pay you earn in a four-hour shift?

12:30 PM

Solve each system. 29. e

5x + y = 0 5x + 2y = 30

30. e

2m = -4n - 4 3m + 5n = -3

2m + 4n = 10 32. e 3m + 5n = 11

-6 = 3x - 6y 33. e 4x = 4 + 5y

0.02a - 1.5b = 4 35. e 0.5b - 0.02a = 1.8

4y = 2x 36. e x 2x + y = 2 + 1

31. e

7x + 2y = -8 8y = 4x

4y x + 3 = 300 3 34. e

3x - 4y = 300

1 x + 23 y = 1 2 37. • 3 1 4x - 3y = 2

38. Select Techniques to Solve Problems (1)(C)  Explain how you decide whether to use substitution or elimination to solve a system. 39. The equation 3x - 4y = 2 and which equation below form a system with no solutions? A. 2y = 1.5x - 2

B. 2y = 1.5x - 1

C. 3x + 4y = 2

D. 4y - 3x = -2

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 81

40. Identify and correct the error shown in finding the solution 3x - 4y = 14 using substitution. of e x + y = -7

x + y = –7 y = –7 – x

41. Apply Mathematics (1)(A)  Jenny’s Bakery sells carrot muffins at $2 each. The electricity to run the oven is $120 per day and the cost of making one carrot muffin is $1.40. How many muffins need to be sold each day to break even? 42. Write a system of equations in which both equations must be multiplied by a number other than 1 or -1 before using elimination. Solve the system. Apply Mathematics (1)(A) Write and solve a system of equations to find a solution for each situation. STEM

3x – 4y = 14 3x – 4(–7 – x) = 14 3x – 28 – 4x = 14 –x – 28 = 14 x = –42 y = –7 – (–42) y = 35

43. A scientist wants to make 6 milliliters of a 30% sulfuric acid solution. The solution is to be made from a combination of a 20% sulfuric acid solution and a 50% sulfuric acid solution. How many milliliters of each solution must be combined to make the 30% solution? 44. In the final round of a singing competition, the audience voted for one of the two finalists, Luke or Sean. Luke received 25% more votes than Sean received. Altogether, the two finalists received 5175 votes. How many votes did Luke receive?

STEM

45. The equation F = 95 C + 32 relates temperatures on the Celsius and Fahrenheit scales. Does any temperature have the same number reading on both scales? If so, what is the number? Find the value of a that makes each system a dependent system. 46. e

y = 3x + a 3x - y = 2

47. e

3y = 2x 6y - a - 4x = 0

x

48. e

y=2+4 2y - x = a

TEXAS Test Practice 49. What is the slope of the line at the right? x+ y = 7 50. What is the x-value of the solution of e ? 3x - 2y = 11

51. Solve 9(x + 7) - 6(x - 3) = 99. What is the value of  x? 52. Georgia has only dimes and quarters in her bag. She has a total of 18 coins that are worth $3. How many more dimes than quarters does she have? 53. The graph of g (x) is a horizontal translation of f (x) = 2| x + 1| + 3, 5 units to the right. What is the x-value of the vertex of g (x)?

82

Lesson 3-2  Solving Systems Algebraically

2

y x

2

O 2 4

4

6

3-3  Systems of Inequalities TEKS FOCUS

VOCABULARY

TEKS (3)(F)  Solve systems of two or more linear inequalities in two variables.

• System of inequalities – A system of inequalities is a set

TEKS (1)(E)  Create and use representations to organize, record, and communicate mathematical ideas.

• Representation – a way to display or describe

of two or more inequalities using the same variables.

Additional TEKS (1)(A), (3)(E), (3)(G)

information. You can use a representation to present mathematical ideas and data.

ESSENTIAL UNDERSTANDING You can solve a system of inequalities in more than one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system.

Problem 1

TEKS Process Standard (1)(E)

Solving a System by Using a Table

Which inequality should you use to build a table? The first inequality has an infinite number of whole number solutions. The second one has a finite number of solutions. Use the second inequality.

Assume that g and m are whole numbers. What is the solution of the system of g+ m#6 inequalities? e 5g + 2m " 20

Make a table of values for g and m that satisfy the second inequality. The values for g and m must be whole numbers. If g = 0, then 5(0) + 2m … 20, and m … 10. If m = 0, then 5g + 2(0) … 20, and g … 4.

In the table, highlight each pair of values that satisfies the first inequality. The highlighted pairs are the solutions of both inequalities.

g

m

0

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

1

0, 1, 2, 3, 4, 5, 6, 7

2

0, 1, 2, 3, 4, 5

3

0, 1, 2

4

0

g

m

0

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

1

0, 1, 2, 3, 4, 5, 6, 7

2

0, 1, 2, 3, 4, 5

3

0, 1, 2

4

0

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Problem 2 Solving a System by Graphing

How can you be sure to shade the correct half-planes? If the inequality is in slope-intercept form, shade above the boundary if y 7 or y Ú and shade below the boundary if y 6 or y … . If not, use a test point.

2x ∙ y # ∙3 y # ∙ 12 x ∙ 1 Graph each inequality. Rewrite 2x - y Ú -3 in slope-intercept form as y … 2x + 3. The overlap is the solution of the system. What is the solution of the system of inequalities? b



y   12 x  1

y  2x  3 4

4

y

4

2x  y  3 y   12 x  1 y

2 x 4

O

2

4

x 4

2

O

2

4

x 4

O

2

2

2

4

4

4

2

4

Check Pick a point in the overlap region, such as (0, 2), and check it in both inequalities of the system. 2x - y Ú -3 2(0) - 2 Ú -3

   y Ú 12 x + 1    2 Ú 12 (0) + 1

-2 Ú -3 ✔   2 Ú 1 ✔

Problem 3

TEKS Process Standard (1)(A)

Using a System of Inequalities Fundraising ​Your city’s cultural center is sponsoring a concert to raise at least $30,000 for the city’s Youth Services. Tickets are $20 for balcony seats and $30 for orchestra seats. If the center has 500 orchestra seats, how many of each type of seat must they sell? Must raise at least $30,000. There are at most 500 orchestra seats.

The possible sales of balcony and orchestra seats

• Model the problem with a system of inequalities. • Graph the inequalities on your calculator.

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84

Lesson 3-3  Systems of Inequalities

Problem 3

continued

Relate  20

#

balcony seats + 30

#

orchestra seats Ú 30,000 orchestra seats … 500



Define Let x = the number of balcony seats sold. Let y = the number of orchestra seats sold. Write 

20

#

x + 30

#

y Ú 30,000 y … 500



Rewrite 20x + 30y Ú 30,000 in slope intercept form as y Ú - 23 x + 1000. The system of inequalities is e What do points in the overlap represent? The points represent combinations of balcony and orchestra seats that have a total value of at least $30,000.

y Ú - 23x + 1000 . y … 500

Use your graphing calculator to graph the inequalities. The solution is the overlap. Test a point. If the cultural center sells 900 balcony and 450 orchestra tickets, will the Youth Services meet its goal? ?

20(900) + 30(450) Ú 30,000 ? 18,000 + 13,500 Ú 30,000 31,500 Ú 30,000 ✔

?

450 … 500 450 … 500 ✔

NLINE

HO

ME

RK

O

Because the number of seats must be a whole number, only the points in the overlap that represent whole numbers are solutions.

WO

PRACTICE and APPLICATION EXERCISES

Scan page for a Virtual Nerd™ tutorial video.

Find all whole number solutions of each system using a table. For additional support when completing your homework, go to PearsonTEXAS.com.

y + 3x … 8 1. b y - 3 7 2x

2. b

x+y68 3x … y + 6

3. b

yÚ x+2 3y 6 -6x + 6

6. b

y…3 y … 12 x + 1

Solve each system of inequalities by graphing. y … 2x + 2 4. b y 6 -x + 1

5. b

y 7 -2 x61

y … 3x + 1 7. b -6x + 2y 7 5

8. b

x + 2y … 10 x+y… 3

11. b

cÚd-3 c 6 12 d + 3

10. b

y 7 -2x 2x - y Ú 2

9. e 12. b

-x - y … 2 y - 2x 7 1 2x + y 6 1 y 7 -2x + 3 PearsonTEXAS.com

 85

13. You want to decorate a party hall with a total of at least 40 red and yellow balloons, with a minimum of 25 yellow balloons. Write and graph a system of inequalities to model the situation. 14. A gardener wants to plant at least 50 tulips and rose plants in a garden, but no more than 20 rose plants. Write and graph a system of inequalities to model the situation. 15. Apply Mathematics (1)(A) ​The food pyramid suggests that you eat 4–6 servings of fruits and vegetables a day for a healthy diet. It also says that the number of servings of vegetables should be greater than the number of servings of fruits. Find the number of servings of fruits and vegetables that could make a healthy diet. Use whole numbers only. 16. Apply Mathematics (1)(A) ​An entrance exam has two sections, a verbal section and a mathematics section. You can score a maximum of 1600 points. For admission, the school of your choice requires a math score of at least 600. Write a system of inequalities to model scores that meet the school’s requirements. Then solve the system by graphing. 17. Create Representations to Communicate Mathematical Ideas (1)(E) Write and graph a system of inequalities for which the solution is bounded by a dashed vertical line and a solid horizontal line. 18. Explain Mathematical Ideas (1)(G)  Explain how you determine where to shade when solving a system of inequalities. 19. Given a system of two linear inequalities, explain how you can pick test points in the plane to determine where to shade the solution set. In Exercises 20–28, identify the inequalities A, B, and C for which the given ordered pair is a solution. 3

A. x + y … 2

2

C. y 7 - 13 x - 2

B. y … 2 x - 1 y

2



y

x O

2

2

y

x 2

O

2

2

2

86

3

20. (0, 0)

21. ( -2, -5)

22. (0, -2)

23. ( -15, 15)

24. (3, 2)

25. (2, 0)

26. ( -6, 0)

27. (4, -1)

28. ( -8, -11)

Lesson 3-3  Systems of Inequalities

O

x

Solve each system of inequalities by graphing. x+y68 29. c x Ú 0 yÚ0

2y - 4x … 0 30. c x Ú 0 yÚ0

y Ú -2x + 4 31. c x Ú -3 yÚ1

Create Representations to Communicate Mathematical Ideas (1)(E)  Write a system of inequalities to describe each shaded figure. 32.

y (0, 2)

(2, 0)

33.

(2, 3)

y

(2, 3)

(2, 0) x (0, 2)

34.

x (3, 0)

(3, 0)

y

(2, 4)

(6, 4)

x (0, 0)

(4, 0)

TEXAS Test Practice 35. Which system of inequalities is shown in the graph? A. b

xÚ4 3x - 2y 7 5

C. b

x64 3x - 2y Ú 5

B. b

x74 3x - 2y … 5

D. b

x…4 3x - 2y 6 5

4 2 4 2

36. What is the equation of the line that passes through the point (4, -3) and has slope 12 ? F. y = 12x + 3

y

O

2

x

2

H. y = 12x - 5

G. y = 12x - 1 J. y = x - 12 37. Which equation is a vertical translation of y = -5x? 5

A. y = - 2x B. y = -5x + 2

C. y = -10x D. y = 5x - 2

38. The cost of renting a pool at an aquatic center is either $30 per hour or $20 per hour with a $40 non-refundable deposit. For how many hours is the cost of renting a pool the same for both plans?

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3-4  Linear Programming TEKS FOCUS

VOCABULARY

TEKS (3)(G)  Determine possible solutions in the solution set of systems of two or more linear inequalities in two variables. TEKS (1)(A)  Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS (3)(E), (3)(F)

• Constraint – restriction

• Objective function – a model

on the variables of the objective function in a linear programming problem

of the quantity that you want to make as large or as small as possible

• Feasible region – contains all the values that satisfy the constraints of the objective function

• Apply – use knowledge or

• Linear programming

– a method for finding a minimum or maximum value of some quantity, given a set of constraints

information for a specific purpose, such as solving a problem

ESSENTIAL UNDERSTANDING Some real-world problems involve multiple linear relationships. Linear programming accounts for all of these linear relationships and gives the solution to the problem.



Key Concept  Vertex Principle of Linear Programming If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region. The constraints in a linear programming situation form a system of inequalities, like the one at the right. The graph of the system is the feasible region. It contains all the points that satisfy all the constraints.

x y d y x

Ú Ú … +

2 3 6 y … 10

The quantity you are trying to maximize or minimize is modeled with an objective function. Often this quantity is cost or profit. Suppose the objective function is C = 2x + y. Graphs of the objective function for various values of C are parallel lines. Lines closer to the origin represent smaller values of C. The graphs of the equations 7 = 2x + y and 17 = 2x + y intersect the feasible region at (2, 3) and (7, 3). These vertices of the feasible region represent the least and the greatest values for the objective function.

Lesson 3-4  Linear Programming

Feasible Region

6 4 2

x O

2

4

8 y

6

8

17  2x  y

6 4 7  2x2  y O

88

8 y

x 2

4

6

8

Problem 1 Testing Vertices

x ∙ 2y " 5 x∙y"2 Constraints d x#0 y#0

Multiple Choice ​What point in the feasible region maximizes P for the objective function P ∙ 2x ∙ y?

What quadrant will the feasible region be in? The constraints x Ú 0 and  y Ú 0 indicate the first quadrant.

(2, 0)

(0, 0)

(3, 1)

(0, 2.5)

Step 1

Step 2

Graph the inequalities.

Form the feasible region.

y

y

4

xy 2 2 x  2y  5 x O 2 2 4

2

The intersections of the boundaries are the vertices of the feasible region.

R

Q

S T

x

4

Step 3

Step 4

Find the coordinates of each vertex. Q (0, 0)

Evaluate P at each vertex. P = 2(0) + 0 = 0

R (0, 2.5)

P = 2(0) + 2.5 = 2.5

S (3, 1)

P = 2(3) + 1 = 7

T (2, 0)

P = 2(2) + 0 = 4

Maximum Value

P has a maximum value of 7 when x = 3 and y = 1. The correct choice is C.

Problem 2

TEKS Process Standard (1)(A)

Using Linear Programming to Maximize Profit Business  You are screen-printing T-shirts and sweatshirts to sell at the Polk County Blues Festival and are working with the following constraints.

•  You have at most 20 hours to make shirts.



• You want to spend no more than $600 on supplies.



• You want to have at least 50 items to sell.

How many T-shirts and how many sweatshirts should you make to maximize your profit? How much is the maximum profit?

T-shirt 1-Coloinr utes to make

m 4 Takes 10 s cost $ Supplie 6 Profit $

irt 3-Color Sweatsh make to Takes 30 minutes Supplies cost $20 Profit $20

continued on next page ▶

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 89

Problem 2

continued

Organize the information in a table. Write the constraints and the objective function.

T-Shirts, x

Sweatshirts, y

Total

10x

30y

1200

Minutes

x y Number 10x + 30y … 1200 4x 20y Cost x + y Ú 50 6x 20y Profit Constraints: e 4x + 20y … 600 xÚ0      Objective Function: P = 6x + 20y yÚ0

600 6x  20y

Step 1

Step 2

Step 3

Graph the constraints to form the feasible region.

Find the coordinates of each vertex.

Evaluate P.

A(50, 0)

P = 6(50) + 20(0) = 300

B(25, 25)

P = 6(25) + 20(25) = 650

C(75, 15)

P = 6(75) + 20(15) = 750

D(120, 0)

P = 6(120) + 20(0) = 720

30 B

24 Sweatshirts

How do you find the coordinates of the vertices if they are hard to read off the graph? Solve the system of equations related to the lines that intersect to form the vertex.

50

18

C

12 6 0

0

A 30 60

90

D 120 150

T-shirts

NLINE

HO

ME

RK

O

You can maximize your profit by selling 75 T-shirts and 15 sweatshirts. The maximum profit is $750.

WO

PRACTICE and APPLICATION EXERCISES

Scan page for a Virtual Nerd™ tutorial video.

Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. For additional support when completing your homework, go to PearsonTEXAS.com.

x+y…8 x + 2y Ú 8 2…x…6 2x + y … 10 1. d 2. • x Ú 2 3. • 1 … y … 5 xÚ0 yÚ0 x+y…8 yÚ0 Maximum for Minimum for Maximum for N = 100x + 40y C = x + 3y P = 3x + 2y 4. Apply Mathematics (1)(A) ​Baking a tray of corn muffins takes 4 cups of milk and 3 cups of wheat flour. Baking a tray of bran muffins takes 2 cups of milk and 3 cups of wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour. He makes $3 profit per tray of corn muffins and $2 profit per tray of bran muffins. How many trays of each type of muffin should the baker make to maximize his profit?

90

Lesson 3-4  Linear Programming

5. Apply Mathematics (1)(A)  A biologist is developing two new strains of bacteria. Each sample of Type I bacteria produces four new viable bacteria, and each sample of Type II produces three new viable bacteria. Altogether, at least 240 new viable bacteria must be produced. At least 30, but not more than 60, of the original samples must be Type I. Not more than 70 of the original samples can be Type II. A sample of Type I costs $5 and a sample of Type II costs $7. How many samples of Type II bacteria should the biologist use to minimize the cost? 6. ​Sometimes two corners of a graph both yield the maximum profit. In this case, many other points may also yield the maximum profit. Evaluate the profit formula P = x + 2y for the graph shown. Find four points that yield the maximum profit.

y D(4, 6) E(0, 6) C(10, 3) x B(10, 0)

O

y

7. Evaluate Reasonableness (1)(B)  Your friend is trying to find the y maximum value of P = -x + 3y subject to the following constraints. 6 y … -2x + 6 cy…x+3 x Ú 0, y Ú 0

6 4

4

2 correct solution? What error did your friend make? What is the 0

2

0

4

2 x

O

6

2

4

6

x

C = 6x + 9y

8. A vertex of a feasible region does not always have whole number coordinates. Sometimes you may need to round coordinates to find the solution. Using the objective function and the constraints at the right, find the whole number values of x and y that minimize C. Then find C for those values of x and y.

x + 2y Ú 50 • 2x + y Ú 60 x Ú 0, y Ú 0

TEXAS Test Practice 9. Which is the graph of y … 0 x - 3 ? A.

4

B.

y

2

4

C.

y

2 2

4

D.

y

x O

2

4

4

y

2

2

x O

4

x

x O

2

4

O

2

4

10. Solve the equation 12(a + b) = c for b. F. b = 12c - a

G. b = 2a - c

H. b = 2c - a

J. b = 2ca

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Technology Lab Linear Programming Use With Lesson 3-4

teks (3)(G), (1)(E)

You can solve linear programming problems using your graphing calculator.

Find the values of x and y that will maximize the objective function P ∙ 13x ∙ 2y for the constraints at the right. What is the value of P at this maximum point?

Step 1 Rewrite the first two inequalities to isolate y. Enter the inequalities. Plot1 Plot2 Y1 = (3/2)X + 4 Y2 = 8X – 48 \Y3 = \Y4 = \Y5 = \Y6 = \Y7 =

∙3x ∙ 2y " 8 • ∙8x ∙ y # ∙48 x # 0, y # 0

Step 2 Use the value option of CALC to find the upper left vertex. Press 0 enter . Y1  (3/2)X  4

Plot3

X0

Step 3 Enter the objective function on the home screen. Press enter for the value of P at the vertex.

Y4

Step 4 U ​ se the intersect option of CALC to find the upper right vertex. Go to the home screen and press enter for the value of P.

13X + 2Y 8

Intersection X=8

Y = 16

Step 5 Use the ZERO option of CALC to find the lower right vertex. Go to the home screen and press enter for the value of P. The objective function has a value of 0 when the vertex is at the origin. The maximum value of P is 136.

13X + 2Y 8 136 78

continued on next page ▶

92 Technology Lab  Linear Programming

Technology Lab

continued

Exercises Graph the following systems of inequalities using a graphing calculator. Identify each vertex of the region. y … -3x + 9 1 1. • y … - 2x + 4 x Ú 0, y Ú 0

3x + 2y … 24 2. • 5x + 6y Ú 60 x Ú 0, y Ú 0

x + 3y Ú 18 3. • 5x + 2y Ú 25 x Ú 0, y Ú 0

5x - y … 40 x + y … 20 4. µ y … 15 x Ú 0, y Ú 0

Find the values of x and y that maximize or minimize the objective function. 4x + 3y Ú 30 5. • x + 3y Ú 21 x Ú 0, y Ú 0

3x + 5y Ú 35 6. • 2x + y … 14 x Ú 0, y Ú 0

Minimum for C = 5x + 8y

Maximum for P = 3x + 2y

x+ yÚ 8 7. • x + 5y Ú 20 x Ú 0, y Ú 2

x + 2y … 24 3x + 2y … 34 8. µ 3x + y … 29 x Ú 0, y Ú 0

Minimum for C = 3x + 4y

Maximum for P = 2x + 3y

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Activity Lab Graphs in Three Dimensions Use With Lesson 3-5

teks (1)(A)

To describe positions in space, you need a three-dimensional coordinate system. You have learned to graph on an xy-coordinate plane using ordered pairs. Adding a third axis, the z-axis, to the xy-coordinate plane creates coordinate space. In coordinate space you graph points using ordered triples of the form (x, y, z).

Points in Space  z-axis (2, 3, 4)

Points in a Plane  y-axis origin 

(2, 3)

ordered pair

3 units up  x-axis O 2 units right

 A two-dimensional coordinate system allows you to graph points in a plane.

ordered triple 

origin  2 units forward

O

4 units up  y-axis

3 units right  A three-dimensional coordinate system allows you to graph points in space.  x-axis

In the coordinate plane, point (2, 3) is two units right and three units up from the origin. In coordinate space, point (2, 3, 4) is two units forward, three units right, and four units up.

1 Define one corner of your classroom as the origin of a three-dimensional coordinate system like the classroom shown. Write the coordinates of each item in your coordinate system. 1. each corner of your classroom 2. each corner of your desk

z

3. one corner of the blackboard 4. the clock 5. the waste-paper basket 6. Pick 3 items in your classroom and write the coordinates of each. x

y

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94

Activity Lab  Graphs in Three Dimensions

Activity Lab

continued

An equation in two variables represents a line in a plane. An equation in three variables represents a plane in space.

2 Given the following equation in three variables, draw the plane in a coordinate space. x + 2y − z = 6 7. Let x = 0. graph the resulting equation in the yz-plane. 8. Let y = 0. graph the resulting equation in the xz-plane. From geometry you know that two non-skew lines determine a plane. 9. Sketch the plane x + 2y - z = 6. (If you need help, find a third line by letting z = 0 and then graph the resulting equation in the xy-plane.)

3 Two equations in three variables represent two planes in space. 10. Draw the two planes determined by the following equations: 2x + 3y - z = 12 2x - 4y + z = 8 11. Describe the intersection of the two planes above.

Exercises Find the coordinates of each point in the diagram. 12. A

13. B

14. C

15. D

16. E

17. F

Sketch the graph of each equation. 18. x - y - 4z = 8

19. x + y + z = 2

20. -3x + 5y + 10z = 15

21. 6x + 6y - 12z = 36

Graph the following pairs of equations in the same coordinate space and describe their intersection, if any. 22. -x + 3y + z = 6 -3x + 5y - 2z = 60

23. -2x - 3y + 5z = 7 2x - 3y - 4z = -4

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3-5  Systems in Three Variables TEKS FOCUS

VOCABULARY

TEKS (3)(B)  Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

• Representation – a way to display or

TEKS (1)(D)  Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

describe information. You can use a representation to present mathematical ideas and data.

Additional TEKS (1)(A), (3)(A)

ESSENTIAL UNDERSTANDING To solve systems of three equations in three variables, you can use some of the same algebraic methods you used to solve systems of two equations in two variables.



Key Concept  Solutions of Systems With Three Variables No solution No point lies in all three planes.

One solution The planes intersect at one point. An infinite number of solutions The planes intersect at all the points along a common line.

Problem 1 Solving a System Using Elimination Which variable do you eliminate first? Eliminate the variable for which the process requires the fewest steps.

What is the solution of the system? Use elimination. The equations are numbered to make the procedure easy to follow.

① 2x ∙ y ∙ z ∙ 4 ② • x ∙ 3y ∙ z ∙ 11 ③ 4x ∙ y ∙ z ∙ 14

Step 1 Pair the equations to eliminate z. Then you will have two equations in x and y. Add. Subtract. ② ① 2x - y + z = 4 x + 3y - z = 11 e e ③  4x + y - z = 14       ② x + 3y - z = 11           ④ 3x + 2y ⑤ -3x + 2y = 15    = -3          

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96

Lesson 3-5  Systems in Three Variables

Problem 1

Does it matter which equation you substitute into to find z? No, you can substitute into any of the original three equations.

continued

Step 2 Write the two new equations as a system. Solve for x and y. Add and solve for y. Substitute y = 3 and solve for x. ④ 3x + 2y = 15 3x + 2y = 15 ④ e   3x + 2(3) = 15 ⑤ -3x + 2y = -3   3x = 9 4y = 12 x = 3 y = 3        Step 3 Solve for z. Substitute the values of x and y into one of the original equations. 2x - y + z = 4 2(3) - 3 + z = 4   6-3+z=4 z=1 ①

Use equation ①. Substitute. Simplify. Solve for z.

Step 4 Write the solution as an ordered triple. The solution is (3, 3, 1).

Problem 2

TEKS Process Standard (1)(D)

Solving an Equivalent System What is the solution of the system? Use elimination. ① x ∙ y ∙ 2z ∙ 3 ② • 2x ∙ y ∙ 3z ∙ 7 ③ ∙x ∙ 2y ∙ z ∙ 10 You are trying to get two equations in x and z. Multiply ① so you can add it to ② and eliminate y. Do the same with ② and ③.

Multiply ④ so you can add it to ⑤ and eliminate x. Substitute z = 3 into ④. Solve for x. Substitute the values for x and z into ① to find y. Check the answer in the three original equations.

① x ∙ y ∙ 2z ∙ 3 e ② 2x ∙ y ∙ 3z ∙ 7



② 2x ∙ y ∙ 3z ∙ 7 e ③ ∙x ∙ 2y ∙ z ∙ 10 ④ x∙ z∙ 4 b ⑤ 3x ∙ 7z ∙ 24

∙x ∙ y ∙ 2z ∙ ∙3 2x ∙ y ∙ 3z ∙ 7 x ∙ z∙ 4 ④





4x ∙ 2y ∙ 6z ∙ 14 ∙x ∙ 2y ∙ z ∙ 10 ⑤ 3x ∙ 7z ∙ 24 ∙3x ∙ 3z ∙ ∙12 3x ∙ 7z ∙ 24 4z ∙ 12 z∙ 3

x∙3∙4 x∙1  x ∙ y ∙ 2z ∙ 3   1 ∙ y ∙ 2(3) ∙ 3 y ∙ ∙4

Check 1 ∙ (∙4) ∙ 2(3) ∙ 3 ✔ 2(1) ∙ (∙4) ∙ 3(3) ∙ 7 ✔ ∙(1) ∙ 2(∙4) ∙ 3 ∙ 10 ✔

The solution is (1, ∙4, 3).

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Problem 3 Solving a System Using Substitution Multiple Choice ​What is the x-value in the solution of the system? ① 2x ∙ 3y ∙ 2z ∙ ∙1 ② • x ∙ 5y ∙ 9 ③ 4z ∙ 5x ∙ 4 1 Which equation should you solve for one of its variables? Look for an equation that has a variable with coefficient 1.

4

6

10

Step 1 Choose equation ②. Solve for x.

② x + 5y = 9 x = 9 - 5y

Step 2 Substitute the expression for x into equations ① and ③ and simplify. ①    2x + 3y - 2z =        2(9 - 5y) + 3y - 2z =           18 - 10y + 3y - 2z =           18 - 7y - 2z = ④         -7y - 2z =

-1 ③     4z - 5x = 4 -1 4z - 5(9 - 5y) = 4 -1    4z - 45 + 25y = 4 -1    4z + 25y = 49 -19 ⑤    25y + 4z = 49

Step 3 Write the two new equations as a system. Solve for y and z. ④ -7y - 2z = -19 e ⑤ 25y + 4z = 49

-14y - 4z = -38 25y + 4z = 49 11y = 11 y=1

④   -7y - 2z = -19 -7(1) - 2z = -19   Substitute the value of y into ④.     -2z = -12      z = 6 Step 4 Use one of the original equations to solve for x. ②   x + 5y = 9   x + 5(1) = 9        Substitute the value of y into ②.      x = 4 The solution of the system is (4, 1, 6), and x = 4. The correct answer is B.

98

Lesson 3-5  Systems in Three Variables

Multiply by 2. Then add.

Problem 4

TEKS Process Standard (1)(A)

Solving a Real-World Problem Business  You manage a clothing store and budget $6000 to restock 200 shirts. You can buy T-shirts for $12 each, polo shirts for $24 each, and rugby shirts for $36 each. If you want to have twice as many rugby shirts as polo shirts, how many of each type of shirt should you buy?

How many unknowns are there? There are three unknowns: the number of each type of shirt.

Relate

T-shirts



rugby shirts

12 #

+ polo shirts =2

T-shirts

#

+ rugby shirts

polo shirts

+ 24 #

+ 36

polo shirts

= 200

#

rugby shirts

= 6000

Define Let  x = the number of T-shirts. y = the number of polo shirts. Let  Let  z = the number of rugby shirts. Write

①



②c z

x + y

③ 12

#

=2 #

+ z y

x + 24

#

= 200 y

+ 36

#

z

= 6000

Step 1 Since 12 is a common factor of all the terms in equation ③, write a simpler equivalent equation.

③ 12x + 24y + 36z = 6000 e ④ x + 2y + 3z = 500

Divide by 12.

Step 2 Substitute 2y for z in equations ① and ④. Simplify to find equations ⑤ and ⑥. ①  x + y + z = 200 ④  x + 2y + 3z = 500   x + y + (2y) = 200  x + 2y + 3(2y) = 500 ⑤   x + 3y = 200 ⑥    x + 8y = 500 Step 3 Write ⑤ and ⑥ as a system. Solve for x and y. ⑤ x + 3y = 200 -x - 3y = -200 e ⑥  x + 8y = 500 x + 8y = 500    5y = 300    y = 60 ⑤  x + 3y = 200   x + 3(60) = 200       x = 20

Multiply by - 1. Then add.

Substitute the value of y into ⑤.

Step 4 Substitute the value of y in ② and solve for z.

② z = 2y z = 2(60) = 120

You should buy 20 T-shirts, 60 polo shirts, and 120 rugby shirts.

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PRACTICE and APPLICATION EXERCISES

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Scan page for a Virtual Nerd™ tutorial video.

Solve each system by elimination. Check your answers. x - y + z = -1 x - y - 2z = 4 -2x + y - z = 2 1. • x + y + 3z = -3 2. • -x + 2y + z = 1 3. • -x - 3y + z = -10 2x - y + 2z = 0 -x + y - 3z = 11 3x + 6z = -24 4. •

a + b + c = -3 6q - r + 2s = 8 x - y + 2z = -7 3b - c = 4 5. • 2q + 3r - s = -9 6. •y+z=1 2a - b - 2c = -5 4q + 2r + 5s = 1 x = 2y + 3z

x + 2y = 2 3x + 2y + 2z = -2 x + 4y - 5z = -7 7. • 2x + 3y - z = -9 8. • 2x + y - z = -2 9. • 3x + 2y + 3z = 7 4x + 2y + 5z = 1 x - 3y + z = 0 2x + y + 5z = 8 STEM

10. Apply Mathematics (1)(A)  In a factory there are three machines, A, B, and C. When all three machines are working, they produce 287 bolts per hour. When only machines A and C are working, they produce 197 bolts per hour. When only machines A and B are working, they produce 202 bolts per hour. How many bolts can each machine produce per hour? 11. In △PQR, the measure of angle Q is three times that of angle P. The measure of angle R is 20° more than that of angle P. Find the measure of each angle.

12. Apply Mathematics (1)(A)  A stadium has 49,000 seats. The number of seats in Section A equals the total number of seats in Sections B and C. Suppose the stadium takes in $1,052,000 from each sold-out event. How many seats does each section hold? A

$25

VISITORS C

B

B

C

$20 $15

$15 $20

HOME A $25

Solve each system by substitution. Check your answers. 13. •

x + 2y + 3z = 6 3a + b + c = 7 5r - 4s - 3t = 3 y + 2z = 0 14. • a + 3b - c = 13 15. •t=s+r z=2 b = 2a - 1 r = 3s + 1

13 = 3x - y x + 3y - z = -4 x - 4y + z = 6 16. • 4y - 3x + 2z = -3 17. • 2x - y + 2z = 13 18. • 2x + 5y - z = 7 z = 2x - 4y 3x - 2y - z = -9 2x - y - z = 1 100

Lesson 3-5  Systems in Three Variables

Solve each system using any method. x - 3y + 2z = 11 x + 2y + z = 4 4x - y + 2z = -6 19. • -x + 4y + 3z = 5 20. • 2x - y + 4z = -8 21. • -2x + 3y - z = 8 2x - 2y - 4z = 2 -3x + y - 2z = -1 2y + 3z = -5 4x - 2y + 5z = 6 2/ + 2w + h = 72 6x + y - 4z = -8 y z 22. • 3x + 3y + 8z = 4 23. • / = 3w 24. • 0 4 - 6 = x - 5y - 3z = 5 h = 2w 2x - z = -2 25. Apply Mathematics (1)(A)  A worker received a $10,000 bonus and decided to split it among three different accounts. He placed part in a savings account paying 4.5% per year, twice as much in government bonds paying 5%, and the rest in a mutual fund that returned 4%. His income from these investments after one year was $455. How much did the worker place in each account? 26. Connect Mathematical Ideas (1)(F) ​Write your own system with three variables. Begin by choosing the solution. Then write three equations that are true for your solution. Use elimination to solve the system. 27. Refer to the regular five-pointed star at the right. Write and solve a system of three equations to find the measure of each labeled angle.

x y y z

28. In the regular polyhedron described below, all faces are congruent polygons. Use a system of three linear equations to find the numbers of vertices, edges, and faces. Every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

TEXAS Test Practice y = - 2x + 10 29. What is the value of z in the solution of the system? • - x + y - 2z = - 2 3x - 2y + 4z = 7

30. What is the x-intercept of the line at the right after it is translated up 3 units?

31. Suppose y varies directly with x, and y = 15 when x = 10. What is y when x = 22?

x

y 2

O

2

2

32. A theater has 490 seats. Seats sell for $25 on the floor, $20 in the mezzanine, and $15 in the balcony. The number of seats on the floor equals the total number of seats in the mezzanine and balcony. Suppose the theater takes in $10,520 from each sold-out event. How many seats does the mezzanine section hold?

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3-6  Solving Systems Using Matrices TEKS FOCUS

VOCABULARY

• Gaussian elimination – the

TEKS (3)(B)  Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution.

• Row operation – an operation

process of using row operations on a matrix that represents a system of equations in order to solve the system

TEKS (1)(E)  Create and use representations to organize, record, and communicate mathematical ideas.

• Matrix – a rectangular array of

Additional TEKS (1)(D)

• Matrix element – Every item

on one or more rows of a matrix

• Representation – a way to display or describe information. You can use a representation to present mathematical ideas and data.

numbers written within brackets listed in a matrix is an element of the matrix.

ESSENTIAL UNDERSTANDING You can use a matrix to represent and solve a system of equations without writing the variables.



Key Concept  Row Operations 2 1 3

Switch any two rows.

Multiply a row by a constant.

3

2 5

3

2 5

2 1 3 3

Add one row to another.

2 5

4 2 6

becomes

becomes

becomes

3

2 5

2 1 3 3

4

Identifying a Matrix Element

102

What is element a23 in matrix A?

-9 5 -2

17 8 10

a23 is 8.

Lesson 3-6  Solving Systems Using Matrices

1 6§ 0



34 22 56

Problem 1

4 A= £ 0 -3

5

2 · 2 1 · 2 3 · 2

Combine any of these steps.

Does the order of the subscript in a23 matter? Yes. a23 and a32 are different elements.

2

A23 is in Row 2 and Column 3.

2

6

3

2 5

4 2 6



7

0 11

4 2 6

Problem 2

TEKS Process Standard (1)(E)

Representing Systems With Matrices Why is the order of elements important in a matrix? Different orders of elements could correspond to different systems of equations.

How can you represent the system of equations with a matrix? A b



2x + y = 9 x − 6y = −1

The matrix c

2 1 9 ` d represents the system above. 1 -6 -1

x − 3y + z = 6 B • x + 3z = 12 y = −5x + 1

Step 1 Write each equation in the same variable order. Line up the variables. Leave space where a coefficient is 0. x - 3y + z = 6 • x + 3z = 12 5x + y = 1



Step 2 Write the matrix using the coefficients and constants. Notice the 1’s and 0’s. 1 £1 5

-3 0 1

1 6 3 † 12 § 0 1

Problem 3

TEKS Process Standard (1)(D)

Writing a System From a Matrix What linear system of equations does this matrix represent? c Each row shows coefficient-coefficientconstant of one equation. Simplify. Write the system.

5 2 7 ` d 0 1 9

5x + 2y = 7 0x + 1y = 9 e

5x + 2y = 7 y=9

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Problem 4 Solving a System in Three Variables Using a Matrix x − 2y − 2z = 6 What is the solution of the system? • 2x − 3y − 5z = 7 3x + 4y + z = 2 Step 1 Write a matrix for the system. Solve using Gaussian elimination. How is solving a system using row operations similar to using elimination? You use the same steps but the variables do not appear in the matrices.

1 £2 3

-2 -3 4

-2 6 -5 † 7 § 1 2

Step 2 Multiply Row 1 by -2 and add Row 2. Replace Row 2 with the sum and rewrite the matrix. -2(1 + 2 0

-2 -3

-2 -5

- 6) - 7)

1

-1

-5)

1 £0 3

-2 1 4

-2 6 -1 † -5 § 1 2

Step 3 Multiply Row 1 by -3 and add Row 3. Replace Row 3 with the sum. -3(1 + 3 0

-2 4

-2 1

- 6) - 2)

10

7

-16)

1 £0 0

-2 1 10

-2 6 -1 † -5 § 7 -16

Note that Steps 2 and 3 are the equivalent of eliminating x from the second and third equations. 1 Step 4 Multiply Row 2 by -10 and add Row 3. Multiply the sum by 17 . Replace Row 3 with the product. -10(0 1 -1 -5) 1 -2 -2 6 + (0 10 7 -16) £0 1 -1 † -5 §

0

0

17

34

1 17( 0

0

17

34)

0

0

1

2

Note that Step 4 is equivalent to eliminating y from the third equation. Now the third equation is equivalent to z = 2. Step 5 Add Row 2 and Row 3. Replace Row 2 with the sum. Multiply Row 2 by 2 and multiply Row 3 by 2. Add both to Row 1. Replace Row 1 with the sum.



+

(0 -1 -1 -5) (0 -0 1 2) 0 -1 0 -3

(1 -2 -2 6) 2(0 1 0 -3) + 2(0 0 1 2) 1 0 0 4

1 £0 0

-2 1 0

1 £0 0

0 1 0

-2 6 0 † -3 § 1 2 0 4 0 † -3 § 1 2

 he solution to the system is (4, -3, 2). You can check your work by substituting these T values into the original equations.

104

Lesson 3-6  Solving Systems Using Matrices

Problem 5 Using a Calculator to Solve a Linear System

How do you enter missing variables into a matrix? If a variable is not present in an equation, enter its coefficient as 0 in the matrix.

2a + 3b − c = 1 What is the solution of the system of equations? • −4a + 9b + 2c = 8 −2a + 2c = 3 Step 1 Enter the system into a calculator as a matrix.

Step 2 Apply the rref( ) function to the matrix. Put the matrix elements in fraction form if some are not integers.

Step 3  List the solution. The solution of the system is a = 12, b = 23, c = 2. Check 2a + 3b - c = 1

() ()

2 12 + 3 23 - 2 ≟ 1 1 + 2 - 2≟1 1=1 ✔

-4a + 9b + 2c = 8

() ()

-4 12 + 9 23 + 2(2) ≟ 8 -2 + 6 + 4 ≟ 8 8=8 ✔

-2a + 2c = 3

()

-2 12 + 2(2) ≟ 3 -1 + 4 ≟ 3 3=3 ✔

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PRACTICE and APPLICATION EXERCISES

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Scan page for a Virtual Nerd™ tutorial video.

3 12 6 Identify the indicated element. A = £ 1 0 9 § 8 7 4

1. a32 2. a21 3. a13 4. a31 Write a matrix to represent each system. r - s + t = 150 y = 3x - 7 5. • 2r + t = 425 6. e x=2 s + 3t = 0

x- y+ z=0 7. • x - 2y - z = 5 2x - y + 2z = 8

Write the system of equations represented by each matrix. 1 0 4 8. c ` d 0 1 -6

2 1 1 1 11. £ 1 1 1 † 2§ 1 -1 1 -2

9. c

5 -2

1 -3 ` d 2 4

0 1 2 4 12. £ -2 3 6 † 9 § 1 0 1 3

10. c

-1 1

2 -6 ` d 1 7

5 2 1 5 13. £ 4 1 2 † 8§ 1 3 -6 2

Solve each system of equations by using Gaussian elimination. x + 3y + 5z = - 5 3x - 5y + z = -20 2x + 4y - z = -33 14. • 2x - 2y - z = -14 15. • 7x - y - 3z = 4 16. • 3x - 2y = 22 x + 5y + 4z = 11 10x + 6y - 2z = 32 x + 3z = 17 17. Apply Mathematics (1)(A)  A manufacturer sells pencils and erasers in packages. Write a system of equations to represent this situation. Then write a matrix to represent the system.

$.41

18. Apply Mathematics (1)(A)  Last year your town invested a total of $25,000 into two separate funds. The return on one fund was 4% and the return on the other was 6%. If the town earned a total of $1300 in interest, how much money was invested in each fund? Solve each system. 19. c

106

x+ y + z=2 2y - 2z = 2 x - 3z = 1

x-y + z= 3 20. c x + 3z = 6 y - 2z = -1

Lesson 3-6  Solving Systems Using Matrices

x + y + z = -1 21. c 3x + 4y - z = 8 6x + 8y - 2z = 16

-2w x - y + 3z = 9 2x + 3y + z = 13 -w 22. • x µ + 2z = 3 23. • 5x - 2y - 4z = 7 24. -2w 2x + 2y + z = 10 4x + 5y + 3z = 25 w

+ + + +

+ + +

x 2x 3x x

y = y + z = 3y + 2z = 2y + z =

0 1 6 5

25. Apply Mathematics (1)(A)  Suppose you want to fill nine 1-lb tins with a snack mix. You have $15 and plan to buy almonds for $2.45 per lb, hazelnuts for $1.85 per lb, and raisins for $.80 per lb. You want the mix to contain twice as much of the nuts as the raisins by weight. a. ​Explain how each equation to the right relates to the problem. What does each variable represent?

x+y+z=9 c 2.45x + 1.85y + 0.8z = 15 x + y = 2z

b. Solve the system. c. How many of each ingredient should you buy?

26. Connect Mathematical Ideas (1)(F) ​The coordinates (x, y) of a point in a plane are 2x + 3y = 13 the solution of the system e . Find the coordinates of the point. 5x + 7y = 31 27. Evaluate Reasonableness (1)(B)  A classmate writes the matrix at the right to represent a system and says that the solution is x = 2, y = 0. Explain your classmate’s error and describe how to correct it.

1 0 2 0 0 0

28. Apply Mathematics (1)(A)  A hardware store mixes paints in a ratio of two parts red to six parts yellow to make two gallons of pumpkin orange. A ratio of five parts red to three parts yellow makes two gallons of pepper red. A gallon of pumpkin orange sells for $25, and a gallon of pepper red sells for $28. Find the cost of 1 quart of red paint and the cost of 1 quart of yellow paint.

TEXAS Test Practice 3

29. Which equation represents a line with a slope of 12 and a y-intercept of 4 ? 3

3

A. y = 12 x - 4

3

B. y = 4 x - 12

3

C. y = 12 x + 4

D. y = 4 x + 12

30. Which graph best represents the solution of the inequality y … 2 0 x - 1 0 - 4? F.

G.

y

O

H.

y

x

O

2

2

x

y

O

2

2

2

4

4

4

x

2

J.

y

O

2

x

4

31. At what point do the graphs of the equations y = 7x - 3 and -6x + y = 2 intersect?

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Topic 3  Review TOPIC VOCABULARY • consistent system, p. 70

• feasible region, p. 88

• linear system, p. 70

• ordered triple, p. 94

• constraint, p. 88

• Gaussian elimination, p. 102

• matrix, p. 102

• row operation, p. 102

• coordinate space, p. 94

• inconsistent system, p. 70

• matrix element, p. 102

• solution of a system, p. 70

• dependent system, p. 70

• independent system, p. 70

• objective function, p. 88

• system of equations, p. 70

• equivalent systems, p. 77

• linear programming, p. 88

Check Your Understanding Choose the correct term to complete each sentence. 1. A consistent system with exactly one solution is a(n) ? . 2. ? is a method for finding a minimum or maximum value, given a system of limits called ? .

3-1  Solving Systems Using Tables and Graphs Quick Review

Exercises

A system of equations has two or more equations. Points of intersection are solutions. A linear system has linear equations. A consistent system can be dependent, with infinitely many solutions, or independent, with one solution. An inconsistent system has no solution.

Without graphing, classify each system of equations as independent, dependent, or inconsistent. Solve independent systems by graphing.

Example

5. e

Solve the system e

3x + 2y = 4 and graph the equations. 2x - 4y = 8 2

4 2

O

7. e

y x

4 The only solution, where the lines intersect, is (2, -1).

108

Topic 3  Review

3. e

6x - 2y = 2 2 + 6x = y 6y + 2x = 8 12y + 4x = 4 2 - 0.25x = 0.5y -1.5y = 1.5x - 3

4. e 6. e 8. e

5 - y = 2x 6x - 15 = -3y 1.5 + 3x = 0.5y 6 - 2y = -12x 1+y=x x+y=1

9. For $7.52, you purchased 8 pens and highlighters from a local bookstore. Each highlighter cost $1.09 and each pen cost $.69. How many pens did you buy?

3-2  Solving Systems Algebraically Quick Review

Exercises

To solve an independent system by substitution, solve one equation for a variable. Then substitute that expression into the other equation and solve for the remaining variable. To solve by elimination, add two equations with additive inverses as coefficients to eliminate one variable and solve for the other. In both cases you solve for one of the variables and use substitution to solve for the remaining variable.

Solve each system by substitution.

Example

14. Roast beef has 25 g of protein and 11 g of calcium per serving. A serving of mashed potatoes has 2 g of protein and 25 g of calcium. How many servings of each are needed to supply exactly 29 g of protein and 61 g of calcium?

Solve e

10 - y = 4x by substitution. x = 4 + 0.5y

Substitute for x: 10 - y = 4(4 + 0.5y) = 16 + 2y.

10. e

x - 2y = 3 3x + y = -5

11. e

14x - 35 = 7y -25 - 6x = 5y

Solve each system by elimination. 12. e

11 - 5y = 2x 5y + 3 = -9x

13. e

2x + 3y = 4 4x + 6y = 9

Solve for y: y = -2. Substitute into the first equation: 10 - ( -2) = 4x. Solve for x: x = 3. The solution is (3, -2).

3-3  Systems of Inequalities Quick Review

Exercises

To solve a system of inequalities by graphing, first graph the boundaries for each inequality. Then shade the region(s) of the plane containing solutions valid for both inequalities.

Solve each system of inequalities by graphing.

Example Solve the system of inequalities by graphing. e

y 7 -3 y … -x - 1

Graph both inequalities and shade the region valid for both inequalities.

y −2

O

x 2

4

15. e

y 6 4x 3x + y Ú 5

2x + 3y 7 6 16. • x … -1 yÚ4

17. For a community breakfast there should be at least three times as much regular coffee as decaffeinated coffee. A total of ten gallons is sufficient for the breakfast. Write and graph a system of inequalities to model the problem.

−4

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3-4  Linear Programming Quick Review

Exercises

Linear programming is used to find a minimum or maximum of an objective function, given constraints as linear inequalities. The maximum or minimum occurs at a vertex of the feasible region, which contains the solutions to the system of constraints.

Graph the system of constraints. Name the vertices. Then find the values of x and y that maximize or minimize the objective function.

Example

Minimum for C = x + 5y

x…8 •y…5 x Ú 0, y Ú 0

Graph the system of constraints and name the vertices. Objective function: P = 2x + y Graph the inequalities and shade the area satisfying all inequalities.

y 6 (0, 5)

(8, 5)

The vertices of the feasible region are (0, 0), (0, 5), (8, 5), and (8, 0).

4

Evaluate the objective function at each vertex:

O

2(0) + 0 = 0 2(8) + 5 = 21

2(0) + 5 = 5 2(8) + 0 = 16

2 (0, 0)

The maximum value occurs at (8, 5).

110

Topic 3 Review

2

4

6

(8, 0) x

xÚ2 18. • y Ú 0 3x + 2y Ú 12

3x + 2y … 12 19. • x + y … 5 x Ú 0, y Ú 0 Maximum for P = 3x + 5y

20. A lunch stand makes $.75 profit on each chef’s salad and $1.20 profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef’s salads and between 35 and 50 Caesar salads. The total number sold has never exceeded 100 salads. How many of each type should be prepared in order to maximize profit?

3-5  Systems in Three Variables Quick Review

Exercises

To solve a system of three equations, either pair two equations and eliminate the same variable from both equations, using one equation twice, or choose an equation, solve for one variable, and substitute the expression for that variable into the other two equations. Then, solve the remaining system.

Solve each system by elimination.

Example

① x + y + z = 10 Solve by elimination. ② c 2x - y + z = 9 ③ -3x + 2y + 2z = 5

-x + y + 2z = -5 22. c 5x + 4y - 4z = 4 x - 3y - 2z = 3 Solve each system by substitution.

Add equations ① and ② to eliminate y. ④ 3x + 2z = 19 Add 2 times ② to ③ to eliminate y.

x + y - 2z = 8 21. c 5x - 3y + z = -6 -2x - y + 4z = -13

⑤   x + 4z = 23

Add -3 times ⑤ to ④ to eliminate x.

z=5

Substitute z = 5 into ⑤.

x=3

Substitute z = 5 and x = 3 into ① or ②.

y=2

The solution to the system is (3, 2, 5).

3x + y - 2z = 22 23. • x + 5y + z = 4 x = -3z x + 2y + z = 14 24. • y = z + 1 x = -3z + 6

3-6  Solving Systems Using Matrices Quick Review

Exercises

A matrix can represent a system of equations where each row stands for a different equation. The columns contain the coefficients of the variables and the constants.

Solve each system using a matrix. 25. b

4x - 12y = -1 6x + 4y = 4

Example

26. b

7x + 2y = 5 13x + 14y = -1

27. c

-5x + 3y + 4z = 2 3x - y - z = 4 x - 6y - 5z = -4

Solve using a matrix. e

6x + 3y = -15 2x + 4y = 10

6 3 -15 ` R. 2 4 10 2 1 -5 Divide the first row by 3 to get J ` R . Subtract the 2 4 10 2 1 -5 first row from the second row to get J ` R . Multiply 0 3 15 2 1 -5 1 ` R . Subtract the second the second row by 3 to get J 0 1 5 2 0 -10 row from the first row to get J ` R . Divide the first row 0 1 5 1 0 -5 by 2 to get J ` R . The solution to the system is ( -5, 5). 0 1 5 Enter coefficients as matrix elements J

x+y+ z= 4 28. • 2x - y + z = 5 x + y - 2z = 13

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Topic 3  TEKS Cumulative Practice Multiple Choice Read each question. Then write the letter of the correct answer on your paper. 1. Which of the following is true about the given system? -4y = 12 - 8x b y = 2x - 3 The system has

4. Josea wants to solve the system using substitution. x = -2y + 4 b 2x - 3y = 5 Which of the following is the best way for Josea to proceed? F. Solve the first equation for y, then substitute into the second equation.

A. zero solutions.

G. Solve the second equation for y, then substitute into the first equation.

B. exactly one solution.

H. Substitute -2y + 4 for x in the second equation.

C. two solutions.

J. Substitute -2y + 4 for y in the second equation.

D. infinitely many solutions.

5. Which graph shows the solution to the given system?

2. Which is the graph of y = - 0 x - 2 0 + 1? F.

H.

y

4

2

y

O

2

2

2

2

2

G.

J.

y

4

2

x O

O

2

2

B.

4

3. Which graph represents the solution of the inequality 0 3x + 12 0 Ú 3? 9 7 5 3 1 3

5

7

9

5 3 1

1

3

5

1

3

5

1

1

1

C. D.

5 3 1

x 2

D.

y 2

O

2

4

5 (3,2 )

4 2

x

O 2

B.

2

2

2

A.

y

1

(3,2 )

4

y

2

2

O

(4, 3) x 4

2

x

2 2

2

C.

2

x

4

y

A.

2

x

O 2

1 x-y=1 b 2 x=3

2

y 1

(3, 2 )

O

2

x 4

2

6. The formula for the area of a trapezoid is A = h2 (b1 + b2). Solve this equation for b2. 2A F. b2 = hb H. b2 = h 2A -b 1

1

G. b2 = 2A - b1 J. b2 = h 2A h + b1 7. What is the equation of the line that passes through ( -2, 4) and (2, 7)? 3

3

3

3

A. y - 7 = 4 (x + 2) C. y - 7 = 4 (x - 2) B. y + 7 = 4 (x - 2) D. y - 2 = 4 (x - 7)

112

Topic 3  TEKS Cumulative Practice

8. Which of the following describes the translation of y = 0 x 0 to y = 0 x + 2 0 - 1?

F. y = 0 x 0 translated 2 units to the left and 1 unit down

19. How can you use the graphs of f (x) = - 0 3x 0 + 6 and g(x) = 2x + 1 to find the solutions of - 0 3x 0 + 6 = 2x + 1?

G. y = 0 x 0 translated 2 units to the right and 1 unit down

20. The equation of line m is y = 3x - 1. What is the equation of a line that goes through the point (3, -2) and is perpendicular to line m? Show your work.

J. y = 0 x 0 translated 1 unit to the right and 2 units down

21. The graph below shows the boundaries for the system of linear inequalities. y … 0.5x + 5 b y … -5x - 6 y B 8

H. y = 0 x 0 translated 1 unit to the left and 2 units down

Gridded Response

9. The nutrition label on a package of crackers shows there are 80 Calories in 16 grams of crackers. How many grams are in a package labeled 100 Calories?

A

10. A family with 4 adults and 3 children spends $47 for movie tickets at the theater. Another family with 2 adults and 4 children spends $36. What is the price of a child’s ticket in dollars?

D

5x = -3y - 7 5y = -4x - 7

14. What is the slope of the line 3y - 4 = 12 x ? 15. The line (y - 2) = 27(x - 1) contains point (a, 4). What is the value of a?

Constructed Response 16. An ice cream shop has regular mix-ins for $.50 each and premium mix-ins for $1 each. You have $2.50 to spend on mix-ins, and you want at least 4 mix-ins. How many of each type of mix-in can you get in your ice cream? 17. Solve the following system by graphing. b

y6 y…

- 13x 2 3x

+1 +4

18. Find f ( -4), f (0), and f (3) for the function f (x) =

1 4x

x 8

4 8

12. The graph of y = x is translated up two units. What is the x-intercept of the new graph?

b

4

8 4

11. What is the sum of the solutions of 0 5 - 3x 0 = x + 1?

13. What is the value of x in the solution of the system of equations? Round your answer to the nearest tenth.

C

O

a. Of the shaded areas A, B, C, and D, which area represents a solution to the first inequality but not the second? b. Which represents a solution to the second inequality but not the first?

c. Which represents a solution to both inequalities?

d. Which represents a solution to neither inequality? 22. Jenna is trying to break her school’s record for doing the most push-ups in ten minutes. The current record holder did 350 push-ups in ten minutes. The table shows the number of push-ups Jenna completed in the first 6 minutes. a. Draw a scatter plot and find the line of best fit. b. Will Jenna beat the current record? Justify your reasoning.

Time (min) Number of push-ups

1

2

3

4

5

6

37

70

99 132 169 207

+ 2.

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Topic 4

Matrices

TOPIC OVERVIEW

VOCABULARY

4-1 Adding and Subtracting Matrices 4-2 Matrix Multiplication 4-3 Determinants and Inverses 4-4 Systems and Matrices

DIGITAL

APPS

English/Spanish Vocabulary Audio Online: English Spanish coefficient matrix, p. 138 matriz de coeficientes constant matrix, p. 138 matriz de constantes corresponding elements, p. 116 elementos correspondientes determinant, p. 130 determinante equal matrices, p. 116 matrices equivalentes matrix equation, p. 116 ecuación matricial multiplicative identity matrix, p. 130 matriz de identidad multiplicativa scalar, p. 123 escalar scalar multiplication, p. 123 multiplicación escalar singular matrix, p. 130 matriz singular square matrix, p. 130 matriz cuadrada variable matrix, p. 138 matriz variable

PRINT and eBook

Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you.

Your Digital Resources PearsonTEXAS.com

Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book.

114

Topic 4 Matrices

3--Act Math

The Big Burger For many people, hamburgers are a hallmark of American food. Nearly every restaurant, from fast-food chains, to diners, to fine dining establishments, offers some kind of hamburger on its menu. Some restaurants offer various types of burgers: beef, salmon, and veggie burgers are all quite popular. You can also often choose extras to add to your burger: double patties of beef, cheese, pickles, onions, lettuce, tomatoes...the options are endless! Think about this as you watch this 3-Act Math video.

Scan page to see a video for this 3-Act Math Task.

If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support.

Learning Animations You can also access all of the stepped-out learning animations that you studied in class.

Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding.

Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities.

Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device.

Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support.

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 115

4-1  Adding and Subtracting Matrices TEKS FOCUS

VOCABULARY

TEKS (3)  The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. TEKS (1)(F)  Analyze mathematical relationships to connect and communicate mathematical ideas.

• Corresponding elements – elements in the same position in each matrix

• Equal matrices – Equal matrices have the same dimensions and equal corresponding elements.

• Matrix equation – an equation in which the variable is a matrix • Zero matrix – The zero matrix O, or Om*n , is the m * n matrix whose elements are all zeroes.

• Analyze – closely examine objects, ideas, or relationships to learn more

Additional TEKS (1)(A)

about their nature

ESSENTIAL UNDERSTANDING ​You can extend the addition and subtraction of numbers to matrices.



Key Concept  Matrix Addition and Subtraction To add matrices A and B with the same dimensions, add corresponding elements. Similarly, to subtract matrices A and B with the same dimensions, subtract corresponding elements.



a a A = c 11 12 d a21 a22

a + b11 A + B = c 11 a21 + b21

a12 + b12 d a22 + b22

b B = c 11 b21

b12 d b22

a - b11 A - B = c 11 a21 - b21

a12 - b12 d a22 - b22

Properties  Properties of Matrix Addition If A, B, and C are m * n matrices, then

Example Property A + B is an m * n matrix Closure Property of Addition A + B = B + A Commutative Property of Addition (A + B) + C = A + (B + C) Associative Property of Addition There is a unique m * n matrix Additive Identity Property O such that O + A = A + O = A For each A, there is a unique Additive Inverse Property opposite, - A, such that A + ( -A) = O

116

Lesson 4-1  Adding and Subtracting Matrices

Problem 1 Adding and Subtracting Matrices

To add matrices they need to have the same dimensions. What are the dimensions of C ? C has 2 rows and 3 columns, so it’s a 2 * 3 matrix.

Given C ∙ c A C ∙ D



c

3 2 4 1 4 3 d and D ∙ c d , what are the following? ∙1 4 0 ∙2 2 4 B C ∙ D

3 2 4 1 4 3 d +c d -1 4 0 -2 2 4 =c =c

3+1 2+4 4+3 d -1 + ( -2) 4 + 2 0 + 4

c

4 6 7 d -3 6 4

3 2 4 1 4 3 d -c d -1 4 0 -2 2 4

=c =c

3-1 2-4 4-3 d -1 - ( -2) 4 - 2 0 - 4

2 -2 1 d 1 2 -4

Problem 2

TEKS Process Standard (1)(A)

Solving a Matrix Equation Sports ​The first table shows the teams with the four best records halfway through their season. The second table shows the full season records for the same four teams. Which team had the best record during the second half of the season? Records for the First Half of the Season

Records for Season

Team

Wins

Losses

Team

Wins

Losses

Team 1

30

11

Team 1

53

29

Team 2

29

12

Team 2

67

15

Team 3

25

16

Team 3

58

24

Team 4

24

17

Team 4

61

21

• Records for the first half of the season • Records for the full season

Records for the second half of the season

• Use the equation: first half records + second half records = season records. • Solve the matrix equation.

Step 1 Write 4 * 2 matrices to show the information from the two tables. Let A = the first half records

B = the second half records



F = the final records

30 29 A= ≥ 25 24

11 53 29 12 67 15 ¥  F = ≥ ¥ 16 58 24 17 61 21

continued on next page ▶

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 117

Problem 2 What are the dimensions of matrix B? B will have 4 rows and 2 columns. It is a 4 * 2 matrix.

continued

Step 2 Solve A + B = F for B. B=F-A



53 67 B= ≥ 58 61



29 30 11 53 - 30 15 29 12 67 - 29 ¥ -≥ ¥ =≥ 24 25 16 58 - 25 21 24 17 61 - 24

29 - 11 23 18 15 - 12 38 3 ¥ =≥ ¥ 24 - 16 33 8 21 - 17 37 4

Team 2 had the best record (38 wins and 3 losses) during the second half of the season.

Problem 3

TEKS Process Standard (1)(F)

Using Identity and Opposite Matrices How is this like adding real numbers? Adding zero leaves the matrix unchanged. Adding opposites give you zero.

What are the following sums? A c

1 2 0 0 d ∙c d 5 ∙7 0 0 ∙c



1∙0 2∙0 1 2 d ∙c d 5 ∙ 0 ∙7 ∙ 0 5 ∙7

B c

2 8 ∙2 ∙8 d ∙c d ∙3 0 3 0 ∙c

2 ∙ (∙2) 8 ∙ (∙8) 0 0 d ∙c d ∙3 ∙ 3 0∙0 0 0

Problem 4 Finding Unknown Matrix Values Multiple Choice ​What values of x and y make the equation true?

c

9 3x + 1 9 16 d =c d 2y - 1 10 -5 10 x = 3, y = 5 17

How can you solve the equation? For the two matrices to be equal, the corresponding elements must be equal.

x = 3 , y = 5 3x + 1 = 16



x = 5, y = -3

Set corresponding elements equal.

2y - 1 = -5

3x = 16 - 1

Isolate the variable term.

2y = -5 + 1

3x = 15



Simplify.

2y = -4

x=5



Solve for x and y.

The correct answer is C.

118

x = 5, y = -2

Lesson 4-1  Adding and Subtracting Matrices

y = -2

HO

ME

RK

O

NLINE

WO

PRACTICE and APPLICATION EXERCISES

For additional support when completing your homework, go to PearsonTEXAS.com.

Scan page for a Virtual Nerd™ tutorial video.

1. Apply Mathematics (1)(A) ​The table shows the number of beach balls produced during one shift at two manufacturing plants. Plant 1 has two shifts per day and Plant 2 has three shifts per day. Write matrices to represent one day’s total output at each plant. Find the difference in daily production totals at the two plants.

Beach Ball Production Per Shift 3-color

1-color Plant 1

Plastic 500

Rubber 700

Plastic 1300

Rubber 1900

Plant 2

400

1200

600

1600

Find each matrix sum or difference if possible. If not possible, explain why. 3 4 −3 1 1 2 5 1 A = £ 6 −2 §   B = £ 2 −4 §   C = c d   D = c d −3 1 0 2 1 0 −1 5

2. A + B 3. B + D 4. B - A 5. C-D 6. Use Representations to Communicate Mathematical Ideas (1)(E) ​The modern pentathlon is a grueling all-day competition. Each member of a team competes in five events: target shooting, fencing, swimming, horseback riding, and cross-country running. Here are scores for the U.S. women at the 2004 Olympic Games. a. Write two 5 * 1 matrices to represent each woman’s scores for each event.

U.S. Women’s Pentathlon Scores, 2004 Olympics Anita Allen

Mary Beth lagorashvili

Shooting

952

760

Fencing

720

832

Swimming

1108

1252

Riding

1172

1144

Running

1044

1064

Event

SOURCE: Athens 2004 Olympic Games

b. Find the total score for each athlete. Find each sum. 2 -3 4 0 0 0 7. c d +c d 5 6 -7 0 0 0 Find the value of each variable.

8. c

6 -3 -6 3 d +c d -7 2 7 -2

2 2 4 -1 x y 9. c d -c d =c d -1 6 0 5 -1 z

10. c

2 4 4x - 6 -10t + 5 d =c d 8 4.5 4x 15t + 1.5x

1 2 5 -6 11. £ 2 1 § + X = £ 1 0§ -3 4 8 5

12. c

2 1 -1 11 3 -13 d -X= c d 0 2 1 15 -9 8

13. X - c

14. X + c

Solve each matrix equation.

1 4 5 -2 d =c d -2 3 1 0

6 1 2 0 d =c d -2 3 -3 1

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 119

U.S. Participation (millions) in Selected Leisure Activities

15. Use Representations to Communicate Mathematical Ideas (1)(E) ​Refer to the table at the right. a. Add two matrices to find the total number of people participating in each activity.

Activity

Male

Female

Movies

59.2

65.4

b. Subtract two matrices to find the difference between the numbers of males and females in each activity.

Exercise Programs

54.3

59.0

Sports Events

40.5

31.1

c. In part (b), does the order of the matrices matter? Explain.

Home Improvement

45.4

41.8

16. Analyze Mathematical Ideas (1)(F)  Given a matrix A, explain how to find a matrix B such that A + B = 0.

SOURCE: U.S. National Endowment for the Arts

Solve each equation for each variable. 4b + 2 -3 4d 11 2c - 1 0 17. C -4a 2 3 S = C -8 2 3S 2f - 1 -14 1 0 3g - 2 1 4c 18. C -3 0

2-d -1 -10

5 2c + 5 4d 2 S = C -3 h 15 0 -4c

g f - gS 15

3 -2 19. Find the sum of E = £ 4 § and the additive inverse of G = £ 0 § . 7 5

20. Prove that matrix addition is commutative for 2 * 2 matrices. 21. Prove that matrix addition is associative for 2 * 2 matrices.

TEXAS Test Practice

22. What is the sum c A. c B. c

5 7 3 -7 4 2 d +c d? -1 0 -4 1 -2 -3

-2 11 5 d 0 -2 -7

C. c

-35 28 6 d -1 0 12

12 3 1 d -2 2 -1

D. The matrices cannot be added.

23. Which arithmetic sequence includes the term 27? I. a(1) = 7, a(n) = a(n - 1) + 5 II. a(n) = 3 + 4(n - 1) III. a(n) = 57 - 6n F. I only

120

G. I and II only

Lesson 4-1  Adding and Subtracting Matrices

H. II and III only

J. I, II, and III

Technology Lab Use With Lesson 4-1

Working With Matrices

Foundational to Teks (3)(B), (1)(E)

You can use a graphing calculator to work with matrices. First you need to know how to enter a matrix into the calculator.

Example 1 Enter matrix A ∙ £

∙3 4 7 ∙5 § into your graphing calculator. 0 ∙2

Select the EDIT option of the

matrix

feature to edit matrix [A].

NAMES MATH EDIT 1: [A] 2: [B] 3: [C] 4: [D] 5: [E]

Specify a 3 * 2 matrix by pressing 3

enter

MATRIX [A] [0 0 [0 0 [0 0

3

2

2

enter

.

] ] ]

1, 1  0

Enter the matrix elements one row at a time, pressing MATRIX [A] [ 3 4 [7 5 [0 2

3

2

enter

after each element.

] ] ]

3, 2  2

Then use the

quit

feature to return to the main screen.

continued on next page ▶

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 121

Technology Lab

Example 2 −3 4 10 −7 Given A = £ 7 −5 § and B = £ 4 −3 § , find A + B and A − B. 0 −2 −12 11

Enter both matrices into the calculator. Use the names option of the feature to select each matrix. Press enter to see the sum. [A]  [B] [[ 7 [ 11 [ 12

3 8 9

] ] ]]

Repeat the corresponding steps to find the difference A - B.

Exercises Find each sum or difference. 1. c

2. c

0 -3 -5 3 d -c d 5 -7 4 10

3 5 -7 -1 6 2 d -c d 0 -2 0 -9 4 0

3 -6 3. c d - c d 5 7

4. [3 5 -8] + [ -6 4 1] 5. c

17 8 0 4 6 5 d -c d 3 -5 2 2 -2 9

6. [ -9 6 4] + [ -3 8 4]

122

Technology Lab  Working With Matrices

matrix

continued

4-2  Matrix Multiplication TEKS FOCUS

VOCABULARY

• Scalar – a real number in a

TEKS (3)  The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions.

special product, such as 3 in the matrix product 3A.

• Scalar multiplication – an operation that multiplies a matrix A by a scalar c. To find the resulting matrix cA,

TEKS (1)(A)  Apply mathematics to problems arising in everyday life, society, and the workplace. Additional TEKS (1)(D), (1)(F)

multiply each element of A by c.

• Apply – use knowledge or information for a specific purpose, such as solving a problem

ESSENTIAL UNDERSTANDING ​ he product of two matrices is a matrix. To find an element in the product matrix, you T multiply the elements of a row from the first matrix by the corresponding elements of a column from the second matrix. Then add the products.



Key Concept  Scalar Multiplication To multiply a matrix by a scalar c, multiply each element of the matrix by c. a a a A = c 11 12 13 d a21 a22 a23

cA = c

ca11 ca12 ca13 d ca21 ca22 ca23

Properties  Scalar Multiplication If A and B are m * n matrices, c and d are scalars, and O is the m * n zero matrix, then Example Property cA is an m * n matrix (cd )A = c(dA) c(A + B) = cA + cB (c + d )A = cA + dA 1 A = A 0 A = O and cO = O

# #



Closure Property Associative Property of Multiplication Distributive Properties Multiplicative Identity Property Multiplicative Properties of Zero

Key Concept  Matrix Multiplication To find element cij of the product matrix AB, multiply each element in the ith row of A by the corresponding element in the jth column of B. Then add the products. a11 AB = J a21

a12 b11 a22 R J b21

b12 a11b11 + a12b21 R =J b22 a21b11 + a22b21

a11b12 + a12b22 R a21b12 + a22b22

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 123

Property  Dimensions of a Product Matrix If A is an m * n matrix and B is an n * p matrix, then the product matrix AB is an m * p matrix. matrix A 1 3 rows 3

5

#

2 4 6

matrix B 2 rows

7 8 9 10 11 12 13 14

2 columns

4 columns

equal dimensions of product matrix

Product matrix AB is a 3 * 4 matrix.

Properties  Matrix Multiplication If A, B, and C are n * n matrices, and O is the n * n zero matrix, then Example Property AB is an n * n matrix

Closure Property

(AB)C = A(BC)

Associative Property of Multiplication

A(B + C) = AB + AC (B + C)A = BA + CA

Distributive Property

OA = AO = O

Multiplicative Property of Zero

Problem 1 Using Scalar Products If A = J What operation should you do first? You should first multiply by the scalars, 4 and 3.

124



2 8 −3 −1 0 5 R and B = J R , what is 4A + 3B? −1 5 2 0 3 −2

4A + 3B = 4J

2 8 -3 -1 0 5 R + 3J R -1 5 2 0 3 -2



=J

8 32 -12 -3 0 15 R +J R -4 20 8 0 9 -6



=J

5 32 3 R -4 29 2

Lesson 4-2  Matrix Multiplication

Problem 2

TEKS Process Standard (1)(F)

Solving a Matrix Equation With Scalars What is the solution of 2X + 3J Where have you seen problems that look like this before? You saw problems like this when you solved one variable equations like 2x + 3(5) = 20.



2X + J

2 −1 8 5 R =J R? 3 4 11 0

6 -3 8 5 R =J R 9 12 11 0



Multiply by the scalar 3.



2X = J

8 5 6 -3 R -J R 11 0 9 12

Subtract J



2X = J

2 8 R 2 -12



Simplify.



X= J

1 4 R 1 -6



Multiply each side by 12 and simplify.

6 9

-3 R from each side. 12

Problem 3 Multiplying Matrices If A = J What relationship must exist between the numbers of elements in a row of A and a column of B? They must be equal.

2 1 −1 3 R and B = J R , what is AB? −3 0 0 4

Step 1 Multiply the elements in the first row of A by the elements in the first column of B. Add the products and place the sum in the first row, first column of AB.

J

2 1 -1 3 -2 __ RJ R =J R -3 0 0 4 __ __

2(1)  1(0)  2

Step 2 Multiply the elements in the first row of A by the elements in the second column of B. Add the products and place the sum in the first row, second column of AB.

J

2 1 -1 3 -2 10 RJ R =J R -3 0 0 4 __ __

2(3)  1(4)  10

Repeat Steps 1 and 2 with the second row of A to fill in row two of the product matrix. Step 3 

J

2 1 -1 3 -2 10 RJ R =J R -3 0 0 4 3 __

(3)(1)  0(0)  3

Step 4 

J

2 1 -1 3 -2 10 RJ R =J R -3 0 0 4 3 -9

(3)(3)  0(4)  9

The product of J

2 1 -1 3 -2 10 R and J R is J R. -3 0 0 4 3 -9

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 125

Problem 4

TEKS Process Standard (1)(A)

Applying Matrix Multiplication Sports  In 1966, Washington and New York (Giants) played the highest-scoring game in National Football League history. The table summarizes the scoring. A touchdown (TD) is worth 6 points, a field goal (FG) is worth 3 points, a safety (S) is worth 2 points, and a point after touchdown (PAT) is worth 1 point. Using matrix multiplication, what was the final score?

TD

FG

S

PAT

WASHINGTON

10

1

0

9

NEW YORK

6

0

0

5

• The number of each type of score • The point value of each score What is the meaning of each number in matrix P? They are the point values for each type of score.

The scoring summary and point values as matrices

Multiply the matrices to find each team’s final score.

Step 1  Enter the information in matrices. 6 10 1 0 9 3 S = c d   P = ≥ ¥ 6 0 0 5 2 1 Step 2  Use matrix multiplication. The final score is the product SP. 6 10 1 0 9 3 SP = J R≥ ¥ 6 0 0 5 2 1

=c

10(6) + 1(3) + 0(2) + 9(1) 72 d =c d 6(6) + 0(3) + 0(2) + 5(1) 41

Step 3  Interpret the product matrix.

The first row of SP shows scoring for Washington, so the final score was Washington 72, New York 41.

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Lesson 4-2  Matrix Multiplication

Problem 5

TEKS Process Standard (1)(D)

Determining Whether Product Matrices Exist Does either product AB or BA exist?

NLINE

HO

ME

RK

O

How can you tell if a product matrix exists without computing it? Compare the dimensions of the matrices.

WO

-2 1 A = £ 3 -2 § 0 1

B= c

-1 0 2 1 d 2 0 0 3

AB BA S (3 * 2)(2 * 4) 3 * 4 product matrix   (2 * 4)(3 * 2) Product AB exists. equal not equal

PRACTICE and APPLICATION EXERCISES

S   no product

Scan page for a Virtual Nerd™ tutorial video.

Use matrices A, B, C, and D. Find each product, sum, or difference. For additional support when completing your homework, go to PearsonTEXAS.com.



3 4 A = £ 6 −2 § 1 0

1. A - 2B

B= £

−3 1 2 −4 § −1 5

3. 4C + 3D

C= J

1 2 R −3 1

D= J

5 1 R 0 2

2. 3A + 2B 4. 2A - 5B

Solve each matrix equation. Check your answers. 5. 3J

2 0 -10 5 R - 2X = J R -1 5 0 17

6. 4X + J

1 3 -3 11 R =J R -7 9 5 -7

Find each product. -3 4 1 0 7. J RJ R 5 2 2 -3

8. J

1 0 -3 4 RJ R 2 -3 5 2

9. J

0 2 0 2 RJ R -4 0 -4 0

10. Apply Mathematics (1)(A) ​A florist makes three special floral arrangements. One uses three lilies. The second uses three lilies and four carnations. The third uses four daisies and three carnations. Lilies cost $2.15 each, carnations cost $.90 each, and daisies cost $1.30 each. a. Write a matrix to show the number of each type of flower in each arrangement. b. Write a matrix to show the cost of each type of flower. c. Find the matrix showing the cost of each floral arrangement.

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11. Apply Mathematics (1)(A) ​A hardware store chain sells hammers for $3, flashlights for $5, and lanterns for $7. The store manager tracks the daily purchases at three of the chain’s stores in a 3 * 3 matrix. What is the total gross revenue from the flashlights sold at all three stores? Number of Items Sold Store A  Store B  Store C 9 8 Hammers 10 14 6§ Flashlights £ 3 5 7 Lanterns 2 12. Apply Mathematics (1)(A) ​Two teams are competing in a two-team track meet. Points for individual events are awarded as follows: 5 points for first place, 3 points for second place, and 1 point for third place. Points for team relays are awarded as follows: 5 points for first place and no points for second place. a. Use matrix operations to determine the score in the track meet.

Relays

Individual Events Team West River

First 8

Second 5

Third 2

First 8

Second 5

6

9

12

6

9

River’s Edge

b. Who would win if the scoring was changed to 5 points for first place, 2 points for second place, and 1 point for third place in each individual event and 5 points for first place and 0 points for second place in a relay? For Exercises 13–20, use matrices D, E, and F. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. 1 2 −1 D = £0 3 1§ 2 −1 −2

2 −5 0 E = £1 0 −2 § 3 1 1

−3 2 F = £ −5 1 § 2 4

13. DE

14. -3F

15. (DE)F

16. D(EF)

17. D - 2E

18. (E - D)F

19. (DD)E

20. (2D)(3F)

21. Explain Mathematical Ideas (1)(G) ​Suppose A is a 2 * 3 matrix and B is a 3 * 2 matrix with elements not all being equal. Are AB and BA equal? Explain your reasoning. Include examples.

128

Lesson 4-2  Matrix Multiplication

Determine whether the product exists. F= J

2 3 R 6 9

G= J

−3 6 R 2 −4

H= J

−5 R 6

J = [0 7]

22. FG 23. GF 24. FH 25. HG For Exercises 26–29, use matrices P, Q, R, S, and I. Determine whether the two expressions in each pair are equal.

P= J

3 4 R 1 2

Q= J

1 0 R 3 −2

R= J

1 4 R −2 1

S= J

0 1 R 2 0

I= J

1 0 R 0 1

26. (P + Q)R and PR + QR 27. (P + Q)I and PI + QI 28. (P + Q)(R + S) and (P + Q)R + (P + Q)S 29. (P + Q)(R + S) and PR + PS + QR + QS

TEXAS Test Practice 30. Which product is NOT defined? A. J

-1 R [ -1 2] 2

C. J

B. J

-1 -1

D. [ -1 2] J

2 R [ -1 2] 2

-1 2 2 -1 RJ R -1 2 2 -1 -1 R 2

31. What is the geometric mean of 8 and 18? F. 12

H. 26

G. 13

J. 36

32. The random number table simulates an experiment where you toss a coin 90 times. Even digits represent heads and odd digits represent tails. What is the experimental probability, to the nearest percent, of the coin coming up heads?

Random Number Table 31504

51648

40613

79321

80927

42404

A. 45%

C. 54%

15594

84675

68591

B. 50%

D. 56%

34178

00460

31754

49676

58733

00884

85400

72294

22551

33. Four percent of the tenants in an apartment building live alone. Suppose five tenants are selected randomly. Which expression represents P(all live alone)? F. (0.04)5

H. (0.96)5

G. (0.4)5

J. (5)0.04

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4-3  Determinants and Inverses TEKS FOCUS

VOCABULARY

TEKS (3)  The student applies mathematical processes to formulate systems of equations and inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. TEKS (1)(C)  Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

• Determinant – The determinant of a square matrix is a real number that can be computed from its elements according to a specific formula.

• Multiplicative identity matrix – For an n * n matrix, the multiplicative identity matrix is an n * n matrix I, or In , with 1’s along the main diagonal and 0’s elsewhere.

Additional TEKS (1)(A), (1)(E)

• Multiplicative inverse of a matrix – If A and B are square matrices and AB = BA = I, then B is the multiplicative inverse of A, written A -1 .

• Singular matrix – A singular matrix is a square matrix with no inverse. Its determinant is zero.

• Square matrix – A square matrix is a matrix with the same number of rows and columns.

• Number sense – the understanding of what numbers mean and how they are related

ESSENTIAL UNDERSTANDING The product of a matrix and its multiplicative inverse matrix is the multiplicative identity matrix. Not all matrices have inverse matrices.



Key Concepts  Identity and Multiplicative Inverse Matrices For an n * n matrix, the multiplicative identity matrix is an n * n matrix I, or In, with 1’s along the main diagonal and 0’s elsewhere. 1 0 0 1 0 I2 = c d , I3 = £ 0 1 0 § , 0 1 0 0 1

and so forth.

If A and B are square matrices and AB = BA = I, then B is the multiplicative inverse

matrix of A, written A -1.

130

Lesson 4-3  Determinants and Inverses



Key Concept  Determinants of 2 × 2 and 3 × 3 Matrices The determinant of a 2 * 2 matrix c

a b a d is det c c d c

b d = ad - bc. d

a1 b1 c1 The determinant of a 3 * 3 matrix £ a2 b2 c2 § is a3 b3 c3

a1b2c3 + b1c2a3 + c1a2b3 - (a3b2c1 + b3c2a1 + c3a2b1)

()* ()* a1 b1 c1 a1 b1 Visualize the pattern this way:  £ a2 b2 c2 § a2 b2 a3 b3 c3 a3 b3



a copy of the first two columns

Key Concept  Area of a Triangle The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is shown below. y

(x 3, y 3) (x 2, y 2) (x1, y1)

O x1

Area = 12 0 det A 0 where A = C x2 x3



y1 y2 y3

x

1 1S 1

Key Concept  Inverse of a 2 × 2 Matrix Let A = c

a b d. c d

If det A = 0, then A is a singular matrix and has no inverse. If det A ≠ 0, then the inverse of A, written A -1, is A -1 = det1 A c

d -b d -b d = ad 1- bc c d. -c a -c a

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Problem 1 Determining Whether Matrices Are Inverses For each of the following, are matrices A and B inverses? How do you determine whether A and B are inverses? Find AB and BA. Each product must equal I.

A A = c

−2 1 1 2 d      B = c 3 − 1 d 3 4 2 2

Since AB = I and BA = I, matrices A and B are inverses.

1 0 2 −1 −2 2 5§ B A = £ −2 1 1 §   B = £ −3 −4 −1 1 2 1 1 −1 Since AB = I and BA = I, matrices A and B are inverses.

C A = c

2 4 2 5 d      B = c d 2 2 −1 −3

Since AB ≠ I (and BA ≠ I ), matrices A and B are not inverses.

Problem 2 Evaluating the Determinants of Matrices

What can you do first to evaluate a 3 × 3 determinant? Copy the first two columns to the right of the matrix. 1 0 -2 1 0 £ 0 4 -1 § 0 4 3 5 2 3 5

132

What are the following determinants? 3 -1 d = (3)(5) - ( -1)(2) = 15 - ( -2) = 17 A detc 2 5

1 0 -2 B det£ 0 4 -1 § = (1)(4)(2) + (0)( -1)(3) + (-2)(0)(5) 3 5 2 - [(3)(4)( -2) + (5)( -1)(1) + (2)(0)(0)] = 8 + 0 + 0 - ( -24 - 5 + 0)

= 37 Check  Use a graphing calculator.

Lesson 4-3  Determinants and Inverses

Problem 3

TEKS Process Standard (1)(A)

Finding the Area of a Polygon Land  One factor in flood safety along a levee is the area that will absorb water should the levee break. The coordinates shown are in miles. What is the area in the pictured California community?

(1.4, 2.0)

(1.9, 0.4) (0, 0)

x1 y1 1 A = £ x2 y2 1 §   Matrix for area formula x3 y3 1

Why must you use absolute value? A determinant can be positive or negative. Area must be positive.

0 0 1 = £ 1.9 0.4 1 §   Substitute coordinates. 1.4 2.0 1

1 Area = 2 0 det A 0

= 1.62

  Area formula   Use a calculator to evaluate.

The area of the triangle is 1.62 mi 2 . Check for Reasonableness  The area is approximately one half the area of a 1.9 by 2 rectangle. Thus, an area of about 1.6 units2 is reasonable.

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Problem 4

TEKS Process Standard (1)(C)

Finding the Inverse of a Matrix Does the matrix A = c

−3 6 d have an inverse? If it does, what is A −1 ? −1 3

Evaluate det A. If det A ≠ 0, the matrix has an inverse.

det A = ad − bc = (−3)(3) − (6)(−1) = −9 − (−6) = −9 + 6 = −3 det A ≠ 0, so A has an inverse.

Find A -1. Switch elements on the main diagonal. Change signs on the other diagonal.

d −b d A −1 = det1 A c −c a

From above, det A = -3. For A, a = - 3, b = 6, c = -1,and d = 3.

3 −6 = 1 c d − 3 1 −3

Multiply by the scalar and simplify the fractions.

= £ −3

3 1

−3

=c

−1 2 − 31 1 d

Check  Use a graphing calculator. Method 1 Check that

Method 2 AA -1

= I. Find A -1.

134

Lesson 4-3  Determinants and Inverses

−6 −3 § −3 −3

Problem 5

TEKS Process Standard (1)(E)

Encoding and Decoding With Matrices How can you use matrix multiplication to encode the A 

Bank on Your Knowledge

account number from the credit card? What size array should you use for the card information? Since the coding matrix has two columns, the information matrix must have two rows, so they can be multiplied.

Step 1  Select a coding matrix.    C = c

2 -1 d 3 5

Step 2 Place the card information in a matrix with appropriate dimensions for multiplication by the coding matrix. A= c

Joe Student

4 1 7 3 1 2 3 4 d 9 8 7 6 1 3 5 7

Step 3 Multiply the coding matrix and the information matrix to encode the information. Use a calculator. CA = c =c

2 -1 4 1 7 3 1 2 3 4 dc d 3 5 9 8 7 6 1 3 5 7

-1 -6 7 0 1 1 1 1 d 57 43 56 39 8 21 34 47

Step 4 The coded account number is -1, -6, 7, 0, 1, 1, 1, 1, 57, 43, 56, 39,8, 21, 34, 47. B How do you use a decoding matrix to recover the account number?

NLINE

HO

ME

RK

O

  Multiply the coded information by the inverse of the coding matrix. 2 -1 -1 -1 -6 7 0 1 1 1 1 4 1 7    c d c d =c 3 5 57 43 56 39 8 21 34 47 9 8 7

WO

PRACTICE and APPLICATION EXERCISES

3 6

1 1

2 3

3 5

4 d 7

Scan page for a Virtual Nerd™ tutorial video.

Determine whether the matrices are multiplicative inverses. For additional support when completing your homework, go to PearsonTEXAS.com.

3 2 3 -2 1. c d, c d 4 3 -4 3

-3 7 -5 7 2. c d, c d -2 5 -2 3

2 -1 4. c d 1 0

5. c

1 5

1 - 10

3. £

0

6. c

2 3 d 2 4

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. 2 3 d 1 1

5

1 §, c0 4

2 d 4

a b d has an inverse. In your c d own words, describe how to switch or change the elements of A to write A -1.

7. Display Mathematical Ideas (1)(G) ​Suppose A = c

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8. Use the map to determine the approximate area of the Bermuda Triangle.

U N I TE D S TATE S

600

Bermuda

400

200 mi.

200

Gul f of Mexi co

Bermuda Triangle

Miami

0 –200

CU BA

–400 –600 1000

800

600

400

San Juan, Puerto Rico 200

0

Car i bbean Sea –200

a b e f d and N = c d . Prove that the product of c d g h the determinants of M and N equals the determinant of the matrix product MN.

11. Let M = c

0

–400

1 2 9. For what value of x will matrix A have no inverse? A = c d 3 x a b 10. Suppose A = c d . For what values of a, b, c, and d will A c d be its own inverse? (Hint: There is more than one correct answer.)

ATLANTIC OCEAN

–800

Evaluate each determinant. 12. c

4 5 d -4 4

0 2 -3 15. £ 1 2 4§ -2 0 1

13. c

-3 10 d 6 20

5 1 0 16. £ 0 2 -1 § -2 -3 1

-1 2 14. c 2 d -2 8

4 6 -1 17. £ 2 3 2§ 1 -1 1

y (2, 5)

18. Use Representations to Communicate Mathematical Ideas (1)(E) ​Use matrices to find the area of the figure at the right.

(4, 3)

19. Analyze Mathematical Relationships (1)(F) ​Evaluate the determinant of each matrix. Describe any patterns.

(4, 1)

1 2 3 a. £ 1 2 3 § 1 2 3

-1 -2 -3 b. £ -3 -2 -1 § -1 -2 -3

1 2 3 c. £ 2 3 1 § 1 2 3

-1 2 -3 d. £ 2 -3 -1 § -1 2 -3

4

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. If it does not exist, explain why not. 20. c

1 4 d 1 3

-2 1 -1 23. £ 2 0 4§ 0 2 5

136

21. c

4 7 d 3 5

2 0 -1 24. £ -1 -1 1§ 3 2 0

Lesson 4-3  Determinants and Inverses

22. c

-3 11 d 2 -7

0 0 2 25. £ 1 4 -2 § 3 -2 1

(4, 3) 2 O

4

(4, 1) 2

4

(1, 4,)

x

Graphing Calculator  Evaluate the determinant of each 3 × 3 matrix. 1 0 0 26. £ 0 1 0 § 0 0 1

0 -2 -3 27. £ 1 2 4§ -2 0 1

12.2 13.3 9 28. £ 1 -4 -17 § 21.4 -15 0

29. Connect Mathematical Ideas (1)(F)  Use matrices to find the area of the figure at the right. Check your result by using standard area formulas.

(0, 6) 4

Evaluate the determinant of each matrix.

(4, 2)

7 2 30. c d 0 -3

6 2 31. c d -6 -2

32. c

33. c

0 0.5 d 1.5 2

(2, 4)

2 x 2

(4, 2)

-1 3 d 5 2

O (2, 2)

-2 4 1 35. £ 3 0 -1 § 1 2 1

5 3 34. c d -2 1

36. Explain Mathematical Ideas (1)(G) ​A student wrote C c

y

1 12 1 3

1 4

S as the inverse of

1 2 d . What mistake did the student make? Explain your reasoning. 3 4

37. Use the coding matrix in Problem 5 to encode the phone number (555) 358-0001. 38. If matrix A has an inverse, what must be true? I. AA -1 = I

II. A -1A = I

III. A -1I = A -1

A. I only

C. I and II only

B. II only

D. I, II, and III

TEXAS Test Practice 39. What is the determinant of c

-2 5

-3 d? 0

4 2 -1 d , and the inverse of A is x c -3 -1 3 Enter your answer as a fraction. 40. If A = c

-2 d , what is the value of x? 4

41. What is the value of 6! 8! ? Give your answer as a fraction in simplest terms.

42. If the equation for a circle is x 2 + y 2 - 2x + 6y - 6 = 0, what is its radius?

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4-4  Systems and Matrices TEKS FOCUS

VOCABULARY

TEKS (3)(B)  Solve systems of three linear equations in three variables by using Gaussian elimination, technology with matrices, and substitution. TEKS (1)(C)  Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

• Coefficient matrix – When representing a system of equations with a matrix equation, the matrix containing the coefficients of the system is the coefficient matrix.

• Constant matrix – A constant matrix is a matrix whose entries are constants.

• Variable matrix – When representing a system of equations with a matrix equation, the matrix containing the variables of the system is the variable matrix.

• Number sense – the understanding of what numbers mean and how they are related

Additional TEKS (1)(A)

ESSENTIAL UNDERSTANDING You can solve some matrix equations AX = B by multiplying each side of the equation by A -1, the inverse of matrix A.



Key Concept  Inverse Matrices If matrix A has an inverse, you can use it to solve the matrix equation AX = B. Multiply each side of the equation by A -1 to find X. AX = B A -1AX = A -1B IX = A -1B X = A -1B



Multiply each side by A -1 . A -1A = I, the identity matrix. IX = X

Key Concept  System of Equations as a Matrix Equation You can write a system of equations as a matrix equation AX = B, using a coefficient matrix, a variable matrix, and a constant matrix.

System of Equations b

Matrix Equation

2x + 3y = 1 5x - 2y = 13

c coefficient matrix, A

138

Lesson 4-4  Systems and Matrices

2 3 x 1 dc d = c d 5 -2 y 13 variable matrix, X

constant matrix, B

Problem 1

TEKS Process Standard (1)(C)

Solving a Matrix Equation Using an Inverse Matrix How do you know the equation has a solution? Check det A. If det A ≠ 0, you can solve the equation.

What is the solution of each matrix equation? A c

5 3 1 dX = c d 3 2 −3

Step 1 Evaluate det A and find A -1 . det A = (5)(2) - (3)(3) = 1 A -1 = det1 A c

d -b 2 -3 2 -3 d = 11 c d =c d -c a -3 5 -3 5

Step 2 Multiply each side of the equation by A -1. Does it matter if you multiply by A−1 on the left or right side of A? Even though A-1A = AA-1 = I, you must multiply each side of the equation by A-1 on the left.

c



2 -3 5 3 2 -3 1 dc dX= c dc d -3 5 3 2 -3 5 -3

(2)(5) + ( -3)(3) (2)(3) + ( -3)(2) (2)(1) + ( -3)( -3) c dX= c d ( -3)(5) + (5)(3) ( -3)(3) + (5)(2) ( -3)(1) + (5)( -3) c



1 0 11 dX= c d 0 1 -18 X= c

Check

11 d -18

Method 1

Method 2

Use paper and pencil.

Use a calculator.

c

c

c

5 3 ≟ 1 dX c d 3 2 -3

5 3 11 ≟ 1 dc d c d 3 2 -18 -3

(5)(11) + (3)( -18) ≟ 1 d c d -3 (3)(11) + (2)( -18) c



B c

1 1 d =c d ✔ -3 -3

3 -9 2 dX= c d -2 6 5

Evaluate det A.

det A = (3)(6) - ( -2)( -9) = 18 - 18 = 0. Matrix A has no inverse. The matrix equation has no solution.

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Problem 2 Writing a System as a Matrix Equation What is the matrix equation that corresponds to each system? A b

4x + 7y = 6 −5x + 3y = 1

Step 1 Identify the coefficient, variable, and constant matrices.

coefficient matrix, A

variable matrix, X

constant matrix, B

x c d y

6 c d 1

4 7 c d -5 3

Step 2 Write the matrix equation. 4 7 x 6 c dc d = c d -5 3 y 1

How is this system different from the one in part A? There are three variables. Some terms have coefficients of 0, and the third equation has a variable on the right side of the = sign.

B •

3a + 5b − 12c = 6 7b + 2c = 8 5a = 3c + 1

Step 1 Rewrite the system so the variables are in the same order in each equation. •

3a + 5b - 12c = 6 3a + 5b - 12c = 6 7b + 2c = 8   S   • 7b + 2c = 8 5a = 3c + 1 5a - 3c = 1

Step 2 Identify the coefficient, variable, and constant matrices.

coefficient matrix, A

variable matrix, X

constant matrix, B

a £b§ c

6 £8§ 1

3 5 -12 £0 7 2§ 5 0 -3

Step 3 Write the matrix equation. 3 5 -12 a 6 £0 7 2§ £b§ = £8§ 5 0 -3 c 1

140

Lesson 4-4  Systems and Matrices

Problem 3

TEKS Process Standard (1)(C)

Solving a System of Two Equations What is the solution of the system b

Write the system as a matrix equation. Write the coefficient, variable, and constant matrices. You need to find A -1. Since det A = 2, A -1  exists.



A   X      =B

5 −4 x 4   c dc d = c d 3 −2 y 3 A −1 =

−2 4 1 c d det A −3 5

−2 4 1 c d (5)( − 2) − (3)( − 4) −3 5



=



= 21 c



= C

£

Multiply each side of the matrix equation by  A -1 on the left. x Solve for c d and y check.

5x − 4y = 4 ? Solve using matrices. 3x − 2y = 3

−1 2 − 32

−2 4 d −3 5

−1 2 − 32

5§ c

2

5S 2

−1 2 5 −4 x 4 d c d = C 3 5S c d 3 −2 y −2 2 3



(−1)(4) + (2)(3) 2 x c d = C S = C 5 3 3S y − 2 (4) + 2 (3) 2

( )

()

The solution is x = 2, y = 32.

() 3(2) − 2 ( 32 ) = 3 ✔

5(2) − 4 32 = 4 ✔

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Problem 4

TEKS Process Standard (1)(A)

Solving a System of Three Equations Multiple Choice ​On a new exercise program, your friend plans to do a run-jog-walk routine every other day for 40 min. She would like to burn 310 calories during each session. The table shows how many calories a person your friend’s age and weight burns per minute of each type of exercise. Calories Burned Running (8 mi/h)

Jogging (5 mi/h)

Walking (3.5 mi/h)

12.5 cal/min

7.5 cal/min

3.5 cal/min

If your friend plans on jogging twice as long as she runs, how many minutes should she exercise at each rate? run 10, jog 5, walk 25

run 5, jog 10, walk 25

run 30, jog 15, walk 5

run 10, jog 20, walk 10

Step 1 Define the variables. Let x = number of minutes running. y = number of minutes jogging. How many equations do you need to solve this problem? Since there are three variables you need three equations.

z = number of minutes walking. Step 2 Write a system of equations for the problem. •

12.5x + 7.5y + 3.5z = 310 12.5x + 7.5y + 3.5z = 310 x+ y + z = 40   S    • x+ y + z = 40 2x =y 2x y + 0z = 0

Step 3 Write the system as a matrix equation. 12.5 7.5 3.5 x 310 £ 1 1 1 § £ y § = £ 40 § 2 -1 0 z 0 Step 4 Use a calculator. Solve for the variable matrix. Step 5 Interpret the solution. Your friend should run for 10 min, jog for 20 min, and walk for 10 min. The correct answer is D.

142

Lesson 4-4  Systems and Matrices

HO

ME

RK

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NLINE

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PRACTICE and APPLICATION EXERCISES

Scan page for a Virtual Nerd™ tutorial video.

Solve each matrix equation. If an equation cannot be solved, explain why. For additional support when completing your homework, go to PearsonTEXAS.com.

12 7 2 -1 1. c dX= c d 5 3 3 2

2. c

0 -4 0 dX= c d 0 -1 4

3. Apply Mathematics (1)(A) ​Your classmate is starting a new fitness program. He is planning to ride his bicycle 60 minutes every day. He burns 7 Calories per minute bicycling at 11 mph and 11.75 Calories per minute bicycling at 15 mph. How long should he bicycle at each speed to burn 600 calories per hour? Write each system as a matrix equation. Identify the coefficient matrix, the variable matrix, and the constant matrix. x + 3y - z = 2 4. • x + 2z = 8 2y - z = 1

5. b

y = 3x - 7 x=2

6. b

x + 2y = 11 2x + 3y = 18

7. Apply Mathematics (1)(A) ​Suppose you want to fill nine 1-lb tins with a snack mix. You plan to buy almonds for $2.45/lb, peanuts for $1.85/lb, and raisins for $.80/lb. You want the mix to contain twice as much nuts as raisins by weight. If you spend exactly $15, how much of each ingredient should you buy? 8. Apply Mathematics (1)(A)  Suppose you are making a trail mix for your friends and want to fill three 1-lb bags. Almonds cost $2.25/lb, peanuts cost $1.30/lb, and raisins cost $.90/lb. You want each bag to contain twice as much nuts as raisins by weight. If you spent $4.45, how much of each ingredient did you buy? Select Tools to Solve Problems (1)(C)  Solve each system. -3x + 4y = 2 9. b x - y = -1 12. •

x=5-y 3y = z x+z=7

10. b

x + 2y = 10 3x + 5y = 26

-x = -4 - z 13. • 2y = z - 1 x=6-y-z

11. b

x - 3y = -1 -6x + 19y = 6

-b + 2c = 4 14. • a + b - c = -10 2a + 3c = 1

Determine whether each system has a unique solution. If it has a unique solution, find it.

15. b

y = 23x - 3 y = -x + 7

16. b

3x + 2y = 10 6x + 4y = 16

x + 2y + z = 4 17. c y = x - 3 z = 2x

18. Connect Mathematical Ideas (1)(F) ​The coordinates (x, y) of a point in a plane are 2x + 3y = 13 the solution of the system b . Find the coordinates of the point. 5x + 7y = 31 19. Connect Mathematical Ideas (1)(F) ​A rectangle is twice as long as it is wide. The perimeter is 840 ft. Find the dimensions of the rectangle. 20. Explain Mathematical Ideas (1)(G)  Substitute each point ( -3, 5) and (2, -1) into the slope-intercept form of a linear equation to write a system of equations. Then use the system to find the equation of the line containing the two points. Explain your reasoning.

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Solve each matrix equation. If the coefficient matrix has no inverse, write no unique solution. 21. c

1 1 x 8 d c d = c d 1 2 y 10

22. c

2 1 x 10 2 -3 a 1 dc d = c d dc d = c d 23. c 4 3 y -2 -4 6 b -2

Complete each system for the given number of solutions.

24. infinitely many x+y=7 b 2x + 2y = ■

25. one solution x+y+z=7 • y + z = ■ z=■

26. no solution x+y+z=7 • y+z=■ y+z=■

27. Apply Mathematics (1)(A) ​A caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 18 lb, costs $29.50, and supplies 850 g of protein.

Paella Nutrition Chart Food

Cost/Ib

Protein/Ib

a. Write a system of three equations to find the weight of each ingredient that the caterer uses.

Chicken

$1.50

100 g

$.40

20 g

b. Solve the system. How many pounds of each ingredient did she use?

Shellfish

$6.00

50 g

Rice

TEXAS Test Practice

28. Which matrix equation represents the system b x 2 -3 -3 A. c d c d =c d y -5 1 14 B. c

2 -3 x -3 dc d = c d -5 1 y 14

C. c

D. c

2x - 3y = -3 ? -5x + y = 14

2 -3 -3 x dc d =c d -5 1 14 y

-3 2 -3 d [x y] = c d 14 -5 1

29. How can you write the three equations below as a matrix equation for a system? Explain your steps. 2x - 3y + z + 10 = 0 x + 4y = 2z + 11 -2y + 3z + 7 = 3x

144

Lesson 4-4  Systems and Matrices

Topic 4  Review TOPIC VOCABULARY • coefficient matrix, p. 138

• matrix equation, p. 116

• singular matrix, p. 130

• constant matrix, p. 138

• multiplicative identity matrix, p. 130

• square matrix, p. 130

• corresponding elements, p. 116

• multiplicative inverse of a matrix, p. 130

• variable matrix, p. 138

• determinant, p. 130

• scalar, p. 123

• zero matrix, p. 116

• equal matrices, p. 116

• scalar multiplication, p. 123

Check Your Understanding Choose the correct term to complete each sentence. 1. If corresponding elements of matrices are equal, the matrices are ? . 2. The additive identity of a matrix is the ? . 3. A(n) ? consists of a coefficient matrix, a variable matrix, and a constant matrix. 4. An n * n matrix is called a(n) ? .

4-1  Adding and Subtracting Matrices Quick Review

Exercises

To perform matrix addition or subtraction, add or subtract the corresponding elements in the matrices.

Find each sum or difference. 1 2 -5 -2 7 -3 d +c d 3 -2 1 1 2 5

Two matrices are equal matrices when they have the same dimensions and corresponding elements are equal. This principle is used to solve a matrix equation.

5. c

Example

Solve each matrix equation.

2 1 If A = £ 1 4 −2 −1 what is A + B? 2+1 A + B = £ 1 + ( -3) -2 + 0 3 -1 = £ -2 2 -2 -1

6. c

0 2 -5 6 d -c d -4 -1 -9 -1

−2 1 −2 4 3 § and B = £ −3 −2 1 § , 5 0 0 5

7. 3 2 -6 8 4 + 3 -1 -2 4 4 = X

1 + ( -2) 4 + ( -2) -1 + 0 2 4§ 10

Find the value of each variable.

-2 + 4 3+1 § 5+5

8. c

9. c 10. c

7 -1 4 9 d +X= c d 0 8 -3 11

x-5 9 -7 w + 1 d =c d 4 t+2 8-r 1

-4 + t 2y 2t 11 d =c d -2r + 12 9 r w+5

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4-2  Matrix Multiplication Quick Review

Exercises

To obtain the product of a matrix and a scalar, multiply each matrix element by the scalar. Matrix multiplication uses both multiplication and addition. The element in the ith row and the jth column of the product of two matrices is the sum of the products of each element of the ith row of the first matrix and the corresponding element of the jth column of the second matrix. The first matrix must have the same number of columns as the second has rows.

Use matrices A, B, C, and D to find each product or difference, if possible. If an operation is not defined, label it undefined.

Example If A = c AB = c =c

1 −3 1 4 d and B = c d , what is AB? −2 0 0 2

(1)(1) + ( -3)(0) (1)(4) + ( -3)(2) d ( -2)(1) + (0)(0) ( -2)(4) + (0)(2) 1 -2 d -2 -8

A= c

6 1 0 −4 3 7

−2 4 C= ≥ 2 1

8 1 3 d d B= c 11 −2 4

1 0 5 −2 ¥ d D= c 2 3 6 1

11. 3A 12. B - 2A 13. AB 14. BA 15. AC - BD

4-3  Determinants and Inverses Quick Review

Exercises

A square matrix with 1’s along its main diagonal and 0’s elsewhere is the multiplicative identity matrix, I. If A and X are square matrices such that AX = XA = I, then X is the multiplicative inverse matrix of A, A -1.

Evaluate the determinant of each matrix and find the inverse, if possible.

You can use a calculator to find the inverse of a matrix. You can find the inverse of a 2 * 2 matrix A= c

a b d by using its determinant. c d

  A-1 = det1 A c

d -b d d = ad 1- bc c -c a -c

Example What is the determinant of c det c

146

2 −3 d? 3 −4

2 -3 d = (2)( -4) - ( -3)(3) 3 -4 = -8 - ( -9) = 1

Topic 4 Review

-b d a

16. c

6 1 d 0 4

18. c

10 1 d 8 5

17. c

5 -2 d 10 -4

1 0 2 19. £ -1 0 1§ -1 -2 0

4-4  Systems and Matrices Quick Review

Exercises

You can use inverse matrices to solve some matrix equations and systems of equations. When equations in a system are in standard form, the product of the coefficient matrix and the variable matrix equals the constant matrix. You solve the equation by multiplying both sides of the equation by the inverse of the coefficient matrix. If that inverse does not exist, the system does not have a unique solution.

Use an inverse matrix to solve each equation or system.

Example What is the matrix equation that corresponds to the 2x − y = 12 following system? b x + 4y = 15 Identify A = c

2 -1 x 12 d , X = c d , and B = c d . 1 4 y 15

The matrix equation is AX = B or c

2 -1 x 12 d c d = c d. 1 4 y 15

20. c

3 5 -2 6 dX= c d 6 2 4 12

21. b 22. c

x-y=3 2x - y = -1

4 1 x 10 dc d = c d 2 1 y 6

23. c 24. b

-6 0 -12 -6 dX= c d 7 1 17 9 x + 2y = 15 2x + 4y = 30

a + 2b + c = 14 25. • b=c+1 a = -3c + 6

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Topic 4  TEKS Cumulative Practice Multiple Choice

5. Which system is represented by the graph below? y

Read each question. Then write the letter of the correct answer on your paper.

4

1. If a person walks toward you, and the expression 0 13 - 3t 0 represents his or her distance from you at time t, what does the 3 represent?

4

2. The graph of y = 0 x - 1 0 is translated up 4 units and to the right 3 units. Which equation represents the translated graph? y = 0x - 40 - 3 F. y = 0 x - 3 0 + 4 H.

G. y = 0 x - 4 0 + 4 J. y = 0x + 20 + 4

2

2y + 6 Ú x A. e 3 y 6 - 2x + 5

2y - 6 Ú x C. e 3 y 6 - 2x + 5

2y + 6 Ú x 2y + 6 Ú x B. e D. e 3 3 y 7 - 2x + 5 -y 6 2x + 5

6. The graph below shows a quadratic function. Which of the following equations is represented by the graph?

3. Which inequality is described by the graph below? 4

y

y 2

2

O

2 O

2

4

x

2

2

x

2 F. y = x 2 - 2x - 8

A. y … 0 x - 2 0 C. y Ú 0x - 20

B. y Ú 0 x + 2 0 D. y Ú 0x0 + 2

4. Which of the following is the compound inequality that describes the range of the following function?

y 2 2 O

2

x

2

G. y = -x 2 + x + 4 H. y = 12 x 2 + x + 8

J. y = - 12 x 2 + x + 4 7. The graph of a quadratic function in the xy-plane opens downward and has x-intercepts at x = -3 and x = 5. For what x-value is the value of this function greatest? A. x = -3 B. x=1 C. x=2

F. -3 6 y … 0 H. 0…y…3 G. -3 … y 6 3 J. 5…y…6

148

x

6

6

A. number of steps C. the walking rate

B. total distance D. number of minutes

O

Topic 4  TEKS Cumulative Practice

D. x=5

8. The area of a rectangle is 6x 3 - 22x2 + 23x - 5. The width is 3x - 5. What is the length? F. 2x 2 - 4x + 1 G. 2x 2

+ 4x - 1

H. 2x 2

+1

Constructed Response 16. What is the equation of the circle below?

y

x

J. 2x 2 - x - 4

O

9. Which relation is the inverse of f (x) = (x - 3)2 ? x2 (3x - 1)2 1 B. g (x) = (3x - 1)2

A. g (x) =

2

4

6

4 17. In a geometric sequence, a1 = 3 and a5 = 768. Explain how to find a2 and a3 .

C. g (x) = { 1x + 3

D. g (x) = { 1x - 3

18. A dietician wants to prepare a meal with 24 g of protein, 27 g of fat, and 20 g of carbohydrates using the three foods shown in the table.

Gridded Response 10. Let f (x) = 2x 2 + 3x - 1 and g (x) = x - 1. Evaluate (g f )(2).

#



11. How many roots does the equation 22 + 1x = 0 have? x

12. A box has 10 items inside. What is the number of combinations possible when selecting 3 of the items? 13. What is the value of det c

4

7 -1 d? 3 2

14. A and B are independent but not mutually exclusive events. If P (A) = 14 and P (B) = 15, what is P (A and B)?

15. What is the value of x in the solution of the matrix equation below? 5 -3 x 2 c dc d = c d 2 -1 y 1

Food

Protein

Fat

Carbohydrates

A

2 g/oz

3 g/oz

4 g/oz

B

3 g/oz

3 g/oz

1 g/oz

C

3 g/oz

3 g/oz

2 g/oz

a. Set up a matrix equation for the data. b. Solve the matrix equation.

c. How many ounces of each food are needed?

19. The coach of a high school debate team must choose 4 of the 6 members to represent the team at a statewide competition. Each of the team members is equally qualified for the competition. Use probability concepts to describe how the coach can make his decision fairly. Describe why your method is fair.

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Additional Practice

Topic 3 Lesson 3-1 Solve each system by graphing. Check your answers. y = 2x + 1 1. b y = 4x - 5

2. b

y= x+4 y = - 2x + 3

3. b

3x - 4y = 13 2x + y = 5

4. b

2x = y - 7 4x - 2y = 14

Without graphing, determine whether each system is independent, dependent, or inconsistent. 2x + 3y = 8 5. b 6x + 9y = 24

6. b

x + 3y = 7 y = - 3x + 7

7. b

3x - 4y = 12 3 y = 4x - 4

8. Carla has $2.40 in nickels and dimes. Deron has $5.50 in dimes and quarters. Deron has as many dimes as Carla has nickels and as many quarters as Carla has dimes. How many of each kind of coin does Carla have? 9. Mr. Chandra bought 2 lbs of cheddar cheese and 3 lbs of chicken loaf. He paid $26.35. Mrs. Hsing paid $18.35 for 1.5 lbs of cheese and 2 lbs of chicken loaf. What was the price per pound of each item?

Lesson 3-2 Solve each system of equations by substitution or elimination. x+y= 5 y = 3x - 1 11. b 10. b x - y = -3 2x + y = 14 13. b

x - 4y = 16 x + 2y = 4

14. b

y = 2x + 5 y=4-x

12. b

3x + 2y = 12 x+ y= 3

15. b

y = 5x - 1 y = 14

16. A kayaker can paddle 12 mi in 2 h moving with the river current. Paddling at the same pace, the trip back against the current takes 4 h. Assume that the river current is constant. Find what the kayaker’s speed would be in still water. 17. Mrs. Mitchell put a total of $10,000 into two accounts. One account earns 6% simple annual interest. The other account earns 6.5% simple annual interest. After 1 year, the two accounts earned $632.50 interest. Find how much money was invested in each account.

Lesson 3-3 Solve each system of inequalities by graphing. 18. b

yÚx-3 y … 3x + 7

19. b

3x + 4y 7 8 y 6 5x

20. b

21. Leyla wants to buy fish, chicken, or some of each for weekend meals. The fish costs $4 per pound and the chicken costs $3 per pound. She will spend at least $11 but no more than $15. a.  Write a system of inequalities to model the situation. b.  Graph the system to show the possible amounts Leyla could buy.

538

Additional Practice

- x - 2y Ú - 5 y63

Additional Practice

Lesson 3-4 Find the values of x and y that maximize or minimize the objective function. x…4 x+y…5 1…x…6 y…3 24. c 2 … y … 4 23. c y Ú x 22. d xÚ0 xÚ0 x+yÚ4 yÚ0 maximum for minimum for maximum for P = 2x + y C = x + y P = 3x + 2y 25. A lunch stand makes $.75 in profit on each chef’s salad and $1.20 in profit on each Caesar salad. On a typical weekday, it sells between 40 and 60 chef’s salads and between 35 and 50 Caesar salads. The total number sold has never exceeded 100 salads. How many of each type of salad should be prepared to maximize profit?

Lesson 3-5 Solve each system of equations. x+y+z=6 26. c x = 2y z=x+1

x - 2y + z = 8 27. c y - z = 4 z=3

3x + y - z = 15 28. c x - y + 3z = - 19 2x + 2y + z = 4

29. Three pumps can transfer 4150 gal of water per day when working at the same time. Pumps A and B together can transfer 3200 gal per day. Pumps A and C together can transfer 2900 gal per day. How many gallons can each pump transfer working alone?

Lesson 3-6 Identify the indicated element. 1 A= c −3

−2 −5

30. a12

10 R 22

2 B= C 4 −1

6 0 3

−2 −4 9

31. a23

−8 5S 15 32. b32

33. b22

Solve each system of equations using a matrix. 34. b

2x - 3y = - 6 x + 2y = 11

35. b

3x + y = 2 - x + 2y = 11

36. b

3x - 10y = - 8 x + 5y = - 1

4x + 2y - 3z = 4 37. c 3x - y + z = 1 x - 3y + 2z = 2

38. The school cafeteria sells three different types of sandwiches: chicken, turkey, and roast beef. Chicken sandwiches sell for $3, turkey sandwiches sell for $3.50, and roast beef sandwiches sell for $4. The cafeteria makes 400 sandwiches in total, and, if all sandwiches are sold, the cafeteria will take in $1375. If the cafeteria makes the same number of chicken sandwiches as it does turkey sandwiches, how many of each type of sandwich does the school make?

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Additional Practice

Topic 4 Lesson 4-1 Find each sum or difference. -8 1. J 19 4 3. J 8

3 12 R + J - 45 -7 -3 10 R + J -1 -9

6 -9

2. J

3.6 4.0

- 9.8 0.8 R - J - 1.7 - 6.1

4. J

-2 0

9 2 R + J 0 -4

61 R 37

6. J

-8 -9

3 6

-5 6

8. X - J

64 R 63 7 2

-3 R 7

3.4 R 7.9

-9 R 4

Solve each matrix equation. - 60 - 37 R +X= J 91 85

25 5. J 42 7. X+ J

6 1

2 5

9 11 R = J 10 3

16 R 8

2.3 9.4

1 5 R -X= J 4 7

8 2

6.5 - 4.7 R = J 9.4 - 8.2

3 R 6 3.6 R - 5.8

Find the values of x and y which make the equations true. 9. 2x - 1 J 6

10. 2 J 5

5 R 10

5 3 R = J 4y + 2 6

-x - 4 2 R = J 2y - 6 5

-5 R 0

Lesson 4-2 Solve each matrix equation. Check your answers. 11. 2 J

2 8

-7 0 R + 4X = J -4 4

-6 R -8

12. 0.5X + J

-5 0

3 -3 R = J -2 -1

3.5 R - 0.5

For Exercises 13–24, use matrices A, B, C, and D shown below. Perform the indicated operations if they are defined. If an operation is not defined, label it undefined. A= J

8 −2

1 −3 R   B = J 5 −2

13. AB

14. BD

17. 0.2A

18. BA

1 −1

9 0 R   C = £ 5 5 2

4 1 1 §   D = J 8 0

7 10

15. 2A 19. 5C

21. A frozen yogurt supplier uses two machines to make chocolate and vanilla frozen yogurt. Both machines can be used in the morning and afternoon. Matrix A shows the maximum hourly output of each machine. Matrix B shows how long the machines are used for production of each flavor. Matrix A:  Output (gal/h)

Matrix B:  Time (h)

    Chocolate Vanilla A.M. P.M. 5 Machine 1   4 3 Chocolate    2 c d c d Machine 2   7  Vanilla    8 1 2 a.  Compute the product AB of these matrices. b.  Describe what this product represents.

540

Additional Practice

3 R −2 16. DA 20. 1 D 2

Additional Practice

Lesson 4-3 Determine whether the matrices are multiplicative inverses. 1 3

22. £

1

2 -6 3 §, £ 9 4 2 3

3

2 23. c 1

3§

-2

1

4 2 d, £ 4 1

1 4 § 1 2

Evaluate the determinant of each matrix. 24. c

2 2

27. c

-2 9

4 30. £ 2 -3

4 d -3

3 d -1 -1 0 2

5 -1

25. c

-2 3

28. c

0 31. £ 1 3

5 -1 § 1

-2 d 3

6 d 3

-2 -1 8

7 0§ -1

-2 0

26. c

0.5 - 1.5

29. c

1 32. £ 3 5.5

-7 d 4

-3 d 4

2.5 -1 -2

-7 0.5 § 0

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. 33. c

3 -6

1 d -2

34. c

2 4

-1 d 2

35. c

6 2

36. What is the area of the triangle with vertices (3, 4), (5, - 2), and (- 1, 8)?

9 d 3

Lesson 4-4 Solve each matrix equation. If an equation cannot be solved, explain why. 37. c

2 -1

1 8 d X= c 7 - 12

1 d 41

38. c

-1 6

0 -9 d X= c d 3 -3

39. c

-3 1

5 29 d X= c d 8 58

Solve each system of equations using a matrix equation. Check your answers. 40. b

x-y=3 x+y=5

41. b

x - 2y = 7 x + 3y = 12

42. b

2x + 5y = 10 x+y=2

43. County economists calculated that, in an average year, 9% of employed people lose their jobs and 86% of the unemployed find new jobs. The remaining people remain employed or unemployed, depending on their previous status. On January 1, the county has an unemployment rate of 7%. Calculate the expected unemployment rate for the next two years to the nearest tenth of a percent. 44. Leona’s Diner offers 8-piece, 12-piece, and 16-piece family chicken meals. The table at the right lists the costs of three different orders. What is the price of each kind of meal?

8-Piece Meals

12-Piece Meals

16-Piece Meals

Total Cost

45. At a diner, two hot dogs and one hamburger cost $10, while three hot dogs and two hamburgers cost $17.25. Write and solve a matrix equation to find the cost of a hot dog and the cost of a hamburger.

2

3

1

$96

4

5

0

$133

2

4

2

$134

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