141
CHAPTER 7 Linear Systems and Matrices Section 7.1 Solving Systems of Equations . . . . . . . . . . . . . . . 549 Section 7.2 Systems of Linear Equations in Two Variables . . . . . . 564 Section 7.3 Multivariable Linear Systems . . . . . . . . . . . . . . . 578 Section 7.4 Matrices and Systems of Equations . . . . . . . . . . . . 603 Section 7.5 Operations with Matrices . . . . . . . . . . . . . . . . . . 620 Section 7.6 The Inverse of a Square Matrix . . . . . . . . . . . . . . 633 Section 7.7 The Determinant of a Square Matrix . . . . . . . . . . . . 645 Section 7.8 Applications of Matrices and Determinants . . . . . . . . 654 Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 © Houghton Mifflin Company. All rights reserved.

CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

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Page 1: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

C H A P T E R 7Linear Systems and Matrices

Section 7.1 Solving Systems of Equations . . . . . . . . . . . . . . . 549

Section 7.2 Systems of Linear Equations in Two Variables . . . . . . 564

Section 7.3 Multivariable Linear Systems . . . . . . . . . . . . . . . 578

Section 7.4 Matrices and Systems of Equations . . . . . . . . . . . . 603

Section 7.5 Operations with Matrices . . . . . . . . . . . . . . . . . . 620

Section 7.6 The Inverse of a Square Matrix . . . . . . . . . . . . . . 633

Section 7.7 The Determinant of a Square Matrix . . . . . . . . . . . . 645

Section 7.8 Applications of Matrices and Determinants . . . . . . . . 654

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687

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Page 2: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

549

C H A P T E R 7Linear Systems and Matrices

Section 7.1 Solving Systems of Equations

■ You should be able to solve systems of equations by the method of substitution.

1. Solve one of the equations for one of the variables.

2. Substitute this expression into the other equation and solve.

3. Back-substitute into the first equation to find the value of the other variable.

4. Check your answer in each of the original equations.

■ You should be able to find solutions graphically. (See Example 5 in textbook.)

1. (a)

No, is not a solution.

(b)

No, is not a solution.��1, �5�

�11 � �6

1 � 1

6��1� � ��5� �?

�6

4��1� � ��5� �?

1

�0, �3�

�3 � �6

3 � 1

6�0� � ��3� �?

�6

4�0� � ��3� �?

1

Vocabulary Check

1. system, equations 2. solution 3. method, substitution

4. point, intersection 5. break-even point

(c)

No, is not a solution.

(d)

Yes, is a solution.��12, �3�

�6 � �6

1 � 1

6��12� � ��3� �

?�6

4��12� � ��3� �

?1

��32, 3�

�6 � �6

�9 � 1

6��32� � �3� �

?�6

4��32� � �3� �

?1

2. (a)

Yes, is a solution.�2, �13�

11 � 11

3 � 3

�2 � ��13� �?

11

4�2�2 � ��13� �?

3 (b)

No, is not a solution.

––CONTINUED––

��2, �9�

11 � 11

7 � 3

���2� � ��9� �?

11

4��2�2 � ��9� �?

3

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Page 3: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

550 Chapter 7 Linear Systems and Matrices

3. (a)

No, is not a solution.

(b)

Yes, is a solution.

(c)

No, is not a solution.

(d)

No, is not a solution.��1, �5�

2 � 2

�5 � �2e�1

3��1� � ��5� �?

2

�5 �?

�2e�1

�0, �3�

3 � 2

�3 � �2

3�0� � ��3� �?

2

�3 �?

�2e0

�0, �2�

2 � 2

�2 � �2

3�0� � ��2� �?

2

�2 �?

�2e0

��2, 0�

�6 � 2

0 � �2e�2

3��2� � 0 �?

2

0 �?

�2e�2 4. (a)

No, is not a solution.

(b)

Yes, is a solution.

(c)

Yes, is a solution.

(d)

No, is not a solution.�1, 1�

109 �

289

3 � 1

19 �1� � 1 �? 28

9

�log10 1 � 3 �?

1

�1, 3�

289 �

289

3 � 3

19 �1� � 3 �? 28

9

�log10 1 � 3 �?

3

�10, 2�

289 �

289

2 � 2

19 �10� � 2 �? 28

9

�log10 10 � 3 �?

2

�100, 1�

1099 �

289

1 � 1

19 �100� � 1 �? 28

9

�log10�100� � 3 �?

1

5.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute

Answer: �2, 2�

x � 2: y � 6 � 2(2) � 2

�3x � 6 � 0 ⇒ x � 2x

�x � (6 � 2x) � 0y

y � 6 � 2xy

Equation 1Equation 2� 2x � y � 6

�x � y � 06.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for

Back-substitute

Answer: ��1, 3�

x � 3 � 4 � �1y � 3:

3y � 4 � 5 ⇒ y � 3y:

� y � 4� � 2y � 5x

x � y � 4x

Equation 1Equation 2�x � y

x � 2y�

�45

2. ––CONTINUED––

(c)

No, is not a solution.��32, 6�

�92 � 11

15 � 3

���32� � �6� �

?11

4��32�2

� �6� �?

3 (d)

Yes, is a solution.��74, �37

4 � 11 � 11

3 � 3

���74� � ��37

4 � �?

11

4��74�2

� ��374 � �

?3

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Page 4: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.1 Solving Systems of Equations 551

9.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute:

Answer: �0, 2�, ��3, 2 � 3�3 �, ���3, 2 � 3�3 �x � ��3: y � 2 � 3�3

x � �3: y � 2 � 3�3

x � 0: y � 2

x � 0, ±�3 ⇒ x�x2 � 3� � 0 ⇒ x3 � 3x � 0x

x3 � 2 � �2 � 3x� � 0y

y � 2 � 3xy

Equation 1Equation 2�3x � y � 2

x3 � 2 � y � 0

10.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for

Back-substitute

Back-substitute

Back-substitute

Answer: �0, 0�, �2, �2�, ��2, 2�

y � ���2� � 2x � �2:

y � �2x � 2:

y � �0 � 0x � 0:

x3 � 4x � 0 ⇒ x�x2 � 4� � 0 ⇒ x � 0, ±2x:

x3 � 5x � ��x� � 0y

y � �xy

Equation 1Equation 2� x � y

x3 � 5x � y�

00

11.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Hence, and Answer: �4, 4�x � �27�4� �

367 � 4.y � 4

�y � 4��49y2 � 180y � 1296� � 0

49y3 � 16y2 � 576y � 5184 � 0

�2y3 � 8� 449 y2 �

14449 y �

362

49 � � 0x

8��27y �

367 �2

� 2y3 � 0x

x � �27 y �

367⇒�

72x � y � 18x

Equation 1Equation 2��

72x � y � �18

8x2 � 2y3 � 0

8.

Solve for y in Equation 1:

Substitute for y in Equation 2:

Solve for x:

Back-substitute

Back-substitute

Answer: �0, �5�, �4, 3�

x � 4: y � 3

x � 0: y � �5

x � 0, 4

5x�x � 4� � 0

5x2 � 20x � 0

x2 � 4x2 � 20x � 25 � 25

x2 � �2x � 5�2 � 25

y � 2x � 5

Equation 1Equation 2��2x � y

x2 � y2

�525

7.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute

Back-substitute

Answer: ��1, 3�, �2, 6�

x � 2: y � 2 � 4 � 6

x � �1: y � �1 � 4 � 3

⇒ x � �1, 2

⇒ �x � 1��x � 2� � 0

x2 � x � 2 � 0 x

x2 � (x � 4) � �2y

y � x � 4y

Equation 1Equation 2� x

x2

yy

�4�2

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Page 5: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

552 Chapter 7 Linear Systems and Matrices

12.

Substitute for in Equation 1:

Solve for

Back-substitute

Back-substitute

Back-substitute

Answer: �0, 4�, �1, 2�, �2, 0�

y � �2�2� � 4 � 0x � 2:

y � �2�1� � 4 � 2x � 1:

y � �2�0� � 4 � 4x � 0:

0 � x�x � 2��x � 1� ⇒ x � 0, 1, 2

0 � x�x2 � 3x � 2�

0 � x3 � 3x2 � 2xx:

�2x � 4 � x3 � 3x2 � 4y

Equation 1Equation 2�y �

y �

x3 � 3x2 � 4�2x � 4

13.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute in Equation 1:

Answer: �5, 5�

y � x � 5

2x � 10 ⇒ x � 5x

5x � 3x � 10y

y � xy

Equation 1Equation 2� x

5x�

y3y

010

14.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for

Back-substitute

Answer: ��3, 2�

x � 1 � 2y � 1 � 4 � �3y � 2:

�14y � �28 ⇒ y � 2y:

5�1 � 2y� � 4y � �23x

x � 1 � 2yx

Equation 1Equation 2� x � 2y

5x � 4y�

1�23

15.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute

Answer: �12, 3�

x �12: y � 2x � 2 � 2� 1

2� � 2 � 3

4x � �2x � 2� � 5 � 0 ⇒ 6x � 3 � 0 ⇒ x �12

x

4x � �2x � 2� � 5 � 0y

y � 2x � 2y

Equation 1Equation 2�2x � y � 2 � 0

4x � y � 5 � 0

16.

Solve for in Equation 2:

Substitute for in Equation 1:

Solve for

Back-substitute

Answer: �43, 43�

x � 4 � 2y � 4 � 2�43� �

43y �

43:

24 � 12y � 3y � 4 � 0 ⇒ �15y � �20 ⇒ y �43y:

6�4 � 2y� � 3y � 4 � 0x

x � 4 � 2yx

Equation 1Equation 2�6x � 3y � 4

x � 2y � 4�

00

17.

Solve for in Equation 2:

Substitute for in Equation 1:

Then, Answer: �1, 1�y �3x � 1

2�

3�1� � 1

2� 1.

x � 1

27x � 27

15x � 12x � 4 � 23

15x � 8�3x � 1

2 � � 23y

�2y �3x � 1

2

�2y � 1 � 3xy

�1.5x � 0.8y � 2.3 ⇒ 0.3x � 0.2y � 0.1 ⇒

15x3x

8y2y

231

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Page 6: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.1 Solving Systems of Equations 553

18.

Multiply both equations by 10:

Solve for in Equation 2: x � �18 � 8yx

2x � 16y � �36

5x � 32y � 90

Equation 1Equation 2�0.5x � 3.2y

0.2x � 1.6y�

9.0�3.6

19.

Solve for in Equation 2:

Substitute for in Equation 1:

Solve for :

Back-substitute

Answer: �203 , 40

3 � � 20 �

403 �

203

x � 20 � yy �403 :

4 �310 y � 8 ⇒ y �

403y

15�20 � y� �

12 y � 8x

x � 20 � yx

Equation 1Equation 2�

15xx

12yy

820

20.

Solve for in Equation 2:

Substitute for in Equation 1:

Solve for

Back-substitute

Answer: �20817 , 88

17�y �

34 �208

17 � � 4 �8817x �

20817 :

12 x �

916 x � 3 � 10 ⇒ 17

16 x � 13 ⇒ x �20817

x:

12 x �

34 �3

4 x � 4� � 10y

y �34 x � 4y

Equation 1

Equation 2�12 x �

34 y

34x � y

10

4

21.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Inconsistent

No solution

15 � 6

�5x � 15 � 5x � 6x

�5x � 3�5 �53x� � 6y

y � 5 �53xy

Equation 1Equation 2��

53x

�5x�

y3y

56

22.

From Equation 1,

In Equation 2,

No solution

2x � 3�2 �23 x� � �6 � 6

y � 2 �23 x

Equation 1Equation 2��

23 x �

y2x � 3y

26

23.

Solve for in Equation 2:

Substitute for in Equation 1:

Answer: �3, 5�, ��2, 0�

x � �2 ⇒ y � 0

x � 3 ⇒ y � 5

x � 3, �2

�x � 3��x � 2� � 0

x2 � x � 6 � 0

x2 � 2x � �x � 2� � 8

y

y � x � 2y

Equation 1Equation 2�x2 � 2x � y

x � y�

8�2

24.

Solve for in Equation 2:

Substitute for in Equation 1:

Answer: �4, 10�, ��2, �2�

x � �2 ⇒ y � �2

x � 4 ⇒ y � 10

x � 4, �2

�x � 4��x � 2� � 0

x2 � 2x � 8 � 0

2x2 � 4x � 16 � 0

2x2 � 2x � �2x � 2� � 14

y

y � 2x � 2y

Equation 1Equation 2�2x2 � 2x � y

2x � y�

14�2

Substitute for in Equation 1:

Solve for

Back-substitute

Answer: (2, 2.5�

y � 2.5: x � �18 � 8�2.5� � 2

y �18072 � 2.5

72y � 180

�90 � 40y � 32y � 90y:

5��18 � 8y� � 32y � 90x

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Page 7: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

554 Chapter 7 Linear Systems and Matrices

25.

Solve for in Equation 2:

Substitute for in Equation 1:

No real solution

2x2 � x � 1 � 0

2x2 � �x � 2� � 1y

y � x � 2y

Equation 1Equation 2�2x2

x�

yy

12

26.

Solve for in Equation 2:

Substitute for in Equation 1:

No real solution

2x2 � x � 1 � 0

2x2 � ��x � 4� � 3y

y � �x � 4y

Equation 1Equation 2�2x2

x�

yy

34

27.

Solve for in Equation 2:

Substitute for in Equation 1:

Solve for

Back-substitute:

Answer: �0, 0�, �1, 1�, ��1, �1�

x � �1 ⇒ y � �1

x � 1 ⇒ y � 1

x � 0 ⇒ y � 0

x�x � 1��x � 1� � 0 ⇒ x � 0, 1, �1x:

x3 � x � 0y

y � xy

Equation 1Equation 2�x3

x�

yy

00

28.

Answer: �0, 0�

x � 0 ⇒ y � 0

x�x2 � 3x � 3� � 0

x3 � 3x2 � 3x � 0

y � �x � x3 � 3x2 � 2x

29.

Point of intersection: �4, 3�

x−2 4 6−1 1 2 3

−2

3

5

2

4

6

3x + y = 15

−x + 2y = 2

(4, 3)

y

��x3x

2yy

215

30.

Point of intersection: �2, �2�

x

−1

1

−1 1 2 4

−2

−3

−4

x y+ = 0

3 2 = 10x y−

y

(2, −2)

� x � y3x � 2y

010

31.

Point of intersection: � 52, 32�

( (,

y

x

x − 3y = −2

5x + 3y = 17

−1−2 1 2 3 5 6

−2

−3

1

2

3

4

5

52

32

� x5x

3y3y

�217

32.

Point of intersection: �5, 3�

x

(5, 3)2

2 4 6

4

6

x y− = 2

−x y+ 2 = 1

y

��xx

2yy

� 1� 2

33.

No solution

−2−3−4 2 3 4

−2

−1

−3

−4

−5

2

3

y

x

x + y = 2x2 + y = 1

�x2 � y � 1x � y � 2

34.

Points of intersection:�2, 0�, ��1, �3�

−1−3−4 3 4

−2

−5

2

1

3

y

x

x − y = 2

x2 − y = 4

�x2 � y � 4x � y � 2

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Page 8: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.1 Solving Systems of Equations 555

35.

Points of intersection: (�3, 0�, �3, 6�

y3 � ��6x � x2 � 27

14126 8 1042

10

86

y

x−6

−6

−8

−10

��x � y � 3 ⇒ y1 � x � 3x2 � 6x � 27 � y2 � 0 ⇒ y2 � �6x � x2 � 27

36.

Points of intersection: �15, 7�, �3, 1�

16 181486 10 122

10

86

4

2x

y

−4

−6

−8−10

�y2 � 4x � 11 � 0�

12 x � y � �

12

⇒ ⇒

y � ±�4x � 11y �

12 x �

12

38.

Point of intersection: �2, 2�

� x � y � 0

5x � 2y � 6

⇒ ⇒

y1 � x

y2 �52x � 3

37.

Point of intersection: �4, �12�

−4

−4

8

4

�7xx

8y8y

248 ⇒ ⇒

y1

y2

�78 x18 x

31

39.

Points of intersection: �8, 3�, �3, �2�

x � y � 5 ⇒ y3 � x � 5

y2 � ��x � 1−5 13

−6

6 x � y2 � �1 ⇒ y2 � x � 1 ⇒ y1 � �x � 1

40.

Points of intersection: �2, �2�, �14, 4�

x � 2y � 6 ⇒ y3 �12

�x � 6�−6 18

−8

8x � y2 � �2 ⇒ y1 � �x � 2, y2 � ��x � 2

41.

Points of intersection: �1.540, 2.372�, ��1.540, 2.372�

y � x2 ⇒ y3 � x2 −6

−4

6

4x2 � y2 � 8 ⇒ y1 � �8 � x2, y2 � ��8 � x2

600

4

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Page 9: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

556 Chapter 7 Linear Systems and Matrices

42.

Points of intersection: �3, 4�, �3, �4�

−8

−8

16

8

y4 � ��41 � �x � 8�2

�x � 8�2 � y2 � 41 ⇒ y3 � �41 � �x � 8�2

y2 � ��25 � x2

x2 � y2 � 25 ⇒ y1 � �25 � x2 43.

Point of intersection: �0, 1�

−3

−1

3

3

� y � ex

x � y � 1 � 0 ⇒ y � x � 1

44.

Point of intersection: ��0.490, �6.530�

−9

−10

9

2

�y � �4e�x

y � 3x � 8 � 0 ⇒ y � �3x � 845.

Point of intersection: �2.318, 2.841�

−1

−1

8

5

�x � 2yy

82 � ln x

⇒ ⇒

y1 � 4 � x2y2 � 2 � ln x

46.

Point of intersection: �5.309, �0.539�

−6

−6

12

6

�y � �2 � ln�x � 1�3y � 2x � 9 ⇒ y �

13 �9 � 2x�

47.

Point of intersection: �94, 11

2 �

−3

−3

15

9

�y � �x � 4y � 2x � 1

48.

Point of intersection: �4, 1�

−2

−4

10

4

� x � y � 3�x � y � 1

⇒ ⇒

y � x � 3 y � �x � 1

49.

Points of intersection: �0, �13�, �±12, 5�

−27

−18

27

18

�x2

x2

y2

8y

169

104

y1 � �169 � x2 andy2 � ��169 � x2

y3 �18x2 � 13

50.

Points of intersection:or �0, �2�, �±1.323, 1.500��0, �2�, �1

2�7, 32�, ��12�7, 32�

2x2 � y � 2 ⇒ y3 � 2x2 � 2

y2 � ��4 � x2

−6

−4

6

4 x2 � y2 � 4 ⇒ y1 � �4 � x2

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Section 7.1 Solving Systems of Equations 557

51.

Substitute for in Equation 2:

Solve for :

Back-substitute

Answer: �1, 2�

x � 1 in Equation 1: y � 2x � 2

x2 � 2x � 1 � �x � 1�2 � 0 ⇒ x � 1x

2x � x2 � 1y

Equation 1Equation 2�y � 2x

y � x2 � 152.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for

No real solutions because the discriminant in the Quadratic Formula is negative.

Inconsistent. No solution

x2 � x � 2 � 0x:

x2 � �4 � x� � 2y

y � 4 � xy

Equation 1Equation 2�x � y

x2 � y�

42

53.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute

Back-substitute

Answers: �2910, 21

10�, ��2, 0�

x � �2: y �3x � 6

7� 0

x �29

10: y �

3x � 6

7�

3(29�10) � 6

7�

21

10

10x2 � 9x � 58 � 0 ⇒ x �9 ± �81 � 40�58�

20 ⇒ x �

29

10, �2

40x2 � 36x � 232 � 0

49x2 � �9x2 � 36x � 36� � 196

x2 � �9x2 � 36x � 36

49 � � 4x

x2 � �3x � 6

7 �2

� 4y

y �3x � 6

7y

Equation 1Equation 2�3x � 7y � 6 � 0

x2 � y2 � 4

54.

Solve for in Equation 2:

Substitute for in Equation 1:

Solve for

Back-substitute

Back-substitute

Answer: �3, 4�, �5, 0�

y � 10 � 2�5� � 0x � 5:

y � 10 � 2�3� � 4x � 3:

⇒ �x � 5��x � 3� � 0 ⇒ x � 3, 5

x2 � 100 � 40x � 4x2 � 25 ⇒ x2 � 8x � 15 � 0x:

x2 � �10 � 2x�2 � 25y

y � 10 � 2xy

Equation 1Equation 2� x2

2x �

y2

y �

2510

55.

Graphing and you see that there are no points of

intersection. No solution

y3 � 4 � x,y1 � �1 � x2, y2 � ��1 � x2

x � y � 4

x2 � y2 � 1 56.

Graphing and you see that there are no points of

intersection.No solution

y3 � x � 5,y1 � �4 � x2, y2 � ��4 � x2

x � y � 5

x2 � y2 � 4

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558 Chapter 7 Linear Systems and Matrices

57.

Point of intersection: �14, 32�

−5 7

−2

6

�y � 2x � 1y � �x � 2

58.

extraneous

Answer: or �1.25, 1.5��54, 32�

x �54 ⇒ y �

32

x � 0

x�4x � 5� � 0

4x2 � 5x � 0

4x2 � 4x � 1 � x � 1

2x � 1 � �x � 1

�yy

2x � 1�x � 1

59.

Point of intersection: Approximately �0.287, 1.751�

6−3

−1

5

� y � e�x � 1 ⇒ y � e�x � 1y � ln x � 3 ⇒ y � ln x � 3

60. Graph and

Point of intersection: �1.262, 3.534�

6−30

6

y � exy � 4 � 2 ln x

61.

Substitute for in Equation 2:

Solve for :

Back-substitute:

Answer: �0, 1�, �1, 0�

x � 1 ⇒ y � 0

x � 0 ⇒ y � 1

x2�x � 1� � 0 ⇒ x � 0, 1

x3 � x2 � 0x

x3 � 2x2 � 1 � 1 � x2y

Equation 1Equation 2�y � x3 � 2x2 � 1

y � 1 � x262.

Substitute for in Equation 1:

Solve for

Back-substitute in Equation 2:

Back-substitute in Equation 2:

Back-substitute in Equation 2:

Answer: �0, �1�, �2, 1�, ��1, �5�

y � ���1�2 � 3��1� � 1 � �5

x � �1

y � �22 � 3�2� � 1 � 1

x � 2

y � �02 � 3�0� � 1 � �1

x � 0

0 � x�x � 2��x � 1� ⇒ x � 0, 2, �1

0 � x�x2 � x � 2�

0 � x3 � x2 � 2xx:

�x2 � 3x � 1 � x3 � 2x2 � x � 1

y

Equation 1Equation 2�y

y�

x3 � 2x2 � x � 1�x2 � 3x � 1

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Page 12: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.1 Solving Systems of Equations 559

63.

Solve for in Equation 1:

Substitute for in Equation 2:

Solve for :

Back-substitute

Back-substitute

Two points of intersection: �1

2, 2�, ��4, �

1

4�

x � �4: y �1

�4� �

1

4

x �1

2: y �

1

12� 2

⇒ �2x � 1��x � 4� � 0 ⇒ x �1

2, �4

2x2 � 4 � 7x � 0x

2x � 4�1

x� � 7 � 0y

y �1

xy

Equation 1Equation 2

� xy � 1 � 02x � 4y � 7 � 0

65.

(Answers will vary.)R $1,910,400

x 192 units

1300x � 250,000

9950x � 8650x � 250,000

00

400

3,500,000

C

R

R � C

C � 8650x � 250,000, R � 9950x

66.

R $968,333 x 233,333 units,

1.50x � 350,000

4.15x � 2.65x � 350,000

R � C

500,00000

2,500,000

R

C

C � 2.65x � 350,000, R � 4.15x

67.

In order for the revenue to break even with the cost, 3133 units must be sold, R $10,308.

x 3133 units

10.8241x2 � 65,830.25x � 100,000,000 � 0

10.8241x2 � 65,800x � 100,000,000 � 30.25x

3.29x � 10,000 � 5.5�x

3.29x � 5.5�x � 10,000

R � C

00

5,000

15,000

C

R

R � 3.29xC � 5.5�x � 10,000,

64.

Solve for in Equation 1:

Substitute for in Equation 2:

(x cannot be 0.)

Two points of intersection: �23

, 3�, ��2, �1�

x � �2 ⇒ y � �1

x �23

⇒ y � 3

�3x � 2��x � 2� � 0

3x2 � 4x � 4 � 0

3x � 2�2

x� � 4 � 0

y

y �2

xy

Equation 1Equation 2

� xy � 2 � 03x � 2y � 4 � 0

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Page 13: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

560 Chapter 7 Linear Systems and Matrices

68.

Quadratic in

units, R $18,798x 1464

�x12.84x � 7.8�x � 18,500 � 0

12.84x � 7.8�x � 18,500

R � C

5,00000

25,000

R

C

C � 7.8�x � 18,500, R � 12.84x

69. Animated film

Horror film

(a)

(b) For

(c)

(d) The answers are the same.

(e) During week 8 the same number (168) were rented.

N � 24 � 18�8� � 168

x � 8

336 � 42x

360 � 24x � 24 � 18x

x � 8, N � 168

N � 24 � 18x

N � 360 � 24x

Week 1 2 3 4 5 6 7 8 9 10 11 12

Animated 336 312 288 264 240 216 192 168 144 120 96 72

Horror 42 60 78 96 114 132 150 168 186 204 222 240

x

70. Sofia

Paige

(a)

(b) In the table, at

(c)

Answer:

(d) The solutions are the same.

(e) Both girls scored 18 points in game 3.

�3, 18�

Ps � Pp � 18

3 � x

12 � 4x

24 � 2x � 12 � 2x

x � 3.Ps � Pp � 18

Pp � 12 � 2x

Ps � 24 � 2x

1 2 3 4 5 6 7 8 9 10 11 12

Sofia 22 20 18 16 14 12 10 8 6 4 2 0

Paige 14 16 18 20 22 24 26 28 30 32 34 36

x

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Page 14: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.1 Solving Systems of Equations 561

73.

To make the straight commission offer better, youwould have to sell more than $11,666.67 per week.

x $11,666.67

0.03x � 350

0.06x � 0.03x � 350 74.

For the second offer to be better, you would haveto sell more than $500,000 per year.

500,000 � x

5000 � 0.01x

25,000 � 0.01x � 20,000 � 0.02x

75. (a)

(b)

As increases, decreases and the amount ofinterest decreases.

(c) The curves intersect at Thus, $5000should be invested at 6.5%.

x � 5000.

yx

00

25,000

20,000 0.065x + 0.085y = 1600

x + y = 20,000

� x � y � 20,0000.065x � 0.085y � 1600

76.

(a)

(b) The two graphs intersect at Algebraically:

Since the scales agree when inches.

(c) V is larger using the Scribner Log Rule whenV is larger using the Doyle Log

Rule when Therefore, for largediameters, you would use the Doyle Log Rule.

24.7 ≤ D ≤ 40.5 ≤ D ≤ 24.7.

D 24.72 5 ≤ D ≤ 40,

D 24.72, 3.9

0.21D2 � 6D � 20 � 0

D2 � 8D � 16 � 0.79D2 � 2D � 4

�D � 4�2 � 0.79D2 � 2D � 4

D � 24.72.

4050

750

ScribnerDoyle

V � 0.79D2 � 2D � 4, 5 ≤ D ≤ 40

V � �D � 4�2, 5 ≤ D ≤ 40

71. (a)

(b)

for units

Algebraically,

x �16,000

20.5 780 units

16,000 � 20.5x

35.45x � 16,000 � 55.95x

x 780C � R

020,000

2,000

100,000

C

R

R � 55.95x

C � 35.45x � 16,000 72. (a)

(b)

3759 units to break even.

Algebraically,

x 3759.4 � 3759 units

5000 � 1.33x

5000 � 2.16x � 3.49x

5,00000

20,000

R

C

R � 3.49x

C � 5000 � 2.16x

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Page 15: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

562 Chapter 7 Linear Systems and Matrices

77.

(a)

(b) From 1998 to 2010

(c)

Point of intersection: �7.87, 5477.0�

45000 20

6500

P � 76.5t � 4875

M � 47.4t � 5104

Year 1990 1994 1998 2002 2006 2010

Missouri 5104 5294 5483 5673 5862 6052

Tennessee 4875 5181 5487 5793 6099 6405

78. (a)

(b)

(c) when or late 2023.

t 23.9,Tpublic � Tprivate 33,495

−1 50

20,000

Tpublic

Tprivate

Tprivate � 810.5x � 14,109

Tpublic � 55.21x2 � 25.4x � 2516

79.

Dimensions: 6 meters � 9 meters

l � w � 3 � 9

w � 6

2w � 12

l � w � 3 ⇒ �w � 3� � w � 15

2l � 2w � 30 ⇒ l � w � 15 80.

Dimensions: centimeters60 � 80

w � l � 20 � 80 � 20 � 60

l � 80

2l � 160

w � l � 20 ⇒ l � �l � 20� � 140

2l � 2w � 280 ⇒ l � w � 140

81.

If the length is supposed to be greater than thewidth, we have miles and miles.w � 8 l � 12

l � 12, w � 8

l � 8 or l � 12

0 � �l � 8��l � 12�

0 � l2 � 20l � 96

20l � l2 � 96

lw � 96 ⇒ l�20 � l� � 96

2l � 2w � 40 ⇒ l � w � 20 ⇒ w � 20 � l 82.

The dimensions areinches and

hypotenuse inches.� 2b � h � �2

a � �2

a2 � 2

1 �12a2

a 2

a

A �12bh

(d)

Using the positive value, or late 2023.

(e) The results are the same.

x 23.9,

�835.9 ± 1805.2

110.42

x �835.9 ± �835.92 � 4�55.21���11,593�

2�55.21�

55.21x2 � 835.9 � 11,593 � 0

55.21x2 � 25.4x � 2516 � 810.5x � 14,109

(d)

(e) The population of Tennessee surpassed that ofMissouri during 1997.

t � 22929.1 7.87

229 � 29.1t

47.4t � 5104 � 76.5t � 4875

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Section 7.1 Solving Systems of Equations 563

90. (a)

(b) Based on the graphs in part (a) it appears thatfor there are three points of intersection for the graphs of and when b isan even number.

y � xby � bxb > 1,

−3

−30

5

300

y = 4x

y = x4

−5

−2

5

20

y = x2 y = 2x

88. Answers will vary. For example,

(a)

(b)

(c)

2x � y � 3

6x � 3y � 9

2x � y � 2

3x � y � 4

3x � y � 5

3x � y � 3

89. Answers will vary. For example,

2y � x � 4

y � x � 3

91.

2x � 7y � 45 � 0

7y � 49 � �2x � 4

y � 7 � �2

7�x � ��2��

m �5 � 7

5 � ��2�� �

2

7

��2, 7�, �5, 5�

92.

2x � 7y � 22 � 0

7y � 28 � 2x � 6

y � 4 �2

7�x � 3�

m �6 � 4

10 � 3�

2

7

�3, 4�, �10, 6� 93.

The line is horizontal.

y � 3 ⇒ y � 3 � 0

m �3 � 3

10 � 6� 0

�6, 3�, �10, 3� 94.

x � 4

�4, �2�, �4, 5�

83. False. You could solve for first.x

85. The system has no solution if you arrive at a falsestatement, ie. , or you have a quadraticequation with a negative discriminant, which wouldyield imaginary roots.

4 � 8

87. (a) The line intersects the parabola at two points, and

(b) The line intersects at only.

(c) The line does not intersect

(Other answers possible.)

y � x2.y � x � 2

�0, 0�y � x2y � 0

�2, 4�.�0, 0�y � x2y � 2x

84. False. There could be four points of intersection.For example, and y � x2 � 3.x2 � y2 � 4

86. The advantage of the method of substitution overthe graphical method is that substitution gives anexact answer.

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564 Chapter 7 Linear Systems and Matrices

Section 7.2 Systems of Linear Equations in Two Variables

■ You should be able to solve a linear system by the method of elimination.

1. Obtain coefficients for either or that differ only in sign. This is done by multiplying all the terms ofone or both equations by appropriate constants.

2. Add the equations to eliminate one of the variables and then solve for the remaining variable.

3. Use back-substitution into either original equation and solve for the other variable.

4. Check your answer.

■ You should know that for a system of two linear equations, one of the following is true.

(a) There are infinitely many solutions; the lines are identical. The system is consistent.

(b) There is no solution; the lines are parallel. The system is inconsistent.

(c) There is one solution; the lines intersect at one point. The system is consistent.

yx

Vocabulary Check

1. method, elimination 2. equivalent 3. consistent, inconsistent

99. Domain: all

Vertical asymptotes:

Horizontal asymptote: y � 1

x � ±4

x � ±4 100.

Domain: all

Vertical asymptote:

Horizontal asymptote: y � 3

x � 0

x � 0

f �x� � 3 �2x2 �

3x2 � 2x2

95.

0 � 30x � 17y � 18

17y � 102 � 30x � 120

y � 6 �30

17�x � 4�

m �6 � 0

4 � 35

�6175

�30

17

�3

5, 0�, �4, 6� 96.

45x � 29y � 127 � 0

29y �29

2� �45x �

225

2

y �1

2� �

45

29�x �5

2�

�152

�296

� �45

29 m �

8 �12

�73 �

52

�� 7

3, 8�, �5

2,

1

2�

97. Domain: all

Vertical asymptotes:

Horizontal asymptote: y � 0

x � 6

x � 6 98. Domain: all

Vertical asymptote:

Horizontal asymptote: y �23

x � �23

x � �23

101. Domain: all real numbers x

Horizontal asymptote: y � 0

102. Domain: all real numbers x

Horizontal asymptote: y � 0

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Section 7.2 Systems of Linear Equations in Two Variables 565

4.

Multiply Equation 1 by 3:

Add this to Equation 2 to eliminate

Substitute in Equation 1:

Answer: �3, 3�

2�3� � y � 3 ⇒ y � 3x � 3

10x � 30 ⇒ x � 3y:

6x � 3y � 9

−4

−8 10

82x − y = 3

4x + 3y = 21

Equation 1Equation 2�2x

4x�

y3y

321

5.

Multiply Equation 1 by 2:

Add this to Equation 2:

There are no solutions.

0 � 9

2x � 2y � 4

Equation 1Equation 2� x

�2x�

y2y

25

−4

−6 6

4x − y = 2−2x + 2y = 5

6.

Multiply Equation 1 by 2 and add to Equation 2:

There are infinitely many solutions.

All points on line .3x � 2y � 5

0 � 0 � 0

−4

−5 7

4

3x − 2y = 5

−6x + 4y = −10Equation 1Equation 2� 3x

�6x�

2y4y

5�10

1.

Add to eliminate

Substitute in Equation 2:

Answer: �2, 1�

2 � y � 1 ⇒ y � 1x � 2

y: 3x � 6 ⇒ x � 2

Equation 1Equation 2�2x

x�

yy

51

−3

−5 7

5x − y = 1

2x + y = 5

2.

Add to eliminate

Substitute in Equation 1:

Answer: ��2, 1�

x � 3�1� � 1 ⇒ x � �2y � 1

5y � 5 ⇒ y � 1x:

−3

−8 4

5x + 3y = 1 −x + 2y = 4Equation 1

Equation 2� x�x

3y2y

14

3.

Multiply Equation 1 by

Add this to Equation 2 to eliminate

Substitute in Equation 1:

Answer: �1, �1�

1 � y � 0 ⇒ y � �1x � 1

y: x � 1

�2: �2x � 2y � 0

Equation 1Equation 2� x

3x�

y2y

01

−4

−6 6

4 x + y = 0

3x + 2y = 1

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Page 19: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

566 Chapter 7 Linear Systems and Matrices

7.

Add to eliminate

Substitute in Equation 1:

Answer: �52, 34�

52 � 2y � 4 ⇒ y �

34

x �52

x �52

2x � 5

y:

Equation 1Equation 2�x

x�

2y2y

41

8.

Adding the equations,

Then

Answer: �3, 2�

3 � 2y � 7 ⇒ y � 2

4x � 12 ⇒ x � 3.

Equation 1Equation 2�3x

x�

2y2y

57

9.

Multiply Equation 2 by 3:

Add this to Equation 1 to eliminate

Substitute in Equation 1:

Answer: �3, 4�

6 � 3y � 18 ⇒ y � 4

x � 3

17x � 51 ⇒ x � 3

y:

15x � 3y � 33

Equation 1Equation 2�2x

5x�

3yy

1811

10.

Multiply Equation 1 by

Add this to Equation 2 to eliminate

Substitute in Equation 1:

Answer: �5, 1�

x � 7 � 12 ⇒ x � 5

y � 1

�26y � �26 ⇒ y � 1

x:

�3x � 21y � �36�3:

Equation 1Equation 2� x

3x�

7y5y

1210

11.

Multiply Equation 1 by 2 and Equation 2 by

Add to eliminate

Substitute in Equation 1:

Answer: �4, �1�

3r � 2��1� � 10 ⇒ r � 4

s � �1

�11s � 11 ⇒ s � �1

r:

�6r � 15s � �9

6r � 4s � 20

�3:

Equation 1Equation 2�3r

2r�

2s5s

103

12.

Multiply Equation 1 by

Add this to Equation 2:

Substitute in Equation 1:

Answer: �r, s� � �56, 56�

2r � 4�56� � 5 ⇒ r �

56

s �56

18s � 15 ⇒ s �56

�16r � 32s � �40�8:

Equation 1Equation 2� 2r

16r�

4s50s

555

13.

Multiply Equation 1 by 3 and Equation 2 by

Add to eliminate

Substitute in Equation 2:

Answer: �127 , 18

7 �3u � 5�18

7 � � 18 ⇒ u �127v �

187

�7v � �18 ⇒ v �187u:

�15u � 25v � �90

15u � 18v � 72��5�:

Equation 1Equation 2�5u

3u�

6v5v

2418

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Page 20: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.2 Systems of Linear Equations in Two Variables 567

22.

Multiply Equation 1 by 5 and Equation 2 by

Adding,

Substituting into Equation 1,

Answer: ��149 , 20

9 �⇒ x � �

149 2x � 5�20

9 � � 8 ⇒ 2x � 8 �1009

9y � 20 ⇒ y �209

�10x � 16y � �20

10x � 25y � 40

�2:

Equation 1Equation 2�2x

5x�

5y8y

810

14.

Multiply Equation 1 by 2 and Equation 2 by 3:

Adding,

Then,

Answer: ��17, 5�

3u � 11�5� � 4 ⇒ u � �17

7v � 35 ⇒ v � 5

� 6u�6u

22v15v

827

Equation 1Equation 2� 3u

�2u�

11v5v

49

15.

Multiply Equation 1 by

Add this to Equation 2:

Inconsistent; no solution

0 � �17

�9x � 6y � �20��5�:

Equation 1Equation 2�1.8x

9x�

1.2y6y

43

16.

Multiply Equation 1 by

Add this to Equation 2:

Substituting this value into Equation 2:

Answer: �13031 , 8�

31x � 12�8� � 34 ⇒ 31x � 130 ⇒ x �13031

17y � 136 ⇒ y � 8.

�31x � 29y � 102�10:

Equation 1Equation 2�3.1x

31x�

2.9y12y

�10.234

17.

Matches (b)

One solution; consistent

x � 5, y � 2

�3x � �15

2x � 5�x � 3� � 0

2x � 5y � 0 x � y � 3 ⇒ y � x � 3

18.

Lines coincide.

Matches (a)

Infinite number of solutions; consistent

14x � 12y � 8

�7x � 6y � �4 19.

One solution; consistent

Matches (c)

y � �2, x � �52x � 3y � �4

�2y � 42x � 5y � 0 �

20.

Parallel lines

Inconsistent; no solution

�7x � 6y � �4

7x � 6y � �6 21.

Multiply Equation 1 by 3 and Equation 2 by

Add to eliminate

Substitute into Equation 1:

Answer: �� 635, 43

35�4x � 3�43

35� � 3 ⇒ x � �635

y �4335

�35y � �43 ⇒ y � 4335x:

� 12x�12x

9y44y

9�52

�4:

Equation 1Equation 2�4x

3x�

3y11y

313

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Page 21: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

568 Chapter 7 Linear Systems and Matrices

23.

Multiply Equation 2 by and add to Equation 1:

Inconsistent; no solution

0 � 8

�2

Equation 1Equation 2�

25 x15 x

32 y34 y

4

�224.

Multiply Equation 1 by and add to Equation 2:

There are an infinite number of solutions. All pointson the line 4x � y � 4.

0 � 0

�6

Equation 1Equation 2�

23 x4x

16 yy

23

4

25.

Multiply Equation 1 by

Add this to Equation 2:

There are an infinite number of solutions.

The solutions consist of all satisfying or 6x � 8y � 1.34 x � y �

18,�x, y�

0 � 0

�94x � 3y � �

38�3:

Equation 1Equation 2�

34 x94 x

y

3y

1838

26.

Multiply Equation 1 by 12 and add to Equation 2:

Inconsistent; no solution

0 � 12

Equation 1

Equation 2� 1

4

�3

x

x

16

2

y

y

1

0

27.

Multiply Equation 1 by 12 and Equation 2 by 4:

Add to eliminate

Substitute into Equation 2:

Answer: �5, �2�

2�5� � y � 12 ⇒ y � �2

x � 5

11x � 55 ⇒ x � 5y:

�3x8x

4y4y

748

Equation 1

Equation 2�x � 3 4

�y � 1 � 1

3 2x � y � 12

28.

Multiply Equation 1 by 4:

Add to eliminate

Substitute into Equation 2:

Answer: �72, �1

2�

7

2� y � 4 ⇒ y � �

1

2

x �7

2

2x � 7 ⇒ x �7

2y:

�xx

yy

34

Equation 1

Equation 2�x � 2 4

�y � 1 � 1

4 x � y � 4

29.

Multiply Equation 1 by 6:

Add this to Equation 2 to eliminate

Substitute in Equation 2:

Answer: �7, 1�

7 � 2y � 5 ⇒ y � 1

x � 7

4x � 28 ⇒ x � 7

y:

3�x � 1� � 2�y � 2� � 24 ⇒ 3x � 2y � 23

Equation 1

Equation 2�x � 1

2x

y � 2

32y

4

5

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Page 22: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.2 Systems of Linear Equations in Two Variables 569

34.

Multiply Equation 1 by 10 and Equation 2 by 2:

Subtract to eliminates

Hence,

Answer: �1, �2�

x � 2 � 0.5�y� � 2 � 0.5��2� � 1

7y � �14 ⇒ y � �2.

x:

�2x2x

6yy

�104

Equation 1Equation 2�0.2x

x�

0.6y0.5y

�12

35.

Multiply Equation 1 by 200 and Equation 2 by 300:

Add to eliminate y:

Substitute in Equation 2:

Answer: �9031, �67

31� y � �

6731

0.07�9031� � 0.02y � 0.16

x �9031

31x �9031

31x � 90

�10x21x

6y6y

4248

Equation 1Equation 2�0.05x

0.07x�

0.03y0.02y

0.210.16

30.

Multiply Equation 1 by 2:

Add to eliminate

Substitute into Equation 2:

Answer: �72, 32�

7

2� y � 2 ⇒ y �

3

2

x �72

2x � 7 ⇒ x �7

2y:

�xx

yy

52

Equation 1

Equation 2�x � 1 2

�y � 2 � 1

2 x � y � 2

31.multiplied by 5

Multiply Equation 1 by

Add this to Equation 2 to eliminate

The solution set consists of all points lying on the line

All points on the line

Let

Answer: where is any real number. a�a, 56a �12�,

x � a, then y �56a �

12.

5x � 6y � 3.

10x � 12y � 6.

0 � 0

x:

�10x � 12y � �6

��4�:

Equation 1Equation 2�2.5x

10x�

3y12y

1.56

32.

There are an infinite number of solutions. All points on the line 7x � 8y � 6.

�Divide by 0.9��Divide by 0.8��7x

7x�

8y8y

66

Equation 1Equation 2�6.3x

5.6x�

7.2y6.4y

5.44.8

33.

Multiply Equation 1 by 40 and Equation 2 by 50:

Adding the equation eliminates

Substitute into Equation 1:

Answer: �101, 96�

8�101� � 20y � �1112 ⇒ y � 96

x � 101

23x � 2323 ⇒ x � 101

y:

� 8x15x

20y20y

�11123435

Equation 1Equation 2�0.2x

0.3x�

0.5y0.4y

�27.868.7

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Page 23: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

570 Chapter 7 Linear Systems and Matrices

36.

Multiply Equation 1 by 10 and Equation 2 by 2:

Subtract to eliminates

Hence,

Answer: ��3, 1�

x � �2.5 � 0.5�y� � �2.5 � 0.5�1� � �3

3y � 3 ⇒ y � 1.

x:

�2x2x

4yy

�2�5

Equation 1Equation 2�0.2x

x�

0.4y0.5y

�0.2�2.5

37. Let and .

Multiply Equation 1 by 4:

Subtract to eliminates

Hence,

Answer: ��1, 1�

x �1X

� �1, y �1Y

� 1

X � 2 � 3Y � 2 � 3�1� � �1

13Y � 13 ⇒ Y � 1.

X:

�4X4X

12YY

8�5

Equation 1Equation 2� X

4X�

3YY

2�5

Y �1y

X �1x

39. Let and .

Multiply Equation 1 by 2:

Adding the equations eliminates

Hence,

Answer: �1, 12�

x �1X

� 1, y �1Y

�12

2Y � 5 � X � 4 ⇒ Y � 2.

5X � 5 ⇒ X � 1.

Y:

�2X3X

4Y4Y

10�5

Equation 1Equation 2� X

3X�

2Y4Y

5�5

Y �1y

X �1x

38. Let and .

Multiply Equation 1 by 2:

Subtract to eliminates

Hence,

Answer: �2, 1�

x �1X

� 2, y �1Y

� 1

X �12

.

Y � 1X:

�4X4X

2Y3Y

0�1

Equation 1Equation 2�2X

4X�

Y3Y

0�1

Y �1y

X �1x

40. Let and .

Adding the equations,

Hence,

Answer: �12, �1�

x �1X

�12

, y �1Y

� �1

Y � 11 � 6X � 11 � 12 � �1.

8X � 16 ⇒ X � 2.

Equation 1Equation 2�2X

6X�

YY

511

Y �1y

X �1x

41.

The system is consistent. There is one solution,

−5

−6

13

6

�5, 2�.

�2x � 5y � 0 ⇒ y �25 x

x � y � 3 ⇒ y � x � 3©

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Page 24: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.2 Systems of Linear Equations in Two Variables 571

48.

Answer: �3, �2�

−7

−8

11

4

� 4y � �8

7x � 2y � 25

⇒ ⇒

y �

y �

�2

�7x � 25��249.

Answer: �6, 5�

−3

−2

15

10

�32 x

�2x�

15 y3y

8 ⇒ y � 5�32x � 8�

3 ⇒ y �13�3 � 2x�

44.

The system is consistent. The solution set consists of all points on the line or 4x � 6y � 9 � 0.y �

23 x �

32,

−5

−4

7

4� 4x � 6y � 9163 x � 8y � 12

⇒ ⇒

y � �4x � 9��6

y � �163 x � 12��8

23 x �

32

23 x �

32

45.

The system is consistent. The solution set consists ofall points on the line or 8x � 14y � 5.y �

47 x �

514,

−6

−4

6

4

�8x2x

14y3.5y

51.25

⇒ ⇒

yy

(8x(2x

5)1.25)

�14 �47 x �

514

�3.5 �47 x �

514

46.

Inconsistent; no solution

−9

−4

9

8

� y �17�x � 3�

y � 5 �17x

47.

Answer: �236 , 7� � �3.833, 7�

903

9

�6x �

6y y

4216

⇒ ⇒

yy

76x � 16

43.

The lines are parallel. The system is inconsistent.

−9

−6

9

6

� 35 x � y � 3 ⇒ y �35 x � 3

�3x � 5y � 9 ⇒ y �15 �3x � 9� �

35 x �

95

42.

The system is consistent. There is one solution,

−8

−5

10

7

�1.8, 1.4�

�2xx

y2y

5�1

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Page 25: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

572 Chapter 7 Linear Systems and Matrices

50.

Answer: �8, 6�

−4

−2

14

10

�x � 6y � 28 ⇒ y � �x � 28��6 � x6

�143

34

x �52

y � �9 ⇒ y �25 �3

4x � 9� �

310

x �185� 51.

Answer: �1, �0.667�

−4

−6 6

4

5x � 3y � 7 ⇒ y �13

�5x � 7�

13

x � y � �13

⇒ y � �13

�13

x

52.

Answer: ��0.4, 2�

−4 2

−1

3

2x �35 y �

25 ⇒ y �

53��2x �

25�

5x � y � �4 ⇒ y � 5x � 4 53.

Answer: ��4, 5�

−12

−2

6

10

�0.5x6x

2.2y0.4y

9 ⇒ y � 1�2.2�9 � 0.5x� �22 ⇒ y � 1�0.4��22 � 6x�

54.

Answer: ��1, �4�

−6

−7

6

1

�2.4x � 3.8y4x � 0.2y

�17.6�3.2

⇒ ⇒

y �

y �

��2.4x � 17.6��3.8�4x � 3.2��0.2

55.

Multiply Equation 2 by 5:

Add this to Equation 1:

Back-substitute into Equation 2:

Answer: �4, 1�

2�4� � y � 9 ⇒ y � 1

x � 4

13x � 52 ⇒ x � 4

10x � 5y � 45

Equation 1Equation 2�3x

2x�

5yy

79

56.

Subtract Equation 2 from Equation 1 to eliminate

Substitute in Equation 1:

Answer: ��2, 5�

���2� � 3y � 17 ⇒ y � 5

x � �2

�5x � 10 ⇒ x � �2

y:

Equation 1Equation 2��x

4x�

3y3y

177

57.

The lines intersect at

��53, �11

3 � � ��1.667, �3.667�

−10 8

−8

4

�yy

4x�5x

312

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Page 26: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.2 Systems of Linear Equations in Two Variables 573

66. There are infinitely many systems that have thesolution One possible system:

6��23� � 1��10� � 6 ⇒ 6x � y � 6

3��23� � 1��10� � �12 ⇒ 3x � y � �12

��23, �10�.

67.

Answer: �80, 10�

p � $10

x � 80 units

50 � 0.625x

50 � 0.5x � 0.125x

Demand � Supply

58.

Answer: �1310, 23

10�y �

1310 � 1 �

2310

x �1310

10x � 13

7x � 3�x � 1� � 16

�7x � 3yy

16 x � 1

59.

Adding the equations,

Back-substituting,

Answer: �6, �3�

�5y � 15 ⇒ y � �3

x � 5y � 6 � 5y � 21 ⇒ 7x � 42 ⇒ x � 6.

� x6x

5y5y

2121

60.

Answer: ��23, 61�

x � �23 ⇒ y � �3��23� � 8 � 61

�x � 23

�3x � 8 � 15 � 2x

�yy

�3x � 815 � 2x

61.

Substitute into Equation 1,

Back-substituting,

Answer: �436 , 25

6 �y � x � 3 �

436 � 3 �

256

⇒ x �436

6x � 43 ⇒ �2x � 8�x � 3� � 19

Equation 1Equation 2��2x � 8y � 19

y � x � 3

62.

Multiply Equation 1 by 5 and Equation 2 by 4.

Adding,

Then,

Answer: �3, 2�

4x � 3�2� � 6 ⇒ x � 3

13y � 26 ⇒ y � 2

�20x � 28y � �4

20x � 15y � 30

Equation 1Equation 2� 4x

�5x�

3y7y

6�1

64. There are infinitely many systems that have as the solution. For example,

� x2x

yy

�110

�3, �4�

63. There are infinitely many systems that have thesolution One possible system is

� x�x

yy

88

�0, 8�.

65. There are infinitely many systems that have thesolution One possible system:

3 � 4�52� � �7 ⇒ x � 4y � �7

2�3� � 2�52� � 11 ⇒ 2x � 2y � 11

�3, 52�.

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Page 27: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

574 Chapter 7 Linear Systems and Matrices

68.

Answer: �500, 75�

p � 75

x � 500

0.15x � 75

25 � 0.1x � 100 � 0.05x

Supply � Demand 69.

Answer: �2,000,000, 100�

p � $100.00

x � 2,000,000 units

60 � 0.00003x

140 � 0.00002x � 80 � 0.00001x

Demand � Supply

70.

Answer: �250,000, 350�

p � 350

x � 250,000

0.0007x � 175

225 � 0.0005x � 400 � 0.0002x

Supply � Demand 71. Let the ground speed and the wind speed.

Substituting in Equation 2:

Answer: x � 550 mph, y � 50 mph

y � 50

550 � y � 600

x � 550

Equation 1

Equation 2�3.6�x � y� � 1800

3�x � y� � 1800

y �x �

x � 550

2x � 1100

x � y � 600

x � y � 500

72. Let the speed of the plane that leaves first and the speed of the plane that leaves second.

Answer: First plane: 880 kilometers per hour; Second plane: 960 kilometers per hour

x � 880

960 � x � 80

y � 960

72 y � 3360

��2x2x

2y32 y

1603200

Equation 1Equation 2� y

2x�

x32 y

803200

y �x �

73. (a)

(b) Multiply Equation 1 by 5:

Subtracting eliminates A:

Hence,

650 adult tickets and 525 child tickets

A � 1175 � 525 � 650.

C � 525

1.5C � 787.5

5A � 3.5C � 5087.5

5A � 5C � 5875

Equation 1

Equation 2� A � C � 1175

5A � 3.5C � 5087.5

(c)

Let

Point of intersection: �A, C� � �650, 525�

C � y2 �1

3.5�5087.5 � 5x�

C � y1 � 1175 � x

120000

1200

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Page 28: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.2 Systems of Linear Equations in Two Variables 575

78. Let number of pairs of $75.50 shoes.

Let number of pairs of $89.95 shoes.

Solving for y in the first equation and substitutinginto the second equation gives:

170 of $75.50 shoes, 80 of $89.95 shoes

y � 250 � x � 80

x � 170

2456.5 � 14.45x

75.50x � 89.95250 � x � 20,031

� x75.50x

y89.95y

25020,031

y �

x �

74. (a)

(b) Multiply Equation 1 by 4 and Equation 2 by 3:

Subtracting,

Hence,

(c) Let

Point of intersection: �C, S� � �1.25, 0.75�

C � y2 � �10.5 � 4S��6

C � y1 � �8.5 � 3S��5

C � $1.25, S � $0.75

S � �8.50 � 5C��3 � 0.75

2C � 2.5 ⇒ C � 1.25:

18C � 12S � 31.5

20C � 12S � 34

Equation 1

Equation 2�5C � 3S � 8.50

6C � 4S � 10.50

76. Family Dollar

Dollar General

(a) Using a graphing utility, the solution is

(b) In 1994, both stores had sales of $905.59 million.

�S, T � � �905.59, 4.38�.

S � 691.48t � �2122.8

S � 440.36t � �1023.0

75. Let number of oranges and let number of grapefruit.

Solving for R in Equation 1:

Substituting into Equation 2:

Hence,

9 oranges and 7 grapefruit

R � 16 � 9 � 7.

M � 9

0.9 � 0.1M

0.95M � 16.8 � 1.05M � 15.9

0.95M � 1.05�16 � M� � 15.9

R � 16 � M.

Equation 1Equation 2� M

0.95M�

R1.05R

16 15.90

R �M �

77. Let number of movies and let number of videos.

Solve for m in Equation 1:

Substitute for m Equation 2:

185 movies and 125 videos

m � 310 � 125 � 185

v � 125

62.5 � 0.5v

3�310 � v� � 2.5v � 867.5

m � 310 � v.

Equation 1Equation 2� m

3m�

v2.5v

310 867.5

v �m �

79.

Least squares regression line: y � 0.97x � 2.10

b � 2.10

a � 0.97

10a � 9.7

� 5b10b

10a30a

20.250.1

⇒ ⇒

�10b10b

20a30a

�40.450.1

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576 Chapter 7 Linear Systems and Matrices

80.

Least squares regression line: y � 0.22x � 1.9

b � 1.9

5b � 10�0.22� � 11.7

a � 0.22

10a � 2.2

� 5b � 10a10b � 30a

� 11.7 ⇒ � 25.6 ⇒

�10b � 20a10b � 30a

�23.425.6

81.

Multiply Equation 1 by

Adding,

y � �2.5x � 5.54

b � �2.7 � 10�2.5���5 � 5.54

a � �2.5

10a � �25.0

10b � 30a � �19.6

�10b � 20a � �5.4

�2:

Equation 1Equation 2� 5b

10b�

10a30a

2.7�19.6

82.

Multiply Equation 1 by

Adding,

y � �1.48x � 2.5

b � ��2.3 � 10�1.48���5 � 2.5

a � �1.48

10a � �14.8

10b � 30a � �19.4

�10b � 20a � 4.6�2:

Equation 1Equation 2� 5b

10b�

10a30a

�2.3�19.4

83. (a)

Adding,

Thus,

(b) Using a graphing utility, you obtain y � 14x � 19.

y � 14x � 19

a � 14, b � 19. ⇒ �5a � �70

�4b7b

7a13.5a

174322

⇒ ⇒

28b�28b

49a54a

1218�1288

(c)

(d) If bushels per acre.y � 14�1.6� � 19 � 41.4

�160 pounds�acre�,x � 1.6,

300

60

84. (a)

Solving this system, you obtain and

(b) y � �44.21x � 89.53

y � �44.21x � 89.53

b � 89.53a � �44.21

�3.00b3.70b

3.70a4.69a

105123.9

(c)

(d) For x � 1.75, y � 12

200

50

85. True. A consistent linear system has either one solution or an infinite number of solutions.

86. True

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Section 7.2 Systems of Linear Equations in Two Variables 577

93.

Multiply the first equation by the second byand add the equations:

v � �tan x.

Hence, v cos x � �u sin x � �sin x

u � 1

u sin2 x � u cos2 x � sec x � cos x

cos x,sin x,

u cos x � v sin x � sec x

u sin x � v cos x � 0 94.

Multiply the first equation by the secondby and add the equations:

u � �12.

Hence, u cos 2x �12 cot 2x � sin 2x � 0

v �12 cot 2x

2v � cot 2x

2v sin2 2x � 2v cos2 2x � csc 2x � cos 2x

cos 2x,2 sin 2x,

u��2 sin 2x� � v�2 cos 2x� � csc 2x

u cos 2x � v sin 2x � 0

88.

Multiply Equation 2 by

Add this to Equation 1 to eliminate

Substitute in Equation 1:

Answer:

The lines are not parallel. The scale on the axes must be changed to see the point of intersection.

�300, 315�

21�300� � 20y � 0 ⇒ y � 315x � 300

� 23x � �200 ⇒ x � 300y:

� 653 x � 20y � �200�� 5

3�:

Equation 1Equation 2�21x

13x�

20y12y

0120

89. No, it is not possible for a consistent system oflinear equations to have exactly two solutions.Either the lines will intersect once or they willcoincide and then the system would haveinfinite solutions.

90. (a)

Subtract Equation 2 from Equation 1:

System is inconsistent no solution

(b)

Multiply Equation 1 by

Add this to Equation 2: (dependent)

The system has an infinite number of solutions.

0 � 0

�2x � 2y � �6��2�:

Equation 1Equation 2� x

2x�

y2y

36

0 � �10

Equation 1Equation 2�x

x�

yy

1020

92.

Multiply Equation 1 by

Add this to Equation 2:

The system is inconsistent if k � �2.

�k � 2�y � 13

ky � 2y � 13

10x � 2y � 423:

Equation 1Equation 2� 15x

�10x�

3yky

69

91.

Multiply Equation 2 by

Add this to Equation 1:

The system is inconsistent if

This occurs when Note that for the two original equations represent parallel lines.

k � �4,k � �4.

�8y � 2ky � 0.

�8y � 2ky � �35

�2: �4x � 2ky � �32

Equation 1Equation 2�4x

2x�

8yky

�316

87.

Subtract Equation 2 from Equation 1 to eliminate x:

Substitute into Equation 1:

Answer:

The lines are not parallel. The scale on the axes must be changed to see the point of intersection.

�39,600, 398�

100�398� � x � 200 ⇒ x � 39,600y � 398

y � 398

Equation 1Equation 2�100y

99y�

xx

200�198

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578 Chapter 7 Linear Systems and Matrices

Section 7.3 Multivariable Linear Systems

■ You should know the operations that lead to equivalent systems of linear equations:

(a) Interchange any two equations.

(b) Multiply all terms of an equation by a nonzero constant.

(c) Replace an equation by the sum of itself and a constant multiple of any other equation in the system.

■ You should be able to use the method of elimination.

95.

−10 −9 −8 −7 −6 −5

3−

x

22

x ≤ �446 � �

223

�6x ≥ 44

�11 � 6x ≥ 33 97.

42 6 8 10 12 14 16 180−2x

�2 < x < 18

�10 < x � 8 < 10

�x � 8� < 10

99.

Critical numbers:Testing the three intervals,

43210−1−2−3−4−5−6x

72

�5 < x < 72.

72, �5.

�2x � 7��x � 5� < 0

2x2 � 3x � 35 < 0

101. ln x � ln 6 � ln 6x

103. log9 12 � log9 x � log9 12x

96.

1 2 3 4 5 6

3 3

x

4 16

43 ≤ x < 163

4 ≤ 3x < 16

�6 ≤ 3x � 10 < 6

98. is true for all x,since

x43210−1−2−3−4

�x � 10� ≥ 0.�x � 10� ≥ �3 100.

Critical numbers:Checking the three intervals,you obtain and

210−1−2−3−4−5−6

x

x > 0.x < �4

0, �4.

3x�x � 4� > 0

3x2 � 12x > 0

102.

� ln x

�x � 3�5

ln x � 5 ln�x � 3� � ln x � ln�x � 3�5

104.

� log6�3x�1�4

14

log6 3 �14

log6 x �14

log6�3x�

105.

� ln� x2

x � 2� 2 ln x � ln�x � 2� � ln x2 � ln�x � 2� 106.

� ln� �x2 � 4

x 12

ln�x2 � 4� � ln x � ln�x2 � 4�1�2 � ln x

107. Answers will vary.

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Section 7.3 Multivariable Linear Systems 579

1. (a) Yes

No

No

No, is not a solution.

(c) No

Yes

No

No, is not a solution.��4, 1, 2�

5�1� � 2�2� �?

8

2��4� � 3�2� �?

�14

3��4� � 1 � 2 �?

1

�3, 5, �3�

5�5� � 2��3� �?

8

2�3� � 3��3� �?

�14

3�3� � 5 � ��3� �?

1 (b) Yes

Yes

Yes

Yes, is a solution.

(d) No

No

Yes

No, is not a solution.�1, 0, 4�

5�0� � 2�4� �?

8

2�1� � 3�4� �?

�14

3�1� � 0 � 4 �?

1

��1, 0, 4�

5�0� � 2�4� �?

8

2��1� � 3�4� �?

�14

3��1� � �0� � 4 �?

1

2. (a) Yes

No

No

No, is not a solution.

(c) Yes

Yes

Yes

Yes, is a solution.�1, 3, �2�

2�1� � 3�3� � 7��2� �?

�21

5�1� � 3 � 2��2� �?

�2

3�1� � 4�3� � ��2� �?

17

�1, 5, 6�

2�1� � 3�5� � 7�6� �?

�21

5�1� � 5 � 2�6� �?

�2

3�1� � 4�5� � 6 �?

17 (b) No

Yes

No

No, is not a solution.

(d) No

No

Yes

No, is not a solution.�0, 7, 0�

2�0� � 3�7� � 7�0� �?

�21

5�0� � 7 � 2�0� �?

�2

3�0� � 4�7� � 0 �?

17

��2, �4, 2�

2��2� � 3��4� � 7�2� �?

�21

5��2� � ��4� � 2�2� �?

�2

3��2� � 4��4� � 2 �?

17

3. (a) Yes

No

No

No, is not a solution.

(c) Yes

Yes

Yes

Yes, is a solution.��12, 34, �5

4� 3��1

2� �34 �

?�

94

�8��12� � 6�3

4� �54 �

?�

74

4��12� �

34 � ��5

4� �?

0

�0, 1, 1�

3�0� � 1 �?

�94

�8�0� � 6�1� � 1 �?

�74

4�0� � 1 � 1 �?

0 (b) No

No

No

No, is not a solution.

(d) Yes

No

No

No, is not a solution.��12, 2, 0�

3��12� � 2 �

?�

94

�8��12� � 6�2� � 0 �

?�

72

4��12� � 2 � 0 �

?0

��32, 54, �5

4� 3��3

2� � �54� �

?�

94

�8��32� � 6�5

4� � ��54� �

?�

74

4��32� �

54 � ��5

4� �?

0

Vocabulary Check

1. row-echelon 2. ordered triple 3. Gaussian

4. independent, dependent 5. nonsquare 6. three-dimensional

7. partial fraction decomposition

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580 Chapter 7 Linear Systems and Matrices

4. (a)

is a solution.

(b)

is not a solution.��332 , �10, 10�

4��332 � � 7��10� �

? 6 No

� 10 � 10 �?

0 Yes

�4��332 � � ��10� � 8�10� �? �6 No

��2, �2, 2�

4��2� � 7��2� �?

6 Yes

� 2 � 2 �?

0 Yes

�4��2� � ��2� � 8�2� �? �6 Yes (c)

is not a solution.

(d)

is a solution.��112 , �4, 4�

4��112 � � 7��4� �

? 6 Yes

�4 � 4 �?

0 Yes

�4��112 � � ��4� � 8�4� �? �6 Yes

�18, �1

2, 12� 4�1

8� � 7��12� �

? 6 No

�12 �

12 �

? 0 Yes

�4�18� � ��1

2� � 8�12� �

? �6 No

5.

Back-substitute into Equation 2:

Back-substitute and into Equation 1:

Answer: �2, �2, 2�

x � 2

2x � 4

2x � ��2� � 5�2� � 16

y � �2z � 2

y � �2

y � 2�2� � 2

z � 2

Equation 1Equation 2Equation 3

2x � yy

5z2zz

1622

7.

Back-substitute into Equation 2:

Back-substitute and into Equation 1:

Answer: �3, 10, 2�

x � 3

2x � 6

2x � 10 � 3(2) � 10

z � 2y � 10

y � 10

y � 2 � 12

z � 2

Equation 1Equation 2Equation 3

2x � yy

3zzz

10122

6.

Back-substitute into Equation 2:

Back-substitute and intoEquation 1:

Answer: ��8, �11�3, �2� x � �8

4x � �32

4x � 3��11�3� � 2��2� � �17

y � �11�3z � �2

y � �11�3

6y � �22

6y � 5��2� � �12

z � �2

Equation 1Equation 2Equation 3

4x � 3y6y

2z5zz

�17�12�2

8.

Answer: �17, �11, �3�

x � �11 � 2��3� � 22 � 17

3y � 8��3� � 9 � �33 ⇒ y � �11

z � �3

x � y3y

2z8zz

22�9�3

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Page 34: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 581

9.

Back-substitute into Equation 2:

Back-substitute and into Equation 1:

Answer: �12, �2, 2�

x �12

4x � 2

4x � 2��2� � 2 � 8

z � 2y � �2

y � �2

�y � 2 � 4

z � 2

Equation 1Equation 2Equation 3

4x �

2yy

zzz

842

10.

Back-substitute in Equation 2:

Back-substitute in Equation 1:

Answer: ��2, � 103 , �4�

5x � 8��4� � 22 ⇒ x � �2

z � �4

3y � 5��4� � 10 ⇒ y � � 103

z � �4

5x 3y

8z5zz

2210

�4

11.

Add Equation 1 to Equation 2.

New Equation 2

This is the first step in putting the system inrow-echelon form.

x

2x

� 2yy

3z2z3z

590

y � 2z � 9

Equation 1Equation 2Equation 3

x�x2x

2y3y

3z5z3z

540

12.

Add times Equation 1 to Equation 3.

This is a step in putting the system in row-echelon form.

4y � 9z � �10

�2

x�x2x

2y3y

3z5z3z

540

13.

Add times Equation 1 to Equation 2.

New Equation 2

This is the first step in putting the system in row-echelon form.

3y � z � �2

�2

Equation 1Equation 2Equation 3

x2x3x

2yyy

z3z4z

101

14.

Add times Equation 1 to Equation 3.

New Equation 3

This eliminates the term 3 in Equation 3.x

�10y � 13z � �5

�3

Equation 1Equation 2Equation 3

x�x3x

2y4y4y

4zzz

2�2

1

15.

Answer: �1, 2, 3�

x � 2 � 3 � 6 ⇒ x � 1

�3y � 3 � �9 ⇒ y � 2

�3z � �9 ⇒ z � 3

��1� Eq. 2 � Eq. 3x �

y3y

zz

3z

6�9�9

��2� Eq. 1 � Eq. 2��3� Eq. 1 � Eq. 3

x �

y3y3y

zz

4z

6�9

�18

Equation 1Equation 2Equation 3

x2x3x

yy

zzz

630

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582 Chapter 7 Linear Systems and Matrices

16.

Answer: �0, 4, �2�

x � 4 � 2 � 2 ⇒ x � 0

12y � 9��2� � 30 ⇒ y � 4

�7z � 14 ⇒ z � �2

Eq. 2 � Eq. 3x � y

12y�

z9z

�7z

23014

3 Eq. 24 Eq. 3

x � y12y

�12y

z9z

16z

230

�16

Eq. 1 � Eq. 2�4 Eq. 1 � Eq. 3

x � y4y

�3y

z3z4z

210

�4

Equation 1Equation 2Equation 3

x�x4x

y3yy

z2z

284

17.

Answer: ��4, 8, 5�

x � 5 � 1 ⇒ x � �4

3y � 5�5� � �1 ⇒ y � 8

��1� Eq. 2 � Eq. 3x

3y

z5zz

1�1

5

��5� Eq. 1 � Eq. 2x

3y3y

z5z4z

1�1

4

�12� Eq. 1 x

5x � 3y3y

z

4z

144

Equation 1Equation 2Equation 3

2x5x � 3y

3y

2z

4z

244

18.

Answer: �6, �3, 0�

x � 6⇒x � ��3� � 3

y � �3⇒�2y � 3�0� � 6

z � 0⇒�5z � 0

3 Eq. 2 � Eq. 3x �

y2y �

3z5z

360

Interchangethe equations.x �

y2y6y

3z4z

36

�18

��2� Eq. 1 � Eq. 3x �

y6y2y

4z3z

3�18

6

13 �New Eq. 1� x

2x

� y6y �

4z3z

3�18

12

Interchange equations1 and 2.3x

2x

� 3y6y �

4z3z

9�18

12

Equation 1Equation 2Equation 3

3x2x

6y3y

4z

3z

�189

12

19.

Answer: �2, �3, �2�

x � ��3� � ��2� � �3 ⇒ x � 2

⇒ z � �2

⇒ �7z � 14

y � �3 ⇒ �3��3� � 7z � 23

��2� Eq. 1 � Eq. 2��4� Eq. 1 � Eq. 3

x � y�5y�3y

z

7z

�31523

x2x4x

y3yy

z2z3z

�39

11

4x2xx

y3yy

3z2zz

119

�3

Equation 1Equation 2Equation 3

Interchange equations1 and 3

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Section 7.3 Multivariable Linear Systems 583

20.

Answer: �12, � 3

2, 1�2x � 4�� 3

2� � 1 � �4 ⇒ x �12

�40y � 25�1� � 85 ⇒ y � � 32

�29z � �29 ⇒ z � 1

�Eq. 2 � Eq. 32x � 4y

�40y�

z25z

�29z

�485

�29

5 Eq. 24 Eq. 3

2x � 4y�40y�40y

z25z4z

�48556

�Eq. 1 � Eq. 2�2 Eq. � Eq. 3

2x � 4y�8y

�10y

z5zz

�41714

Equation 1Equation 2Equation 3

2x2x4x

4y4y2y

z6zz

�4136

21.

Inconsistent; no solution.

�Eq. 2 � Eq. 3x �

y5y

2z10z

0

3�810

�3 Eq. 1 � Eq. 2�2 Eq. 1 � Eq. 3

x �

y5y5y

2z10z10z

3�8

2

Interchangethe equations. x

3x2x

y2y3y

2z4z6z

318

22.

Inconsistent; no solution.

�12 Eq. 2 � Eq. 3

x � 11y52y

4z18z

0

3�12

7

�5 Eq. 1 � Eq. 2�2 Eq. 1 � Eq. 3

x � 11y52y26y

4z18z9z

3�12

1

Interchange rows x5x2x

11y3y4y

4z2zz

337

Equation 1Equation 2Equation 3

5x2xx

3y4y

11y

2zz

4z

373

23.

Answer: �1, �32, 12�

x � 3��32� � 7�1

2� � 2 ⇒ x � 1

84y � 180� 12� � �36 ⇒ y � �

32

2z � 1 ⇒ z �12

�Eq. 2 � Eq. 3x � 3y

84y�

7z180z

2z

2�36

1

6 Eq. 23.5 Eq. 3

x � 3y84y84y

7z180z182z

2�36�35

�3 Eq. 1 � Eq. 2�5 Eq. 1 � Eq. 3

x � 3y14y24y

7z30z52z

2�6

�10

�Eq. 3 � Eq. 1 x3x5x

3y5y9y

7z9z

17z

200

2 Eq. 16x3x5x

6y5y9y

10z9z

17z

200

3x3x5x

3y5y9y

5z9z

17z

100

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Page 37: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

584 Chapter 7 Linear Systems and Matrices

24.

Answer: � 310, 25, 0�

2x �25 � 3�0� � 1 ⇒ x �

310

5y � 5�0� � 2 ⇒ y � 25

4z � 0 ⇒ z � 0

�Eq. 2 � Eq. 32x � y

5y�

3z5z4z

120

�Eq. 1 � Eq. 2�3 Eq. 1 � Eq. 3

2x � y5y5y

3z5z9z

122

Equation 1Equation 2Equation 3

2x2x6x

y6y8y

3z8z

18z

135

25.

Answer: ��3a � 10, 5a � 7, a�

x � �3a � 10

y � 5a � 7

Let z � a, then:

�2 Eq. 2 � Eq. 1x y

3z5z

10�7

�13 Eq. 2x � 2y

y�

7z5z

�4�7

Eq. 2 � Eq. 3x �

2y3y

7z15z

0

�4210

�2 Eq. 1 � Eq. 2�3 Eq. 1 � Eq. 3

x �

2y3y3y

7z15z15z

�421

�21

Equation 1Equation 2Equation 3

x2x3x

2yy

9y

7zz

36z

�413

�33

26.

Answer: �� 12a �

52, 4a � 1, a�

x � � 12a �

52

y � 4a � 1

z � a

�Eq. 2 � Eq. 12x y

z4z

5�1

�12 Eq. 2

2 Eq. 2 � Eq. 32x � y

y�

3z4z0

4�1

0

�2 Eq. 1 � Eq. 2Eq. 1 � Eq. 3

2x � y�2y

4y

3z8z

16z

42

�4

Equation 1Equation 2Equation 3

2x4x

�2x

y

3y

3z2z

13z

410

�8

27.

Answer: �1, �1, 2�

x � y � 2z � 6 ⇒ x � 6 � ��1� � 2�2� � 1

y � z � �3 ⇒ y � 2 � 3 � �1

��2� Eq. 2 � Eq. 3x � yy

2zzz

6�3

2

�13� Eq. 2 x � y

y2y

2zzz

6�3�4

��2� Eq. 1 � Eq. 2��1� Eq. 1 � Eq. 3x � y

3y2y

2z3zz

6�9�4

Equation 1Equation 2Equation 3

x2xx

yyy

2zzz

632

28.

Answer: �0, 1, �2�

x � y � z � 3 ⇒ x � 3 � 1 � 2 � 0

�3y � z � �5 ⇒ �3y � 2 � 5 � �3 ⇒ y � 1

z � �2

��2� Eq. 1 � Eq. 2��1� Eq. 1 � Eq. 3x � y

�3y�

zz

�z

3�5

2

Equation 1Equation 2Equation 3

x2xx

yyy

zz

2z

315

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Page 38: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 585

29.

Let then:

Answer: ��a � 3, a � 1, a�

x � �a � 3

y � a � 1

z � a,

x y

zz

31

x � yy

2zz0

210

��1� Eq. 1 � Eq. 2��5� Eq. 1 � Eq. 3

x �

y3y3y

2z3z3z

23

�3

�13� Equation 1 x

x5x

y2y8y

2zz

13z

257

Equation 1Equation 2Equation 3

3xx

5x

3y2y8y

6zz

13z

657

31.

No solution; inconsistent.

�Eq. 2 � Eq. 3x � 2y

5y�

3z7z0

4�12

6

�3 Eq. 1 � Eq. 2�1 Eq. 1 � Eq. 3

x � 2y5y5y

3z7z7z

4�12�6

Equation 1Equation 2Equation 3

x3xx

2yy

3y

3z2z4z

40

�2

33.

No solution; inconsistent.

Eq. 2 � Eq. 3x

y�

4z6z0

192

�Eq. 1 � Eq. 2�2 Eq. � Eq. 3

x

yy

4z6z6z

19

�7

xx

2x�

yy

4z10z2z

110

�5

30.

Answer: ��4a � 13, � 152 a �

452 , a�

x � �4a � 13

y � � 152 a �

452

z � a

�Eq. 2 � Eq. 3x

�2y

4z15z

0

13�45

0

�4 Eq. 1 � Eq. 2�2 Eq. 1 � Eq. 3

x

�2y�2y

4z15z15z

13�45�45

Equation 1Equation 2Equation 3

x4x2x

2y2y

4zz

7z

137

�19

32.

No solution; inconsistent.

�Eq. 2 � Eq. 3�x � 3y

10y�

zz0

49

11

4 Eq. 1 � Eq. 22 Eq. 1 � Eq. 3

�x � 3y10y10y

zzz

49

20

Equation 1Equation 2Equation 3

�x4x2x

3y2y4y

z5z3z

4�712

34.

No solution; inconsistent.

Eq. 2 � Eq. 3x � y

y�

5z9z0

�35

�3

�3 Eq. 1 � Eq. 23 Eq. 1 � Eq. 3

x � yy

�y

5z9z9z

�35

�8

Interchange theequations x

3x�3x

y2y2y

5z6z6z

�3�4

1

Equation 1Equation 2Equation 3

3x�3x

x

2y2yy

6z6z5z

�41

�3

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Page 39: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

586 Chapter 7 Linear Systems and Matrices

35.

Answer: ��1, 1, 0�

x � 2y � z � 1 ⇒ x � 1 � 2 � �1

y � 1 � 0 � 1

z � 0

��3� Eq. 2 � Eq. 3x � 2y

y�

z12z52z

11

0

��14� Eq. 2

��1� Eq. 3 x � 2y

y3y

z12zz

113

��1� Eq. 1 � Eq. 2��2� Eq. 1 � Eq. 3x � 2y

�4y�3y

z2zz

1�4�3

Equation 1Equation 2Equation 3

xx

2x

2y2yy

z3zz

1�3�1

36.

Answer: �0, �12, 1�

x � 2y � z � 2 ⇒ x � 2 � 2��12� � 1 � 0

6y � 5z � �8 ⇒ 6y � �8 � 5 � �3 ⇒ y � �12

z � 1

��53� Eq. 2 � Eq. 3

x � 2y

6y

z

5z133 z

2

�8133

��2� Eq. 1 � Eq. 2��5� Eq. 1 � Eq. 3x � 2y

6y10y

z5z4z

2�8�9

Equation 1Equation 2Equation 3

x2x5x

2y2y

z3zz

2�4

1

37.

Let Then and

Answer: �14 a, 21

8 a � 1, a�x �

14 ay �

218 a � 1z � a.

2 Eq. 2 � Eq. 1x

y

14 z

218 z

0

�1

18 Eq. 2x � 2y

y�

5z218 z

2�1

�4 Eq. 1 � Eq. 2x � 2y8y

5z21z

2�8

x4x

� 2y �

5zz

20

39.

Answer: ��3

2a �

1

2, �

2

3a � 1, a

x � �32

a �12

y � �23

a � 1

Let z � a, then:

Eq. 2 � Eq. 12x 3y

3z2z

13

2 Eq. 1 � Eq. 22x � 3y3y

z2z

�23

2x�4x

3y9y

� z �

�27

38.

To avoid fractions, let then:

Answer: �9a, �35a, 67a�

x � 9a

x � 6��35a� � 3�67a� � 0

y � �35a

�67y � 35�67a� � 0

z � 67a,

2 Eq. 2 � Eq. 1�12 Eq. 1 � Eq. 2x � 6y

�67y�

3z35z

00

Interchangethe equations.23x

12x�

4y5y

zz

00

40.

Infinite number of solutions. Let Then

Answer: or �137

a, 487

a, a�13a, 48a, 7a�

x � y � 5z � 48a � 5�7a� � 13a

y �96z14

�96�7a�

14� 48a

z � 7a.

�19 Eq. 1 � Eq. 2x � y14y

5z96z

00

2 Eq. 1 � Eq. 2 x19x

y5y

5zz

00

Equation 1Equation 210x

19x�

3y5y

2zz

00

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Page 40: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 587

41.

Let then and

Answer: ��5a � 3, �a � 5, a�

x � �5a � 3.y � �a � 5,z � a,

3 Eq. 2 � Eq. 1x y �

5zz

3�5

12 Eq. 2x � 3y

y�

2zz

18�5

�5 Eq. 1 � Eq. 2x � 3y2y

2z2z

18�10

Equation 1Equation 2 x

5x�

3y13y

2z12z

1880

42.

Let then:

Answer: ��38a �

14, �3

4a �52, a�

2x �34a � �1

2 ⇒ x � �38a �

14

12y � 9a � 30 ⇒ y � �34 a �

52

z � a,

�14 Eq. 2 � Eq. 12x

12y�

34z9z

�12

30

�2 Eq. 1 � Eq. 22x � 3y12y

3z9z

730

Equation 1Equation 22x

4x�

3y18y

3z15z

744

43.

Answer: �12, 0, 12, 32�

x � y � 2z � w � 0 ⇒ x � �2�12� �

32 �

12

2y � 2z � �1 ⇒ 2y � �1 � 2�12� � 0 ⇒ y � 0

�2z � 2w � 2 ⇒ �2z � 2 � 2�32� � �1 ⇒ z �

12

w �32

��32� Eq. 2 � Eq. 3

��1� Eq. 2 � Eq. 4

x � y2y

2z2z

�2z

w

w2w

0�1

32

2

InterchangeEq. 2 and 3

x � y2y3y2y

2z2z3z4z

w

w2w

0�1

01

��2� Eq. 1 � Eq. 2��1� Eq. 1 � Eq. 3��1� Eq. 1 � Eq. 4

x � y

3y2y2y

2z3z2z4z

ww

2w

00

�11

Equation 1Equation 2Equation 3Equation 4

x

2xxx

yyyy

2zz

2z

wwww

00

�11

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Page 41: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

588 Chapter 7 Linear Systems and Matrices

44.

Answer: �1, 0, �1, 2�

x � 2y � z � 2w � 6 ⇒ x � 6 � 2�0� � ��1� � 2�2� � 1

y � 3z � 3w � �9 ⇒ y � �9 � 3��1� � 3�2� � 0

2z � w � �4 ⇒ 2z � �4 � 2 � �2 ⇒ z � �1

w � 2

�2� Eq. 3 � Eq. 4

x � 2yy

z3z2z

2w3ww

2w

6�9�4

4

��1� Eq. 2 � Eq. 4

x � 2yy

z3z2z

�4z

2w3ww

4w

6�9�412

��2� Eq. 1 � Eq. 2��1� Eq. 1 � Eq. 3

x � 2yy

y

z3z2zz

2w3www

6�9�4

3

Equation 1Equation 2Equation 3Equation 4

x

2xx

2y3y2yy

zzzz

2wwww

6323

45. Let

Answer: �1, �2, �1�

x �1X

� 1, y �1Y

� �2, z �1Z

� �1

X � 2Y � 3Z � 3 ⇒ X � 3 � 2��1

2 � 3��1� � 1

2Y � 2Z � 1 ⇒ 2Y � 1 � 2Z � �1 ⇒ Y � �1

2

Eq. 2 � Eq. 3X � 2Y

2Y�

3Z2ZZ

31

�1

��12� Eq. 2X � 2Y

2Y�2Y

3Z2Z3Z

31

�2

��1� Eq. 1 � Eq. 2��2� Eq. 1 � Eq. 3

X � 2Y�4Y�2Y

3Z4Z3Z

3�2�2

Equation 1Equation 2Equation 3

XX

2X

2Y2Y2Y

3ZZ

3Z

314

X �1x, Y �

1y, Z �

1z. 46. Let

Answer: ��1, �1, �1�

x �1X

� y � z � �1

Z � �1, Y � �1, X � �1

�� 310� Eq. 2 � Eq. 3

X � 2Y�10Y

2Z9Z

�1710Z

�11

1710

��4� Eq. 1 � Eq. 2��2� Eq. 1 � Eq. 3

X � 2Y�10Y�3Y

2Z9ZZ

�112

InterchangeEq. 1 and 2 X

4X2X

2Y2YY

2ZZ

3Z

�1�3

0

Equation 1Equation 2Equation 3

4XX

2X

2Y2YY

Z2Z3Z

�3�1

0

X �1x, Y �

1y, Z �

1z.

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Page 42: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 589

47. Let

Answer: �1, �1, 2�

x �1X

� 1, y �1Y

� �1, z �1Z

� 2

X � 2Y � 2Z � �2 ⇒ X � �2 � 2��1� � 2�1

2 � 1

�5Y � 6Z � 8 ⇒ �5Y � 8 � 6�1

2 � 5 ⇒ Y � �1

Z �12

��35�Eq. 2 � Eq. 3

X � 2Y�5Y

2Z6Z

325 Z

�28

165

��2� Eq. 1 � Eq. 2��3� Eq. 1 � Eq. 3

X � 2Y�5Y�3Y

2Z6Z

10Z

�288

InterchangeEq. 1 and 2 X

2X3X

2YY

3Y

2Z2Z4Z

�242

Equation 1Equation 2Equation 3

2XX

3X

Y2Y3Y

2Z2Z4Z

4�2

2

X �1x, Y �

1y, Z �

1z.

48. Let

Answer: �1, �1, 2�

x � 1, y � �1, z � 2

Z �12

, Y � �1, X � 1

��1� Eq. 2�4� Eq. 2 � Eq. 3

X � YY

2Z6Z

22Z

12

11

��2� Eq. 1 � Eq. 2��3� Eq. 1 � Eq. 3

X � Y�Y

�4Y

2Z6Z2Z

1�2

3

Equation 1Equation 2Equation 3

X2X3X

YYY

2Z2Z4Z

106

X �1x, Y �

1y, Z �

1z. 49. There are an infinite number of linear systems that

have as their solution. One such system isas follows:

3�4�

�4�

��4�

��1�

2��1�

��1�

�2�

�2�

3�2�

9

0

1

3x

x

�x

y

2y

y

z

z

3z

� 9

� 0

� 1

�4, �1, 2�

50. There are an infinite number of linear systems thathave as their solution. One such system is:

2�1� � 2

2��2� � 1 � �3

1��5� � 1��2� � 1 � �6 ⇒

��5, �2, 1�51. There are an infinite numbers of linear systems that

have as their solution. One such system is:

4z � 7 ⇒ 4�74� � 7

4y � 8z � 12 ⇒ 4��12� � 8�7

4� � 12

x � 2y � 4z � 9 ⇒ 1�3� � 2��12� � 4�7

4� � 9

�3, �12, 74�

x � y � z2y � z

2z

�6�3

2

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Page 43: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

590 Chapter 7 Linear Systems and Matrices

52. There are an infinite number of linear systems that have as their solution. One such system is:

�2�� 32� � 5�4� � 3��7� � 44 ⇒

4��32� � 2�4� � ��7� � �5 ⇒

2�� 32� � 4 � ��7� � 8 ⇒

�� 32, 4, �7�

2x

4x

�2x

y

2y

5y

z

z

3z

8

�5

44

53.

x y

4

22

6 64

6

z

�6, 0, 0�, �0, 4, 0�, �0, 0, 3�, �4, 0, 1�

2x � 3y � 4z � 12

55.

x y

4

6 644

6

z

�2, 0, 0�, �0, 4, 0�, �0, 0, 4�, �0, 2, 2�

2x � y � z � 4

57.7

x2 � 14x�

7

x�x � 14��

A

x�

B

x � 14

59.12

x3 � 10x2�

12

x2�x � 10��

A

x�

B

x2�

C

x � 10

61.4x2 � 3�x � 5�3 �

A�x � 5� �

B�x � 5�2 �

C�x � 5�3

54.

x y

4

2

22

6 644

6

z

�6, 0, 0�, �0, 6, 0�, �0, 0, 6�, �1, 1, 4�

x � y � z � 6

56.

x y

4

6 64

6

2

z

�6, 0, 0�, �0, 3, 0�, �0, 0, 3�, �2, 1, 1�

x � 2y � 2z � 6

58.x � 2

x2 � 4x � 3�

A

x � 3�

B

x � 1

60.x2 � 3x � 24x3 � 11x2 �

x2 � 3x � 2x2�4x � 11� �

Ax

�Bx2 �

C4x � 11

62.6x � 5

�x � 2�4 �A

x � 2�

B�x � 2�2 �

C�x � 2�3 �

D�x � 2�4

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Page 44: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 591

65.

1x2 � x

�1x

��1

x � 1�

1x

�1

x � 1

A � BA

01 ⇒ B � �1

1 � A�x � 1� � Bx � �A � B�x � A

1

x2 � x�

1x�x � 1� �

Ax

�B

x � 1

63.

1x2 � 1

��1�2x � 1

�1�2

x � 1�

12�

1x � 1

�1

x � 1�2B � 1 ⇒ B �

12 ⇒ A � �

12

A�A

BB

01

1 � A�x � 1� � B�x � 1� � �A � B�x � �B � A�

1

x2 � 1�

Ax � 1

�B

x � 164.

Let :

1

4x2 � 9�

1

6�1

2x � 3�

1

2x � 3�

Let x �3

2: 1 � 6B ⇒ B �

1

6

1 � �6A ⇒ A � �1

6x � �

3

2

1 � A�2x � 3� � B�2x � 3�

1

4x2 � 9�

A

2x � 3�

B

2x � 3

66.

Let :

Let :

3

x2 � 3x�

1

x � 3�

1

x

3 � �3B ⇒ B � �1x � 0

3 � 3A ⇒ A � 1x � 3

3 � Ax � B�x � 3�

3

x2 � 3x�

A

x � 3�

B

x

67.

12x2 � x

��2

2x � 1�

1x

�1x

�2

2x � 1

A � 2BB

01 ⇒ A � �2

1 � Ax � B�2x � 1� � �A � 2B�x � B

12x2 � x

�1

x�2x � 1� �A

2x � 1�

Bx

68.

Let :

Let :

5

x2 � x � 6�

1

x � 2�

1

x � 3

5 � 5B ⇒ B � 1x � 2

5 � �5A ⇒ A � �1x � �3

5 � A�x � 2� � B�x � 3�

5

x2 � x � 6�

A

x � 3�

B

x � 2

69.

and

5 � x2x2 � x � 1

�3

2x � 1�

�2x � 1

A � 3��1 � 2B� � B � 5 ⇒ B � �2

AA

2BB

�15 ⇒ A � �1 � 2B

� �A � 2B�x � �A � B�

5 � x � A�x � 1� � B�2x � 1�

�A

2x � 1�

Bx � 1

5 � x2x2 � x � 1

�5 � x

�2x � 1��x � 1� 70.

Solving for A and B,

x � 2x2 � 4x � 3

�5�2

x � 3�

3�2x � 1

A �52, B � �

32

A � BA � 3B

1�2

(A � B�x � �A � 3B� � x � 2

A(x � 1� � B�x � 3� � x � 2

x � 2x2 � 4x � 3

�A

x � 3�

Bx � 1

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592 Chapter 7 Linear Systems and Matrices

71.

x2 � 12x � 12x3 � 4x

��3x

��1

x � 2�

5x � 2

2C � 10 ⇒ C � 5 ⇒ B � �1

BB

CC

46

A

4A

B2B

C2C

11212 ⇒ A � �3

� �A � B � C�x2 � ��2B � 2C�x � ��4A�

x2 � 12x � 12 � A�x � 2��x � 2� � Bx�x � 2� � Cx�x � 2�

x2 � 12x � 12

x3 � 4x�

x2 � 12x � 12x�x � 2��x � 2� �

Ax

�B

x � 2�

Cx � 2

72.

Solving,

x2 � 12x � 9x3 � 9x

�1x

�2

x � 3�

2x � 3

A � 1, B � 2 and C � �2

A

�9A

� B3B

C3C

112

�9

A�x2 � 9� � Bx�x � 3� � Cx�x � 3� � x2 � 12x � 9

�Ax

�B

x � 3�

Cx � 3

x2 � 12x � 9x3 � 9x

�x2 � 12x � 9

x�x � 3��x � 3�

73.

4x2 � 2x � 1x2�x � 1� �

3x

��1x2 �

1x � 1

B � �1 ⇒ A � 3 ⇒ C � 1

AA � B

B

� C �

42

�1

� �A � C�x2 � �A � B�x � B

4x2 � 2x � 1 � Ax�x � 1� � B�x � 1� � Cx2

4x2 � 2x � 1

x2�x � 1� �Ax

�Bx2 �

Cx � 1

74.

Let :

2x � 3

�x � 1�2�

2

x � 1�

1

�x � 1�2

2 � A

�3 � �A � 1

Let x � 0: �3 � �A � B

�1 � Bx � 1

2x � 3 � A�x � 1� � B

2x � 3

�x � 1�2�

A

x � 1�

B

�x � 1�2

75.

27 � 7xx�x � 3�2 �

3x

��3

x � 3�

2�x � 3�2

A � 3 ⇒ B � �3 ⇒ C � �7 � 18 � 9 � 2

A6A9A

B3B � C

0�727

� �A � B�x2 � ��6A � 3B � C�x � 9A 27 � 7x � A�x � 3�2 � Bx�x � 3� � Cx

27 � 7xx�x � 3�2 �

Ax

�B

x � 3�

C�x � 3�2

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Section 7.3 Multivariable Linear Systems 593

76.

Solving,

x2 � x � 2x�x � 1�2 �

2x

�1

x � 1�

2�x � 1�2

A � 2, B � �1 and C � 2

A�2A

A

BB � C

1�1

2

A�x � 1�2 � Bx�x � 1� � Cx � x2 � x � 2

x2 � x � 2

x�x � 1�2�

A

x�

B

x � 1�

C

�x � 1�277.

2x3 � x2 � x � 5x2 � 3x � 2

� 2x � 7 �1

x � 1�

17x � 2

A � 1 ⇒ B � 17

A2A

BB

1819

� �A � B�x � �2A � B�

18x � 19 � A�x � 2� � B�x � 1�

18x � 19

�x � 1��x � 2� �A

x � 1�

Bx � 2

2x3 � x2 � x � 5

x2 � 3x � 2� 2x � 7 �

18x � 19�x � 1��x � 2�

78.

Let

Let

x3 � 2x2 � x � 1x2 � 3x � 4

� x � 1 �27

5�x � 4� �3

5�x � 1�

x � �4: �27 � �5A ⇒ A �275

x � 1: 3 � 5B ⇒ B �35

6x � 3 � A�x � 1� � B�x � 4�

6x � 3�x � 4��x � 1� �

Ax � 4

�B

x � 1

x3 � 2x2 � x � 1x2 � 3x � 4

� x � 1 �6x � 3

�x � 4��x � 1�

79.

x4

�x � 1�3 �6

x � 1�

4�x � 1�2 �

1�x � 1�3 � x � 3

A � 6 ⇒ B � �8 � 2�6� � 4 ⇒ C � 3 � 6 � 4 � 1

A�2A

A�

BB � C

6�8

3

� Ax2 � ��2A � B�x � �A � B � C� 6x2 � 8x � 3 � A�x � 1�2 � B�x � 1� � C

6x2 � 8x � 3

�x � 1�3 �A

x � 1�

B�x � 1�2 �

C�x � 1�3

x4

�x � 1�3 � x � 3 �6x2 � 8x � 3

�x � 1�3

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594 Chapter 7 Linear Systems and Matrices

80.

Answer:x2

�34

�3

2�2x � 1� �1

�2x � 1�2 �1

4�2x � 1�3

A � B � C � 3 ⇒ C � 1

�4A � 2B � �16 ⇒ 2B � 8 ⇒ B � 4

4A � 24 ⇒ A � 6

24x2 � 16x � 3 � A�2x � 1�2 � B�2x � 1� � C � A4x2 � ��4A � 2B�x � �A � B � C�

24x2 � 16x � 34�2x � 1�3 �

14�

A2x � 1

�B

�2x � 1�2 �C

�2x � 1�3�

4x4

�2x � 1�3 �x2

�34

�24x2 � 16x � 3

4�2x � 1�3

81.

Vertical asymptotes: Vertical asymptotes:and and

The combination of the vertical asymptotes of theterms of the decompositions are the same as thevertical asymptotes of the rational function.

x � 4x � 0x � 4x � 0

–6 2 8 10

–8

2

8

x

y = 3x

y = 3x

y = 2x − 4

y = 2x − 4

y

–6 –4 2 8 10

–8

2

4

6

8

x

y

y �3

x, y � �

2

x � 4y �

x � 12

x�x � 4�

x � 12

x�x � 4��

3

x�

2

x � 4

⇒ A � 3, B � �2 A�4A

� B �

1�12

x � 12 � A(x � 4� � Bx

x � 12

x�x � 4��

A

x�

B

x � 482.

Let

Let

Vertical asymptotes: Vertical asymptotes:

The combination of the vertical asymptotes of the terms of the decompositions are the same as the vertical asymptotes of the rational function.

x � 3, x � �3x � ±3

–4 2 4 6 8

–8

–6

–4

6

8

x

y = 5x + 3

y = 5x + 3

y = 3x − 3

y = 3x − 3

y

–4 4 6 8

–8

–6

–4

4

6

8

x

y

y �3

x � 3, y �

5

x � 3y �

2�4x � 3�x2 � 9

2�4x � 3�x2 � 9

�3

x � 3�

5

x � 3

x � �3: 1�30 � �6B ⇒ B � 5

x � 3: 18 � 6A ⇒ A � 3

2�4x � 3� � A(x � 3� � B�x � 3�

2�4x � 3�x2 � 9

�A

x � 3�

B

x � 3

83.

Solving the system,

Thus,

� �16t2 � 144.

s �12��32�t2 � �0�t � 144

a � �32, v0 � 0, s0 � 144.

128800

12 a2a92 a

v0

2v0

3v0

s0

s0

s0

⇒ ⇒ ⇒

a2a9a

2v0

2v0

6v0

2s0

s0

2s0

256800

�1, 128�, �2, 80�, �3, 0�

s �12at2 � v0t � s0 84.

Solving the system,

Thus,

� �16t2 � 64t.

s �12��32�t2 � 64t � 0

s0 � 0.a � �32, v0 � 64,

486448

12a2a92a

v0

2v0

3v0

s0

s0

s0

⇒ ⇒ ⇒

a2a9a

2v0

2v0

6v0

2s0

s0

2s0

966496

�1, 48�, �2, 64�, �3, 48�

s �12at2 � v0t � s0

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Page 48: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 595

87. passing through

Answer:

The equation of the parabola is y �12x2 � 2x.

a �12, b � �2, c � 0

� �0, 0�: 0�2, �2�: �2

�4, 0�: 0

4a4a

16a

2b2b4b

ccc

⇒ ⇒ ⇒

c�1

0

�4a � 2b2a � b4a � b

−15

−10

15

10�0, 0�, �2, �2�, �4, 0�y � ax2 � bx � c

88. passing through

Answer:

The equation of the parabola is y � �x2 � 2x � 3.

a � �1, b � 2, c � 3

��0, 3�: 3�1, 4�: 4�2, 3�: 3

a4a

b2b

ccc

⇒ ⇒

10

a � b2a � b

−15

−10

15

10�0, 3�, �1, 4�, �2, 3�y � ax2 � bx � c

89. passing through

Answer:

The equation of the parabola is y � x2 � 6x � 8.

a � 1, b � �6, c � 8

� �2, 0�: 0�3, �1�: �1

�4, 0�: 0

4a9a

16a

2b3b4b

ccc

⇒ ⇒ ⇒

c�1

0

�4a � 2b5a � b

12a � 2b

−15

−10

15

10�2, 0�, �3, �1�, �4, 0�y � ax2 � bx � c

90. passing through

Answer:

The equation of the parabola is y � �2x2 � 5x.

a � �2, b � 5, c � 0

��1, 3�: 3�2, 2�: 2�3, �3�: �3

a4a9a

b2b3b

ccc

⇒ ⇒

�1�6

3a � b 8a � 2b

−15

−10

15

10�1, 3�, �2, 2�, �3, �3�y � ax2 � bx � c

85.

Solving the system,

Thus,

� �16t2 � 32t � 500

s �12 ��32�t2 � 32t � 500

a � �32, v0 � �32, s0 � 500

�452372260

12 a2a92 a

v0

2v0

3v0

s0

s0

s0

⇒ ⇒ ⇒

a2a9a

2v0

2v0

6v0

2s0

s0

2s0

904372520

�1, 452�, �2, 372�, �3, 260�

s �12 at2 � v0t � a0 86.

Solving the system

Thus,

� �16t2 � 16t � 132.

s �12��32�t2 � 16t � 132

s0 � 132.a � �32, s0 � 16,

�13210036

12a2a92a

v0

2v0

3v0

s0

s0

s0

⇒ ⇒ ⇒

a2a9a

2v0

2v0

6v0

2s0

s0

2s0

26410072

�1, 132�, �2, 100�, �3, 36�

s �12at2 � v0t � s0

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596 Chapter 7 Linear Systems and Matrices

91. passing through

The equation of the circle is

To graph, let y1 � �4x � x2 and y2 � ��4x � x2.

x2 � y2 � 4x � 0.

�4, 0�: 16 � 4D � 2E � F � 0 ⇒ D � �4 and E � 0

�2, 2�: 18 � 2D � 2E � F � 0 ⇒ D � E � �4

�0, 0�: 16 � 4D � 2E � F � 0

�0, 0�, �2, 2�, �4, 0�x2 � y2 � Dx � Ey � F � 0

−6

−4

6

4

92. passes through

The equation of the circle is

To graph, complete the square first, then solve for

Let and y2 � 3 � �9 � x2.y1 � 3 � �9 � x2

y � 3 ± �9 � x2

y � 3 � ±�9 � x2

�y � 3�2 � 9 � x2

x2 � �y � 3�2 � 9

−9

−2

9

10x2 � y2 � 6y � 9 � 9

y.

x2 � y2 � 6y � 0.

��0, 0�:�0, 6�:�3, 3�: 18 �

363D

6E3E

F � 0F � 0F � 0

⇒ ⇒

E � �6D � 0

�0, 0�, �0, 6�, �3, 3�.x2 � y2 � Dx � Ey � F � 0

93. passes through

Answer:

The equation of the circle is To graph, complete the squares first, then solve for

Let y1 � 4 � �25 � �x � 3�2 and y2 � 4 � �25 � �x � 3�2.

y � 4 ±�25 � �x � 3�2

y � 4 � ±�25 � �x � 3�2

�y � 4�2 � 25 � �x � 3�2

�x � 3�2 � �y � 4�2 � 25

�x2 � 6x � 9� � �y2 � 8y � 16� � 0 � 9 � 16

y.x2 � y2 � 6x � 8y � 0.

D � 6, E � �8, F � 0

��6, �8�: 100 � 6D � 8E � F � 0 ⇒ 100 � �6D � 8E � F

��2, �4�: 120 � 2D � 4E � F � 0 ⇒ 20 � �2D � 4E � F

��3, �1�: 110 � 3D � 8E � F � 0 ⇒ 10 � �3D � 8E � F

−9

−2

9

10��3, �1�, �2, 4�, ��6, 8�.x2 � y2 � Dx � Ey � F � 0

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Section 7.3 Multivariable Linear Systems 597

95. Let amount at 8%.

Let amount at 9%.

Let amount at 10%.

Solving the system, at 8%,at 9%, at 10%.z � $91,666.67y � $316,666.67

x � $366,666.67

� x8xx

y9y

z10z4z

775,0006,700,000

0

� x0.08x

x

y0.09y

z0.10z

775,00067,000

4z

z �

y �

x � 96. Let amount at 8%.

Let amount at 10%.

Let amount at 12%.

Solving the system,z � $316,000 at 12%.y � $228,000 at 10%,

x � $456,000 at 8%,

�0.08xxx

0.10yy

2y

0.12zz

97,200 1,000,0000

z �

y �

x �

98. Let amount in certificates of deposit.

Let amount in municipal bonds.

Let amount in blue-chip stocks.

Let amount in growth or speculative stocks.

The system has infinitely many solutions.

Let then

Answer:

One possible solution is to let $100,000.

Certificates of deposit: $356,250

Municipal bonds: $18,750

Blue-chip stocks: $25,000

Growth or speculative stocks: $100,000

s �

�406,250 �12s, �31,250 �

12s, 125,000 � s, s�

C � 406,250 �12s.

M �12 s � 31,250

B � 125,000 � sG � s,

� C � M � B � G0.09C � 0.05M � 0.12B � 0.14G

B � G

500,000 0.10�500,000�14�500,000�

G �

B �

M �

C �

94. passes through

Solving the system and the circle is

y2 � �1 � �25 � �x � 1�2

y1 � �1 � �25 � �x � 1�2

�x � 1�2 � � y � 1�2 � 25

�x2 � 2x � 1� � �y2 � 2y � 1� � 23 � 1 � 1−9

−7

9

5 x2 � y2 � 2x � 2y � 23 � 0

D � 2, E � 2, F � �23,

���6, �1�:��4, 3�:�2, �5�:

36164

19

25

6D4D2D

E � F � 03E � F � 05E � F � 0

⇒ ⇒ ⇒

6D4D2D

E3E5E

� F� F� F

� 37� 25� �29

��6, �1�, ��4, 3�, �2, �5�.x2 � y2 � Dx � Ey � F � 0

97. Let amount in certificates of deposit.

Let amount in municipal bonds.

Let amount in blue chip stocks.

Let amount in growth or speculative stocks.

Solving the system

G � s

B � 218,750 � 1.75s

M � 125,000

C � 156,250 � 0.75s

� C � M � B � G0.08C � 0.09M � 0.12B � 0.15G

M

500,000 0.10�500,000�14�500,000�

G �

B �

M �

C �

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598 Chapter 7 Linear Systems and Matrices

99. Let number of 1-point free throws.

Let number of 2-point field goals.

Let number of 3-point basket.

Solving the system,

18 free throws, 24 2-point field goals,6 3-point baskets.

x � 18, y � 24, z � 6.

� x�x

2yyy

3z

4z

8460

z �

y �

x � 100. Let number of 1-point free throws.

Let number of 2-point field goals.

Let number of 3-point baskets.

Solving the system,

12 free throws, 18 2-point field goals,9 3-point baskets.

x � 12, y � 18, z � 9.

�xxx

2y

y

3zz

753

�6

z �

y �

x �

101. Let number of touchdowns.

Let number of extra-point kicks.

Let number of field goals.

Solving the system,

4 touchdowns, 4 extra-points and 1 field goal

x � 4, y � 4, z � 1.

�x

6xxx

yy

y

z3z4z

93100

z �

y �

x � 102. Let number of touchdowns.

Let number of extra points.

Let number of field goals.

Solving the system, touchdowns,extra points, field goalsz � 4y � 3

x � 4

� x6xx

yy

z3zz

11390

z �

y �

x �

103.

Answer: ampere, amperes,ampereI3 � 1

I2 � 2I1 � 1

I1 � 2 � 1 � 0 ⇒ I1 � 1

10I2 � 6�1� � 14 ⇒ I2 � 2

26I3 � 26 ⇒ I3 � 1

�Eq. 2 � Eq. 3�I1 � I2

10I2

I3

6I3

26I3

01426

2 Eq. 25 Eq. 3

�I1 � I2

10I2

10I2

I3

6I3

20I3

01440

�3 Eq. 1 � Eq. 2�I1 � I2

5I2

2I2

I3

3I3

4I3

078

Equation 1Equation 2Equation 3

� I1

3I1

I2

2I2

2I2

I3

4I3

078

104.

Answer: a � �16 ft�sec2t2 � 48 lb,t1 � 96 lb,

t1 � 96

t2 � 48

a � �16

�4a � 64

�t1t1

� 2t2

t2

2aa

012832

⇒ ⇒ ⇒

2t2�2t2

2a2a

128�64

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Page 52: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 599

107. Least squares regression parabola through

Solving the system,

Thus, y � �524x2 �

310x �

416 .

a � �524, b � �

310, and c �

416 .

� 4c

40c

40b

40a

544a

19�12160

��4, 5�, ��2, 6�, �2, 6�,�4, 2�108. Least squares regression parabola through

Solving the system,

Thus, y �37 x2 �

65x �

2635.

c �2635.b �

65,a �

37,

� 5c

10c

10a10b34a

81222

�2, 5��1, 2�,�0, 1�,��1, 0�,��2, 0�,

109. Least squares regression parabola through

Solving the system,Thus, y � x2 � x.

a � 1, b � �1, and c � 0.

� 4c9c

29c

9b29b99b

29a99a

353a

2070

254

�0, 0�, �2, 2�, �3, 6�, �4, 12�110. Least squares regression parabola through

Solving the system,

Thus, y � � 54x2 �

920x �

19920 .

a � � 54, b �

920, c �

19920 .

� 4c6c

14c

6b14b36b

14a36a98a

252133

�0, 10�, �1, 9�, �2, 6�, �3, 0�

111. (a)

Solving the system,and

(b)

(c) For feet.y � 453x � 70,

6000

500

y � 0.165x2 � 6.55x � 103

c � 103.b � �6.55a � 0.165,

�a�30�2

a�40�2

a�50�2

b�30�b�40�b�50�

ccc

55105188

112. (a)

Solving the system,and

Parabola:

(b)

(c) For x � 170, y � 13%

1701100

100

y � �0.015x2 � 3.25x � 106

c � �106.a � �0.015, b � 3.25

�a�120�2

a�140�2

a�160�2

b�120�b�140�b�160�

ccc

685530

105. Let number of par-3 holes.

Let number of par-4 holes.

Let number of par-5 holes.

Solving the system,

4 par-3 holes, 10 par-4 holes, and 4 par-5 holes

x � 4, y � 10, z � 4.

�3x

x

� 4yy

5z2zz

7220

z �

y �

x � 106. Let number of par-3 holes.

Let number of par-4 holes.

Let number of par-5 holes.

Solving the system,

2 par-3 holes, 14 par-4 holes, and 2 par-5 holes

x � 2, y � 14, z � 2.

�3x

x

� 4yy

5z7zz

7204

z �

y �

x �

©H

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Page 53: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

600 Chapter 7 Linear Systems and Matrices

113. (a)

�2000

7 � 4x�

2000

11 � 7x

2000�4 � 3x�

�11 � 7x��7 � 4x��

�2000

11 � 7x�

2000

7 � 4x

A � �2000B � 2000

⇒ ��60008000

�4A7A

7B11B

2000�4 � 3x� � A�7 � 4x� � B�11 � 7x�

2000�4 � 3x��11 � 7x��7 � 4x�

�A

11 � 7x�

B

7 � 4x, 0 ≤ x ≤ 1 (b)

100

700 20007 − 4x

200011 − 7x

y2 �2000

11 � 7x

y1 �2000

7 � 4x

114.

Hence, and 60

100 � p�

60100 � p

�120p

10,000 � p2.A � 60, B � �60

� A100A

B100B

1200

A�100 � p� � B�100 � p� � 120p

120p�100 � p��100 � p� �

A100 � p

�B

100 � p

0 ≤ p < 100C �120p

10,000 � p2 �120p

�100 � p��100 � p� ,

115. False. The coefficient of in the second equationis not 1.

y

117. False. The correct form is

Ax � 10

�B

x � 10�

C�x � 10�2.

116. True. A common point would be a solution.

118. No, the problem was notworked correctly. You must first divide theimproper fraction.

A � �1 ⇒ B � 1.

119.

1a2 � x2 �

1�2aa � x

�1�2aa � x

�12a�

1a � x

�1

a � x�

A �12a

, B �12a

⇒ ��AAa

BBa

01

1 � A�a � x� � B�a � x� � ��A � B�x � �Aa � Ba�

1

a2 � x2 �1

�a � x��a � x� �A

a � x�

Ba � x

120.

Let :

Let :

1

�x � 1��a � x��

1

a � 1 � 1

x � 1�

1

a � x�

B�a � 1� ⇒ B �1

a � 1x � a

1 � A�a � 1� ⇒ A �1

a � 1x � �1

1 � A�a � x� � B�x � 1�

1

�x � 1��a � x��

A

x � 1�

B

a � x121.

1

y�a � y��

1

a�1

y�

1

a � y

A �1a

, B �1a

1 � A�a � y� � By � ��A � B�y � aA

1

y�a � y��

A

y�

B

a � y

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Page 54: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.3 Multivariable Linear Systems 601

122.

Let :

Let :

1

x�x � a��

1

a �1

x�

1

x � a�

1 � �aB ⇒ B � �1

ax � �a

1 � aA ⇒ A �1

ax � 0

1 � A�x � a� � Bx

1

x�x � a��

A

x�

B

x � a, a is a constant. 123. No, they are not equivalent. In the second system,

the constant in the second equation should be and the coefficient of in the third equationshould be 2.

z�11

124. When using Gaussian elimination to solve a system of linear equations, a system has no solution when there is a row representing a contradictory equation such as where

is a nonzero real number.

For instance:

No solution

Eq. 1 � Eq. 2�x � y0

36

Equation 1Equation 2� x

�x�

yy

33

N0 � N,

125.

� � �5

y � 5

x � 5

⇒ x � y � ��

⇒ 2x � 10 � 0 �y

x

xy

10

126.

� � �4

y � 2

x � 2

2x � 4

� 2x � � � 02y � � � 0

x � y � 4 � 0

x � y � �

2x � 4 � 0

��2 127.

From the first equation,

If then and

If then

Thus, the solutions are:

(1)

(2)

(3) x � ��2

2, y �

1

2, � � 1

x ��2

2, y �

1

2, � � 1

x � y � � � 0

� � 1 ⇒ y �12 and x � ±�1

2.x � 0,

� � 0.y � 02 � 0x � 0,

x � 0 or � � 1.

�2x � 2x� � 0�2y � � � 0

y � x2 � 0

⇒ ⇒ ⇒

x � x�

2y � �

y � x2

128.

Then, and

Answer: x � 0, y � 100, � � �1.

y � 100 � 2x � 100� � �1

x � 0

�2x � 0

2 � 2x � 2��2x � 1� � 0

� 2 � 2x � 2� � 02x � 1 � � � 0

2x � y � 100 � 0

� � �2x � 1

129.

7654321−1−2−3

5

4

3

2

1

6

7

x

y

y � �3x � 7

©H

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Page 55: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

602 Chapter 7 Linear Systems and Matrices

130.

9854321−1

9

8

5

4

3

2

1x

y132.

9

8

5

4

3

2

6

7

54321−1−2−3−4−5x

y

y �14x2 � 1131.

5432−2−3−4−5

1

−5

−6

−7

−8

−9

x

y

y � �2x2

133.

65421−1−2−3−4

7

6

5

−2

−3

x

y

y � �x2�x � 3� 134.

5432−1−2−3−4−5

5

4

3

2

1

−3

−4

−5

x

y

y �12x3 � 1

135. (a)

(b)25

20

−5

−10

−15

642−2−6x

y

⇒ x � 0, �4, 3

� x�x2 � x � 12� � x�x � 4��x � 3�

f�x� � x3 � x2 � 12x

137. (a)

(b)

642−2−6

20

10

−30

−40

−50

−60

x

y

⇒ x � �32, �4, 3

� �2x � 3��x � 4��x � 3�

f �x� � 2x3 � 5x2 � 21x � 36

136. (a)

(b)

5431−1−3−4−5

35

x

y

⇒ x � 0, 0, �2, 2

� 8x2��x2 � 4� � 8x2�2 � x��2 � x� f �x� � �8x4 � 32x2

138. (a)

(b)

8642−2−4

10x

y

⇒ x � 5, �12, 13

� �x � 5��2x � 1��3x � 1� f �x� � 6x3 � 29x2 � 6x � 5

©H

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Page 56: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 603

Section 7.4 Matrices and Systems of Equations

■ You should be able to use elementary row operations to produce a row-echelon form (or reducedrow-echelon form) of a matrix.

1. Interchange two rows

2. Multiply a row by a nonzero constant

3. Add a multiple of one row to another row

■ You should be able to use either Gaussian elimination with back-substitution or Gauss-Jordan eliminationto solve a system of linear equations.

139.

−1−2 1 2 3 4 5 6 7 8

1

2

3

4

8

9

x

yy � �12�x�2

x 0 1 2

y 16 8 4 2 1

�1�2 140.

−2−4−6−8−10 2 4 6 8 10

2

16

18

x

yy � �13�x

� 1

x 0 1

y 10 4 2 1.33

�1�2

141.

−1

−2

1 2 3

1

2

3

4

5

6

7

8

x

yy � 2x�1

� 1

x 0 1 2

y 0 1 3 7�12

�1�2142.

−1−2−3−4−5 1 2 3 4 5

1

3

4

56

7

8

9

x

yy � 3x�1

� 2

x 0 1 2

y 2.037 2.33 3 5

�2

143. Answers will vary.

Vocabulary Check

1. matrix 2. square 3. row matrix, column matrix

4. augmented matrix 5. coefficient matrix 6. row-equivalent

7. reduced row-echelon form 8. Gauss-Jordan elimination

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Page 57: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

604 Chapter 7 Linear Systems and Matrices

1. Since the matrix has one rowand two columns, its order is1 � 2.

2. Since the matrix has one rowand four columns, its order is1 � 4.

3. Since the matrix has threerows and one column, its orderis 3 � 1.

5. Since the matrix has two rowsand two columns, its order is2 � 2.

4. Since the matrix has threerows and four columns, itsorder is 3 � 4.

6. Since the matrix has two rowsand three columns, its order is2 � 3.

10. �100

�340

107

���

10

�5�

14.

�6x

�x4x

2y

y8y

z7z

10zz

5w3w6w

11w

�257

23�21

16.

13R1→

�1

4

2

�3

83

6�

�3

4

6

�3

8

6�

18.

�R1 � R2→�2R1 � R2→ �

1

0

0

2

�3

2

4

�7

�4

3212

6

12R1→�

1

1

2

2

�1

6

4

�3

4

32

2

9

�2

1

2

4

�1

6

8

�3

4

3

2

9�

7.

� 6

�2

�7

5��

11

�1�

6x�2x

7y5y

11�1

8.

�7

5

4

�9��

22

15�

�7x5x

4y9y

2215

9.

�1

5

2

10

�3

1

�2

4

0

���

2

0

6�

� x5x2x

10y3yy

2z4z

206

11.

�314

�1��

9

�3�

3xx

4yy

9�3

12.

�7

8

�5

3��

0

�2�

�7x8x

5y3y

0�2

13.

� 9x�2x

x

12y18y7y

3z5z8z

010

�4

�9

�21

12187

35

�8

���

010

�4�

15.

�2R1 � R2→ �1

0

4

2

3

�1�

�1

2

4

10

3

5�

17.

15R2→�1

0

0

1

1

3

4

�25

20

�165

4�

�3R1 � R2→2R1 � R3→

�1

0

0

1

5

3

4

�2

20

�1

6

4�

�1

3

�2

1

8

1

4

10

12

�1

3

6�

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Page 58: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 605

30.

(a) (b) (c) (d)

(e) (f) This matrix is in reduced row-echelon form.�1

0

0

0

0

1

0

0��

1

0

0

0

5

1

19

�34�

�1

0

0

0

5

2

19

�34��

1

0

0

7

5

2

19

1��

1

0

�3

7

5

2

4

1��

7

0

�3

1

1

2

4

5�

�7

0

�3

4

1

2

4

1�

19. Add times Row 2 to Row 1.�3 20. 3 times Row 1 added to Row 2.

21. Interchange Rows 1 and 2. 22. 5 times Row 1 added to Row 3.

23.

This matrix is in reduced row-echelon form.

�1

0

0

0

1

0

0

1

0

0

5

0� 24.

This matrix is in reduced row-echelon form.

�1

0

0

3

0

0

0

1

0

0

8

0�

25.

The first nonzero entries in rows one and two are not one. The matrix is not in row-echelon form.

�3

0

0

0

�2

0

3

0

1

7

4

5� 26.

This matrix is in row-echelon form, but not reducedrow-echelon form.

�1

0

0

0

1

0

2

�3

1

1

10

0�

27.

The first nonzero entry in row three is two, not one.The matrix is not in row-echelon form.

�1

0

0

0

1

0

0

0

0

1

�1

2� 28.

The matrix is in row-echelon form, but not reducedrow-echelon form. There is a one above the leadingone of row three.

�1

0

0

0

1

0

1

0

1

0

2

0�

29.

(a) (b) (c)

(d) (e) This matrix is in reduced row-echelon form.�1

0

0

0

1

0

�1

2

0��

1

0

0

2

1

0

3

2

0�

�1

0

0

2

�5

0

3

�10

0��

1

0

0

2

�5

�5

3

�10

�10��

1

0

3

2

�5

1

3

�10

�1�

�1

2

3

2

�1

1

3

�4

�1�

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Page 59: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

606 Chapter 7 Linear Systems and Matrices

36.

�2R2 � R3→�1

0

0

�3

1

0

0

1

0

�7

2

0�

3R1 � R2→�4R1 � R3→

�1

0

0

�3

1

2

0

1

2

�7

2

4�

�1

�3

4

�3

10

�10

0

1

2

�7

23

�24�

33.

(Answers may vary.)

�14R3→ �

1

0

0

2

1

0

312

1

0

�56

�3512�

�2R2 � R3→�1

0

0

2

1

0

312

�4

0

�56

353

16R2→�1

0

0

2

1

2

312

�3

0

�56

10�

R1 � R2→�2R1 � R3→

�1

0

0

2

6

2

3

3

�3

0

�5

10�

�1

�1

2

2

4

6

3

0

3

0

�5

10

� 34.

�3R2 � R3 → �

1

0

0

2

1

0

�1

�2

1

3

5

�1�

�3R1 � R2 →2R1 � R3 →

�1

0

0

2

1

3

�1

�2

�5

3

5

14�

�1

3

�2

2

7

�1

�1

�5

�3

3

14

8�

31. (See Exercise 29.) (Answer is a series of screens.)

(a) (b)

(c) (d)

(e)

32. (a) (b)

(c) (d)

(e) (f)

35.

�2R2 � R3→

�1

0

0

�1

1

0

�1

6

0

1

3

0�

�5R1 � R2→ 6R1 � R3→

�1

0

0

�1

1

2

�1

6

12

1

3

6�

�1

5

�6

�1

�4

8

�1

1

18

1

8

0�

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Page 60: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 607

40.

5R2 � R1 → �1

001

02

2�6�

��1�R1 →

126R2 →�1

0�5

1�10

232

�6�

5R1 � R2 → ��10

526

1052

�32�156�

� 5�1

15

210

4�32�

R1

R2 ��1

551

102

�324�

38.

�1R2→ �1

0

0

3

0

0

0

1

0�

2R2 � R1→

6R2 � R3→ �1

0

0

3

0

0

0

�1

0�

�5R1 � R2→�2R1 � R3→ �

1

0

0

3

0

0

2

�1

6�

�1

5

2

3

15

6

2

9

10�37.

�4R3 � R1 →

3R3 � R2 → �100

010

001�

12R3 →�

100

010

4�3

1�

�R2 � R1 →

�2R2 � R3 → �100

010

4�3

2�

R1 � R2 →�2R1 � R3 → �

100

112

1�3�4�

13R1 →

�1

�12

104

1�4�2�

�3

�1

2

3

0

4

3

�4

�2�

39.

2R2 � R1 → �1

0

0

1

�37

�127

�87

107�

17R2 →�1

0�2

13

�127

�4107�

4R1 � R2 → �1

0�2�7

312

�4�10�

R1 →R2 → � 1

�4�2

130

�46�

��41

1�2

03

6�4�

41.

Answer: �x, y� � ��2, �3�

x � 2y � 4 � 2��3� � 4 � �2

y � �3

x � 2y � 4 42.

Answer: ��12, 3�

x � �12

x � 8�3� � 12

y � 3

x � 8y � 12

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Page 61: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

608 Chapter 7 Linear Systems and Matrices

44.

Answer: �x, y, z� � ��31, 12, �3�

x � �2y � 2z � 1 � �24 � 6 � 1 � �31

y � �z � 9 � 3 � 9 � 12

z � �3

y � z � 9

x � 2y � 2z � �1

50.

Answer: ��1, 3�

y � 3, x � �1

�20

01

��

�23�

�20

6�3

��

16�9�

�22

63

��

167�

43.

Answer: �8, 0, �2�

x � 8

x � 0 � 2��2� � 4

y � 0

y � ��2� � 2

�x � yy

2zzz

42

�2

45.

Answer: �7, �5�

y � �5

x � 7

�1

0

0

1��

7

�5� 46.

Answer: ��2, 4�

y � 4

x � �2

�10

01

��

�24�

47.

Answer: ��4, �8, 2�

z � 2

y � �8

x � �4

�1

0

0

0

1

0

0

0

1

���

�4

�8

2� 48.

Answer: �3, �1, 0�

z � 0

y � �1

x � 3

�1

0

0

0

1

0

0

0

1

���

3

�1

0�

49.

Answer: �3, 2�

x � 2�2� � 7 ⇒ x � 3

y � 2

�1

3R2→�1

0

2

1��

7

2� �2R1 � R2→

�1

0

2

�3��

7

�6�

�1

2

2

1��

7

8�� x2x

2yy

78

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Page 62: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 609

54.

Answer: ��1, 0, 6, 4�

x � 9 � 4�0� � 3�6� � 2�4� � �1

y � �4 �45�6� �

15�4� � 0

w �16040

� 4, z � 50 � 4�11� � 6

�1000

�4100

3�

45

10

�215

1140

����

9�450

160�

�1

0

0

0

�4

1

0

0

3

�45

�25

�185

�215

�22525

����

9

�4

�20

�20�

�1000

�410

�13�7

3�8102

�22

�7�1

����

9�40

328�

�13

�4�2

�4�2

31

31

�2�4

�2�4

13

����

9�13�4

�10�

52.

No solution, inconsistent

�R2→�8R2 � R3→ �

1

0

0

2

1

0

���

0

6

�40�

�R1 � R2→�3R1 � R3→ �

1

0

0

2

�1

�8

���

0

6

8�

�1

1

3

2

1

�2

���

0

6

8�

� xx

3x

2yy

2y

068

51.

No solution, inconsistent

�14R2 →

�7R2 � R3 → ��1

00

110

���

�2214

�160�

R2 →R3 → �

�100

1�4

7

���

�22�56�62�

3R1 � R2 → 4R1 � R3 → �

�100

17

�4

���

�22�62�56�

��1

34

14

�8

���

�224

32���x

3x4x

y4y8y

�224

32

53.

Answer: �3, �2, 5, 0�

x � 25 � ��2� � 4�5� � 3

w � 0, z �18537 � 5, y � �52 � 10�5� � �2

�1000

�1100

4�10

370

2�3101537

����

25�52185

0�

�1000

�1100

4�10

3717

2�3105

����

25�5218585

��1000

�15

�12

4�13

10�3

2�5

3�1

����

25�75

52�19

��

31

�21

2�1

11

�1421

12

�11

����

02526�

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Page 63: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

610 Chapter 7 Linear Systems and Matrices

58.

Answer: �� 32a �

32, 13a �

13, a�

x � � 32a �

32

y �13a �

13

z � a

12R1 →

�13R2 → �1

0

0

0

1

0

32

�13

0

3213

0�

�3R2 � R3 → �

200

0�3

0

310

���

3�1

0�

�2R1 � R2 →�4R1 � R3 → �

200

0�3�9

313

���

3�1�3�

�248

0�3�9

37

15

���

359�

�2x4x8x

3y9y

3z7z

15z

359

56.

Answer: �8, 10, 6�

�5R3 � R1 →

�100

010

001

���

8106�

12R1 →

�100

010

501

���

38106�

R2 � R1 →

�7R3 � R2 → �200

010

1001

���

76106�

17R1 →

�115R3→ �

200

�110

371

���

24526�

�1400

�710

217

�15

16852

�90��1400

�72

�3

21�1

�21

16814

�156��140

14

�72

�10

21�1

0

1681412�

�207

�12

�5

3�1

0

���

24146�55.

Answer: �4, �3, 2�

3R3 � R1→�7R3 � R2→�

1

0

0

0

1

0

0

0

1

���

4

�3

2�

�17R3→

�1

0

0

0

1

0

�3

7

1

���

�2

11

2�

�2R2 � R3→�1

0

0

0

1

0

�3

7

�7

���

�2

11

�14�

�3R1 � R2→�2R1 � R3→

�1

0

0

0

1

2

�3

7

7

���

�2

11

8�

�1

3

2

0

1

2

�3

�2

1

���

�2

5

4�

� x3x2x

y2y

3z2zz

�254

57.

Let any real number

Answer: �2a � 1, 3a � 2, a�

x � 2a � 1 ⇒ x � 2a � 1

y � 3a � 2 ⇒ y � 3a � 2

z � a,

R2 � R1→�R2→�

1

0

0

0

1

0

�2

�3

0

���

1

2

0�

�3R2 � R3→�1

0

0

1

�1

0

�5

3

0

���

3

�2

0�

�R1 � R2→�2R1 � R3→

�1

0

0

1

�1

�3

�5

3

9

���

3

�2

�6�

�1

1

2

1

0

�1

�5

�2

�1

���

3

1

0�

� xx

2x

y

y

5z2zz

310

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Page 64: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 611

59.

Answer: �7, �3, 4�

R2 � R1→

�100

010

001

���

7�3

4�

� R3 � R1 →

R3 � R2 → �100

�110

001

���

10�3

4�

13 R3 →�

100

�110

1�1

1

���

14�7

4�

�R1 →

�5R2 � R3 → �100

�110

1�1

3

���

14�712�

2R1 � R2 →3R1 � R3 → �

�100

115

�1�1�2

���

�14�7

�23�

��1

23

1�1

2

�111

���

�142119�

��x2x3x

yy

2y

zzz

�142119

61.

Let any real number

Answer: �0, �4a � 2, a�

x � 0

y � �4a � 2

z � a,

�3

1

2

�1

3

1

5

2

12

4

20

8

����

6

2

10

4� ⇒ �

1

0

0

0

0

1

0

0

0

4

0

0

����

0

2

0

0�

�3xx

2x�x

3yy

5y2y

12z4z

20z8z

62

104

60.

Answer: �22, 36, 114�

3R2 � R1 →

�12R2 →�

100

010

001

2236

114�

�R3 � R1 →�R3 � R2 → �

100

�3�2

0

001

�86�72114�

�100

�3�2

0

111

2842

114��100

�38

�2

1�3

1

28�54

42��

21

�1

2�3

1

�110

���

22814�

62.

Let any real number

Answer: ��2a, a, a�

x � �2a

y � a

z � a,

�123

135

111

���

000� ⇒ �

100

010

2�1

0

���

000�

� x2x3x

y3y5y

zzz

000

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Page 65: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

612 Chapter 7 Linear Systems and Matrices

66. The solutions are not the same.

(a)

Answer:

(b)

Answer: ��3, �2, 2�

x � �4��2� � 11 � �3.y � �3�2� � 4 � �2,z � 2,

��25, �2, 2�

x � 3��2� � 4�2� � 11 � �25.y � 2 � 4 � �2,z � 2,

68. The solutions are not the same.

(a)

Answer:

(b)

Answer: �3, 6, �4�

x � 6 � 3��4� � 15 � 3.y � 2��4� � 14 � 6,z � �4,

��3, 6, �4�

x � �3�6� � ��4� � 19 � �3.y � �6��4� � 18 � 6,z � �4,

64. row reduces to

Answer: �1, 0, 4, �2�

�1000

0100

0010

0001

����

104

�2��

2315

1452

�102

�1

216

�1

����

�61

�33�

63.

Answer: ��5a, a, 3�

x � �5ay � a,z � 3,

�2x

xx

�3x

10y5y5y

15y

2z2zz

3z

663

�9�

2

1

1

�3

10

5

5

�15

2

2

1

�3

����

6

6

3

�9� ⇒ �

1

0

0

0

5

0

0

0

0

1

0

0

����

0

3

0

0�

65. Yes, the systems yield the same solutions.

(a)

Answer:

(b)

Answer: ��1, 1, �3�

x � �1 � 2��3� � 6 � �1y � �3��3� � 8 � 1,z � �3,

��1, 1, �3�

x � 2�1� � ��3� � 6 � �1y � 5��3� � 16 � 1;z � �3;

67. No, solutions are different.

(a)

Answer:

(b)

Answer: �19, 2, 8�

x � 6�2� � 8 � 15 � 19y � �5�8� � 42 � 2,z � 8,

��5, 2, 8�

x � 4�2� � 5�8� � 27 � �5y � 7�8� � 54 � 2,z � 8,

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Page 66: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 613

70.

Answer: y � �x2 � 2x � 8

a � 8 � 2 � 9 ⇒ a � �1

b �32�8� � 14 ⇒ b � 2

c � 8

�12R2→

�3R2 � R3→�1

0

0

1

1

0

132

1

���

9

14

8�

�4R1 � R2→�9R1 � R3→

�1

0

0

1

�2

�6

1

�3

�8

���

9

�28

�76�

�1

4

9

1

2

3

1

1

1

���

9

8

5�

� f�1�f�2�f�3�

a4a9a

b2b3b

ccc

985

f�x� � ax2 � bx � c69.

Answer: y � x2 � 2x � 5

a � 2 � 5 � 8 ⇒ a � 1

b �32 �5� �

192 ⇒ b � 2

c � 5

�12R2 →

�3R2 � R3 → �1

0

0

1

1

0

132

1

���

8192

5�

�4R1 � R2 →�9R1 � R3 → �

1

0

0

1

�2

�6

1

�3

�8

���

8

�19

�52�

�1

4

9

1

2

3

1

1

1

���

8

13

20�

� f �1� � a � b � c � 8f �2� � 4a � 2b � c � 13f �3� � 9a � 3b � c � 20

f �x� � ax2 � bx � c

71.

Answer: y � 2x2 � x � 1

a � b � c � 2 ⇒ a � 2 � 1 � 1 � 2

�6b � 3c � 3 ⇒ �6b � 3 � 3 � 6 ⇒ b � �1

�5c � �5 ⇒ c � 1

��1�R2 � R3→ �1

0

0

1

�6

0

1

�3

�5

���

2

3

�5�

�4R1 � R2 →�9R1 � R3 → �

1

0

0

1

�6

�6

1

�3

�8

���

2

3

�2�

�1

4

9

1

�2

3

1

1

1

���

2

11

16�

� f �1� � a � b � c � 2f ��2� � 4a � 2b � c � 11f �3� � 9a � 3b � c � 16

f�x� � ax2 � bx � c 72.

Answer: y � �x2 � 2x � 2

a � b � c � �1 ⇒ a � �1 � 2 � 2 � �1

2b � c � �2 ⇒ 2b � �2 � 2 � �4 ⇒ b � �2

�2c � �4 ⇒ c � 2

R2 � R3→ �

1

0

0

1

2

0

1

1

�2

���

�1

�2

�4�

�13R2→�

1

0

0

1

2

�2

1

1

�3

���

�1

�2

�2�

�4R1 � R2→�4R1 � R3→

�1

0

0

1

�6

�2

1

�3

�3

���

�1

6

�2

�1

4

4

1

�2

2

1

1

1

���

�1

2

�6�

� f �1� � a � b � c � �1f ��2� � 4a � 2b � c � 2f �2� � 4a � 2b � c � �6

f �x� � ax2 � bx � c

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Page 67: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

614 Chapter 7 Linear Systems and Matrices

74.

Solving the system,

f �x� � �2x2 � 2x � 1

a � �2, b � �2, c � 1.

f ��2�f �1�f �2�

4aa

4a

2bb

2b

ccc

�3�3

�11

f �x� � ax2 � bx � c

76.

Solving the system,

f �x� � x3 � x2 � 2x � 1

a � 1, b � �1, c � 2, d � �1.

f ��2�f ��1�

f �1�f �2�

�8a�a

a8a

4bbb

4b

2ccc

2c

dddd

�17�5

17

f �x� � ax3 � bx2 � cx � d

73.

Solving the system,

f�x� � �9x2 � 5x � 11

a � �9, b � �5, c � 11.

f ��2�f ��1�

f �1�

4aaa

2bbb

ccc

�157

�3

f�x� � ax2 � bx � c

75.

Solving the system,

f�x� � x3 � 2x2 � 4x � 1

a � 1, b � �2, c � �4, d � 1.

f ��2�f ��1�

f �1�f �2�

�8a�a

a8a

4bbb

4b

2ccc

2c

dddd

�72

�4�7

f�x� � ax3 � bx2 � cx � d

77. amount at 7%, amount at 8%, amount at 10%

Answers: $150,000 at 7%, $750,000 at 8%, $600,000 at 10%

x � 750,000 � 600,000 � 1,500,000 ⇒ x � 150,000

y � 3�600,000� � 2,550,000 ⇒ y � 750,000

7z � 4,200,000 ⇒ z � 600,00

100R2→4R2 � R3→ �

100

110

137

���

1,500,0002,550,0004,200,000�

�0.07R1 � R2→�4R1 � R3→ �

100

10.01�4

10.03�5

���

1,500,00025,500

�6,000,000�

�1

0.074

10.08

0

10.1�1

���

1,500,000130,500

0�� x

0.07x4x

y0.08y

z0.1z

z

1,500,000130,500

0

z �y �x �

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Page 68: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.4 Matrices and Systems of Equations 615

78. amount at 9%, amount at 10%, amount at 12%

Answers: $100,000 at 9%, $250,000 at 10%, $150,000 at 12%

x � 100,000y � 250,000,z � 150,000,

116R3→

�1

0

0

0

1

0

�2

3

1

���

�200,000

700,000

150,000�

�R2 � R1→

7R2 � R3→ �1

0

0

0

1

0

�2

3

16

���

�200,000

700,000

2,400,000�

100R2→2R3→ �

1

0

0

1

1

�7

1

3

�5

���

500,000

700,000

�2,500,000�

�0.09R1 � R2→�2.5R1 � R3→ �

1

0

0

1

0.01

�3.5

1

0.03

�2.5

���

500,000

7,000

�1,250,000�

�1

0.09

2.5

1

0.10

�1

1

0.12

0

���

500,000

52,000

0�

� x0.09x2.5x

y0.10y

y

z0.12z

500,00052,000

0

z �y �x �

79.

amperes; amperes; amperesI1 �115 �

910 � 0 ⇒ I1 �

1310I2 � 2� 9

10� � 4 ⇒ I2 �115I3 �

910

� 1

10R3→�1

0

0

�1

1

0

1

2

1

���

0

4910�

�4R2 � R3→ �1

0

0

�1

1

0

1

2

�10

���

0

4

�9�

12R2→�1

0

0

�1

1

4

1

2

�2

���

0

4

7�

R3→R2→ �

1

0

0

�1

2

4

1

4

�2

���

0

8

7�

�2R1 � R2→ �1

0

0

�1

4

2

1

�2

4

���

0

7

8�

�1

2

0

�1

2

2

1

0

4

���

0

7

8�

� I1

2I1

I2

2I2

2I2

I3

4I3

078

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Page 69: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

616 Chapter 7 Linear Systems and Matrices

80. (a)

(b)

(c) Maximum height feet

Strikes ground when feet.

(d) Complete the square:

Maximum height feet

Range: � 103.793 feety � 0 ⇒ x ��0.367 ± �0.3672 � 4�0.004�5

�0.008

� 13.418

�0.004�x2 � 91.75x � 2104.5� � 5 � 8.418

x � 104�y � 0�

� 13

12000

18

y � �0.004x2 � 0.367x � 5

�0

225900

01530

111

���

5.09.6

12.4� ⇒ 1

�00

010

001

���

�0.0040.367

5�� f�0�

f�15�f�30�

c225a900a

5.015b30b

cc

9.612.4

f�x� � ax2 � bx � c

81. (a)

Solving the system using matrices,

you obtain

(b)

(c) For 2005, and dollars

For 2010, and dollars

For 2015, and dollarsy � 103.08t � 15

y � 86.33t � 10

y � 67.58t � 5

0 50

80

y � �0.04t2 � 4.35t � 46.83

a � �0.04, b � 4.35, c � 46.83.

�49

16

234

111

���

55.3759.5263.59� ⇒

1

�00

010

001

���

�0.044.35

46.83�

� 4a9a

16a

2b3b4b

ccc

55.3759.5263.59

�2, 55.37��3, 59.52��4, 63.59�

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Section 7.4 Matrices and Systems of Equations 617

82. (a)

Solving the system using matrices,

you obtain

(b)

(c) For 2005, and thousand

For 2010, and thousand

For 2015, and thousand

(d) Answers will vary.

y � 12.6t � 15

y � 36.6t � 10

y � 45.6t � 5

0 50

60

y � �0.3t2 � 2.7t � 39.6

a � �0.3, b � 2.7, c � 39.6.

�49

16

234

111

���

43.845.045.6� ⇒

1

�00

010

001

���

�0.32.7

39.6�

� 4a9a

16a

2b3b4b

ccc

43.845.045.6

�2, 43.8��3, 45.0��4, 45.6�

84. Let number of adults

Let number of students

Let number of children

Solving the system,

120 adults, 80 students and 60 children

x � 120, y � 80 and z � 60.

� 5xx

�12x

3.5y

y

2.5z2z

10300

20

z �

y �

x �

83. Let number of pounds of glossy.

Let number of pounds of semi-glossy

Let number of pounds of matte

Solving the system,

50 pounds of glossy, 35 pounds of semi-glossy and 15 pounds of matte

x � 50, y � 35 and z � 15.

� x5.5x

y4.25y

y

z3.75z

z

10048050

z �

y �

x �

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618 Chapter 7 Linear Systems and Matrices

x1 � x3 � 600x1 � x2 � x4 ⇒ x1 � x2 � x4 � 0x2 � x5 � 500x3 � x6 � 600x4 � x7 � x6 ⇒ x4 � x6 � x7 � 0x5 � x7 � 500

85. (a)

�R3 � R2→�R4 � R3→

�R4→

1

0�0

0

0

0

0

�1

0

0

0

0

1

0

�1

0

0

0

0

0

0

1

0

0

0

�1

0

�1

1

0

0

0

�1

�1

0

0

0

0

0

0

1

0

������

600

�500

�600

�500

500

0

�R1 � R2→R2 � R3→R3 � R4→R4 � R5→

�R5 � R6→�

1

0

0

0

0

0

0

�1

0

0

0

0

1

�1

�1

0

0

0

0

�1

�1

�1

0

0

0

0

1

1

1

0

0

0

0

1

0

0

0

0

0

0

1

0

������

600

�600

�100

500

500

0

�1

1

0

0

0

0

0

�1

1

0

0

0

1

0

0

1

0

0

0

�1

0

0

1

0

0

0

1

0

0

1

0

0

0

1

�1

0

0

0

0

0

1

1

������

600

0

500

600

0

500

Let then:

(b) If

x6 � x7 � 0

x5 � 500

x4 � 0

x3 � 600

x2 � 0

x1 � 0

x6 � x7 � 0, then s � t � 0, and

x1 � 600 � �600 � s� � s

x2 � 500 � �500 � t� � t

x3 � 600 � s

x4 � �500 � s � �500 � t� � s � t

x5 � 500 � t

x7 � t and x6 � s,

(c) IfThus:

x7 � �500

x6 � 0

x5 � 1000

x4 � 500

x3 � 600

x2 � �500

x1 � 0

x5 � 1000 and x6 � 0, then s � 0 and t � �500.

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Section 7.4 Matrices and Systems of Equations 619

86. (a)

Let

x1 � 200 � s � t � 300 ⇒ x1 � 500 � s � t

x2 � s � 350 � t � 150 ⇒ x2 � �200 � s � t

Let x3 � s.

x4 � t � 350 ⇒ x4 � 350 � t

x5 � t.

�R2→�R3→

R3 � R4→�1

0

0

0

1

1

0

0

0

�1

0

0

0

1

1

0

0

0

1

0

����

300

150

350

0�

R2 � R3→ �

1

0

0

0

1

�1

0

0

0

1

0

0

0

�1

�1

1

0

0

�1

1

����

300

�150

�350

350�

�R1 � R2→ �

1

0

0

0

1

�1

1

0

0

1

�1

0

0

�1

0

1

0

0

�1

1

����

300

�150

�200

350�

�1

1

0

0

1

0

1

0

0

1

�1

0

0

�1

0

1

0

0

�1

1

����

300

150

�200

350�

�x1

x1

x2

x4

x2

x3

200x5

300150

x3

350

� x4

� x5

⇒ x1 � x3 � x4 � 150 ⇒ x2 � x3 � x5 � �200

(b) When and

(c) When and

x1 � 150

x2 � 150,x3 � 0,x4 � 0,x5 � 350,

150 � �200 � 0 � t ⇒ t � 350.

x2 � �200 � s � t

x3 � 0:x2 � 150

x1 � 100

x2 � 200,x3 � 50,x4 � 0,x5 � 350,

200 � �200 � 50 � t ⇒ t � 350.

x2 � �200 � s � t

x3 � 50:x2 � 200

87. False. It is a matrix.2 � 4 88. False. Gauss-Jordan elimination reduces a matrixto reduced row-echelon form.

89.

� xx

2x

y2yy

7z11z10z

�10

�3

2�Equation 1� � �Equation 2� → new Equation 3

�Equation 1� � 2�Equation 2� → new Equation 2

�Equation 1� � �Equation 2� → new Equation 1

Equation 1Equation 2�x

y�

3z4z

�21

90. (a) In the row-echelon form of an augmentedmatrix that corresponds to an inconsistentsystem of linear equations, there exists a rowconsisting of all zeros except for the entry inthe last column.

(b) In the row-echelon form of an augmentedmatrix that corresponds to a system with aninfinite number of solutions, there are fewerrows with nonzero entries than there are variables. Nor does the last row consist of allzeros except for the entry in the last column.

91. The row operation was not performedon the last column. Nor was �R2 � R1.

�2R1 � R2 92. Answers will vary.

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620 Chapter 7 Linear Systems and Matrices

93.

Asymptotes: x � �1, y � 0

128−4−8−12

4

−4

−12

−8

x

y

f �x� �7

�x � 1

95.

Asymptotes: x � 4, y � x � 2

4

−4

−8 128

12

16

8

x

y

f �x� �x2 � 2x � 3

x � 4� x � 2 �

5x � 4

94.

Horizontal asymptote: y � 0

2−4

0.4

−0.8

−2 4 6 8

0.2

0.6

0.8

x

y

f �x� �4x

5x2 � 2

96.

Vertical asymptote:

Slant asymptote: y � x � 1

x � �1,

8 1−4− 28

4

8

−12

−8

−12

x

y

f �x� �x2 � 36x � 1

■ if and only if they have the same order and

■ You should be able to perform the operations of matrix addition, scalar multiplication, and matrixmultiplication.

■ Some properties of matrix addition and scalar multiplication are:

(a) (b)

(c) (d)

(e) (f)

■ Some properties of matrix multiplication are:

(a) (b)

(c) (d)

■ You should remember that in general.AB � BA

c�AB� � �cA�B � A�cB��A � B�C � AC � BC

A�B � C� � AB � ACA�BC� � �AB�C

�c � d�A � cA � dAc�A � B� � cA � cB

1A � A�cd�A � c�dA�A � �B � C� � �A � B� � CA � B � B � A

aij � bij.A � B

Section 7.5 Operations with Matrices

Vocabulary Check

1. equal 2. scalars 3. zero, 0 4. identity

5. (a) iii (b) i (c) iv (d) v (e) ii 6. (a) ii (b) iv (c) i (d) iii

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Section 7.5 Operations with Matrices 621

1. x � �4, y � 22

3.

3z � 14 � 4 ⇒ z � 6

3y � 12 ⇒ y � 4

2x � 7 � 5 ⇒ x � �1

5. (a)

(b)

(c)

(d) 3A � 2B � �3

6

�3

�3� �2 � 2

�1

�1

8� � �3

6

�3

�3� � ��4

2

2

�16� � ��1

8

�1

�19�

3A � 3�1

2

�1

�1� � �3�1�3�2�

3��1�3��1�� � �3

6

�3

�3�

A � B � �1

2

�1

�1� � � 2

�1

�1

8� � �1 � 2

2 � 1

�1 � 1

�1 � 8� � ��1

3

0

�9�

A � B � �1

2

�1

�1� � � 2

�1

�1

8� � �1 � 2

2 � 1

�1 � 1

�1 � 8� � �3

1

�2

7�

2. x � 13, y � 12

4.

z � 2 � 11 ⇒ z � 9

2y � �8 ⇒ y � �4

x � 4 � 2x � 9 ⇒ x � �5

6. (a)

(b)

(c)

(d) 3A � 2B � �3

6

6

3� � 2��3

4

�2

2� � �3 � 6

6 � 8

6 � 4

3 � 4� � � 9

�2

10

�1�

3A � 3�1

2

2

1� � �3�1�3�2�

3�2�3�1�� � �3

6

6

3�

A � B � �1

2

2

1� � ��3

4

�2

2� � �1 � 3

2 � 4

2 � 2

1 � 2� � � 4

�2

4

�1�

A � B � �1

2

2

1� � ��3

4

�2

2� � �1 � 3

2 � 4

2 � 2

1 � 2� � ��2

6

0

3�

7.

(a) (b)

(c) (d) 3A � 2B � �246

�12

�39

15� � �2

�22

12�10

20� � �228

�14

�1519

�5�3A � �246

�12

�39

15�

A � B � �73

�5

�78

�5�A � B � �91

�3

5�215�

A � �82

�4

�135�, B � �

1�1

1

6�510�

8. (a)

(b)

(c)

(d) 3A � 2B � �30

�318

927� � ��4

�608

�10�14� � �7

6�310

1941�

3A � �30

�318

927�

A � B � �33

�12

816�

A � B � �10

�16

39� � ��2

�304

�5�7� � ��1

�3�110

�22�

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622 Chapter 7 Linear Systems and Matrices

9.

(a)

(b)

(c)

(d)

� �1015

15�10

�1�10

73

1214�

3A � 2B � �123

156

�3�6

9�3

120� � � 2

�120

16 �2

42

�60

�14�

3A � �123

156

�3�6

9�3

120�

A � B � �37

5�6

0�4

22

47�

A � B � � 5�5

510

�20

4�4

4�7�

B � � 1�6

08

�12

1�3

0�7�A � �4

152

�1�2

3�1

40�,

10. (a)

(c) 3A � ��3

9150

�12

12�61224

�3

06

�3�18

0�

A � B � ��4

5153

�4

9�6�5100

1�5�2

�10�2

� (b)

(d) 3A � 2B � �35

�5�6

�12

22

3020

�5

�220

�1�10

4�

A � B � �21

�5�3�4

�12

136

�2

�190

�22�

11.

(a) is not possible.

(b) is not possible.

(c)

(d) is not possible.3A � 2B

3A � � 18�3

0�12

90�

A � B

A � B

B � �84

�1�3�A � � 6

�10

�430�, 12. (a) is not defined.

(b) is not defined.

(c)

(d) is not defined.3A � 2B

3A � 3�32

�1� � �96

�3�A � B

A � B

13. � ��53

0�6� � ��3

12�7

5� � ��815

�7�1���5

30

�6� � � 7�2

1�1� � ��10

14�8

6�

14. ��6

�17

901� � �

0�2

3

5�1�6�� � �

�134

�6

�7�1

0� � �6

�310

14�1�5� � �

�134

�6

�7�1

0� � ��7

14

7�2�5�

15. ��24�12

�432

1212�4���4

002

13� � �2

31

�6�2

0�� � 4��6�3

�18

33� �

16. �12 �19 4 �14 9� � �19

2 2 �7 92�1

2�5 �2 4 0� � �14 6 �18 9��

17. � 2�1

5�4� � ��3

202� � ��1

15

�2� 18. ��8�9

925�

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Page 76: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.5 Operations with Matrices 623

19. �12�

3.211�1.004

0.055

6.8294.914

�3.889� � 8�1.6305.256

�9.768

�3.0908.3354.251� � �

�14.645�41.546

78.117

21.305�69.137�32.064�

20.

� ��4

2�9

�111

�3� � �13

�1

1

2312

2� � ��113

1

�8

�31332

�1�� �

�42

�9

�111

�3� �16�

2�6

6

43

12���1��4

�29

11�1

3� �16��

�530

�14

13� � �7

�96

5�1�1��

21. X � 3��2

1

3

�1

0

�4� � 2 �

0

2

�4

3

0

�1� � �

�6

3

9

�3

0

�12� � �

0

4

�8

6

0

�2� � �

�6

�1

17

�9

0

�10�

22.

X � A �12B � �

�2

1

3

�1

0

�4� �

12�

0

2

�4

3

0

�1� � �

�2

1

3

�1

0

�4� � �

0

1

�2

32

0� 1

2� � �

�2

0

5

� 52

0� 7

2�

2X � 2A � B

23. X � �32 A �

12 B � �

32�

�2

1

3

�1

0

�4� �

12 �

0

2

�4

3

0

�1� � �

3

�12

�132

3

0112

�24.

X � �A � 2B � �1��2

1

3

�1

0

�4� � 2�

0

2

�4

3

0

�1� � �

2

�1

�3

1

0

4� � �

0

�4

8

�6

0

2� � �

2

�5

5

�5

0

6�

2A � 4B � �2X

25. is and is is not defined.3 � 3 ⇒ ABB3 � 2A 26. AB � �067

�10

�1

238� �

241

�1�5

6� � ��21518

171246�

27. AB � ��1�4

0

653� �2

039� � �

�2�8

0

513327� 28. is is is

�1

0

0

0

4

0

0

0

�2� �

3

0

0

0

�1

0

0

0

5� � �

3

0

0

0

�4

0

0

0

�10�

3 � 3.3 � 3 ⇒ AB3 � 3, BA

29.

AB � �5

0

0

0

�8

0

0

0

7� �

15

0

0

0

�18

0

0

012� � �

1

0

0

0

1

0

0

072�

A is 3 � 3, B is 3 � 3 ⇒ AB is 3 � 3.

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624 Chapter 7 Linear Systems and Matrices

30. �0

0

0

0

0

0

5

�3

4� �

6

8

0

�11

16

0

4

4

0� � �

0

0

0

0

0

0

0

0

0�

31. AB � �56���3 �1 �5 �9� � ��15

�18�5 �6

�25�30

�45�54�

32. is is is not defined.2 � 2 ⇒ AB2 � 4, BA

33. (a)

(b)

(c) A2 � �15

22��

15

22� � �1 � 10

5 � 102 � 4

10 � 4� � �1115

614�

BA � � 2�1

�18��

15

22� � � 2 � 5

�1 � 404 � 2

�2 � 16� � ��339

214�

� �08

1511�AB � �1

522��

2�1

�18� � � 2 � 2

10 � 2�1 � 16�5 � 16�

34. (a) (b)

(c) A2 � AA � � 30�4

610�

BA � ��124

�6�10�AB � � 6

�23

�4���2

204� � ��6

�412

�16�

35. (a)

(b)

(c) A2 � �3

1

�1

3� �3

1

�1

3� � �9 � 1

3 � 3

�3 � 3

�1 � 9� � �8

6

�6

8�

BA � �1

3

�3

1� �3

1

�1

3� � �3 � 3

9 � 1

�1 � 9

�3 � 3� � � 0

10

�10

0�

AB � �3

1

�1

3� �1

3

�3

1� � �3 � 3

1 � 9

�9 � 1

�3 � 3� � � 0

10

�10

0�

36. (a)

(b)

(c) A2 � �1

1

�1

1� �1

1

�1

1� � �1�1� � ��1��1�1�1� � �1��1�

1��1� � ��1��1�1��1� � 1�1�� � �0

2

�2

0�

BA � � 1

�3

3

1� �1

1

�1

1� � � 1�1� � �3�1�3�1� � �1��1�

1��1� � 3�1��3��1� � 1�1�� � � 4

�2

2

4�

AB � �1

1

�1

1� � 1

�3

3

1� � �1�1� � ��1���3�1�1� � 1��3�

1�3� � ��1��1�1�3� � 1�1�� � � 4

�2

2

4�

37. (a)

(b)

(c) is not defined.A2

BA � �1 1 2� �78

�1� � �7 � 8 � 2� � �13�

AB � �78

�1� �1 1 2� � �78

�1

78

�1

1416

�2�

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Section 7.5 Operations with Matrices 625

38. (a)

(b)

(c) The number of columns of does not equal the number of rows of the multiplication is not possible.A;A

BA � �2

3

0� �3 2 1� � �

2�3�3�3�0�3�

2�2�3�2�0�2�

2�1�3�1�0�1�� � �

6

9

0

4

6

0

2

3

0�

AB � �3 2 1� �2

3

0� � �3�2� � 2�3� � 1�0�� � �12�

39. AB � �703216

�1711

�38

736

70� 40. AB � �124228192

�70452

�72�

41. ��3

�12

5

8

15

�1

�6

9

1

8

6

5� �

3

24

16

8

1

15

10

�4

6

14

21

10� � �

151

516

47

25

279

�20

48

387

87�

42. is is is not defined.

4 � 2 ⇒ AB3 � 3, BA 43.is not defined.A is 2 � 4 and B is 2 � 4 ⇒ AB 44. AB � �

�238119210

50115135

�484342555�

45. ��30

1�2��

1�2

02���

12

04� � �1

42

�4��12

04� � � 5

�48

�16�

46. � 27�6

�6�27�

47. � ��43

1014��0

421

�22���

40

�1

0�1

2� � ��2�3

0

35

�3�� � �04

21

�22��

2�3�1

34

�1�

48. �3

�157��4 2� � �

12�42028

6�21014

�49.

(a) is not a solution.

(b) is a solution.

(c) is not a solution.

(d) is not a solution.� 2�3� ⇒ �1

322��

2�3� � ��4

0�

��44� ⇒ �1

322��

�44� � � 4

�4�

��23� ⇒ �1

322��

�23� � �4

0�

�21� ⇒ �1

322��

21� � �4

8�

�13

22

��

40�

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Page 79: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

626 Chapter 7 Linear Systems and Matrices

50. Let

(a) is a solution.

(b) is not a solution.

(c) is not a solution.

(d) is not a solution.A��39� � � 0

48� ⇒ ��39�

A� 3�9� � � 0

�48� ⇒ � 3�9�

A� 2�6� � � 0

�32� ⇒ � 2�6�

A��13� � � 0

16� ⇒ ��13�

A � � 6�1

25�� 6

�125

��

016�. 51.

(a) is not a

solution.

(b) is a

solution.

(c) is not

a solution.

(d) is not

a solution.

�42� ⇒ ��2

4�3

2��42� � ��14

20�

��66� ⇒ ��2

4�3

2���6

6� � � �6�12�

� 6�2� ⇒ ��2

4�3

2��6

�2� � ��620�

�30� ⇒ ��2

4�3

2��30� � ��6

12�

��24

�32

��

�620�

52. Let

(a) is a solution.

(b) is not a solution.

(c) is not a solution.

(d) is not a solution.A� 211� � ��67

17� ⇒ � 211�

A��4�5� � � 15

�17� ⇒ ��4�5�

A�52� � �11

17� ⇒ �52�

A�45� � ��15

17� ⇒ �45�

A � �53

�71��5

3�7

1��

�1517�. 53. (a)

(b) By Gauss-Jordan elimination on

we have

Thus, X � �48�.x1 � 4 and x2 � 8.

�R2 � R1→

�R2→ �1

0

0

1��

4

8�,

�R1→

2R1 � R2→ �1

0

�1

�1��

�4

�8�

��1

�2

1

1 ��

4

0�

A � ��1

�2

1

1�, X � �x1

x2�, B � �4

0�

54. (a)

(b) By Gauss-Jordan elimination on

we have

Answer: ��23�

x1 � �2 and x2 � 3.

�4R2 � R1→

�15R2→ �

1

0

0

1

..

.

..

.�2

3�,

�2R1 � R2→ �1

0

4

�5

..

.

..

.10

�15�

�1

2

4

3

..

.

..

.10

5�

A � �2

1

3

4�, X � �x1

x2�, B � � 5

10� 55. (a)

(b)

Answer: X � ��76�

x1 � �7, x2 � 6.

12R1 �1

001

��

�76�

3R2 � R1 ��2

001

��

146�

��18�R2

��20

�31

��

�46�

3R1 � R2

��20

�3�8

��

�4�48�

��26

�31

��

�4�36�

A � ��26

�31�, X � �x1

x2�, B � � �4

�36�

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Page 80: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.5 Operations with Matrices 627

56. (a)

(b)

Answer: ��23�

353�

x2 � �353x1 � �23,

�13 R2 →�1

001

��

�23�

353�

��1�R2 � R1 → �1

00

�3��

�2335�

4R1 � R2 → �10

�3�3

��

1235�

��41

9�3

�1312�

R1

R2 � 1

�4�3

912

�13�

B � ��1312�x � �x1

x2�,A � ��4

19

�3�,57. (a)

(b)

Answer: X � �1

�12�.

x1 � 1, x2 � �1, x3 � 2.

�7R3 � R1 →�2R3 � R2 → �

1

0

0

0

1

0

0

0

1

���

1

�1

2�

2R2 � R1 →

R2 � R3 → �

1

0

0

0

1

0

7

2

1

���

15

3

2�

R1 � R2 →�2R1 � R3 →

�1

0

0

�2

1

�1

3

2

�1

���

9

3

�1�

�1

�1

2

�2

3

�5

3

�1

5

���

9

�6

17�

A � �1

�1

2

�2

3

�5

3

�1

5�, X � �

x1

x2

x3�, B � �

9

�6

17�

58. (a)

(b) row reduces to

Answer: �23232�

�100

010

001

21.51.5��

1�1

1

12

�1

�301

���

�112�

A � �1

�11

12

�1

�301�, X � �

x1

x2

x3�, B � �

�112�

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Page 81: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

628 Chapter 7 Linear Systems and Matrices

60. (a)

(b)

Answer: �4

�52�x1 � 4x2 � �5,x3 � 2,

R2 � R1 →

�100

010

001

���

4�5

2�

4R3 � R1 →

�R3 � R2 →�R3 → �

100

�110

001

���

9�5

2�

6R2 � R3 → �100

�110

4�1�1

���

17�7�2�

14 R2 →�100

�11

�6

4�1

5

���

17�740�

�110

�13

�6

405

���

17�11

40�

��1�R1 � R2 → �100

�14

�6

4�4

5

���

17�28

40�B � �

17�11

40�x � �x1

x2

x3�,A � �

110

�13

�6

405�,

59. (a)

(b)

Answer: X � ��1

3�2�x1 � �1, x2 � 3, x3 � �2

5R2 � R1

�100

010

001

���

�13

�2� ��1

2�R2 �100

�510

001

���

�163

�2� �2R3 � R1

�5R3 � R2 �100

�5�2

0

001

���

�16�6�2�

130R3

�100

�5�2

0

251

���

�20�16�2�

�7R2 � R3

�100

�5�2

0

25

�30

���

�20�16

60� R2

R3

�100

�5�2

�14

255

���

�20�16�52�

3R1 � R2 �100

�5�14�2

255

���

�20�52�16��

1�3

0

�51

�2

2�1

5

���

�208

�16�A � �

1�3

0

�51

�2

2�1

5�, X � �x1

x2

x3�, B � �

�208

�16�

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Page 82: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.5 Operations with Matrices 629

61. (a)

(b)

The answers are the same.

AB � AC � �7

�616

�21311

5�8�3�

A�B � C� � �7

�616

�21311

5�8�3� 62. (a)

(b)

The answers are the same.

BA � CA � �94

�10

5�5�8

05

13�

�B � C�A � �94

�10

5�5�8

05

13�

63. (a)

(b)

The answers are the same.

A2 � AB � BA � B2 � �261111

112014

0�3

0�

�A � B�2 � �261111

112014

0�3

0� 64. (a)

(b)

The answers are the same.

A2 � AB � BA � B2 � �0

4125

9�12

0

�6�1

�26�

�A � B�2 � �0

4125

9�12

0

�6�1

�26�

65. (a)

(b)

The answers are the same.

�AB�C � �25

�53�76

�343430

28�721�

A�BC� � �25

�53�76

�343430

28�721� 66. (a)

(b)

The answers are the same.

�cA�B � �33

�33�18

91239

�121218�

c�AB� � �33

�33�18

91239

�121218�

67. (a)

(b) � ��1�5

10�1

�40�A � cB � � 1

�121

�20� � 2��1

�24

�1�1

0�

A � cB � ��1�5

10�1

�40�

68. Not possible

B and C have different orders.

�a�, �b� A�B � C� 69. Not possible

Number of columns of A (3) does not equal thenumber of rows of B (2).

�a�, �b� c�AB�

70. (a)

(b) � ��41

�2�4

50�B � dA � ��1

�24

�1�1

0� � ��3�� 1�1

21

�20�

B � dA � ��41

�2�4

50�

71. Not possible

is is 2 � 2.BC3 � 3,CA

�a�, �b� CA � BC 72. Not possible

not defined, nor B2AB

�a�, �b� dAB2

73. (a)

(b) � ��66

�12�6

120�cd A � 2��3�� 1

�121

�20�

cd A � ��66

�12�6

120�

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Page 83: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

630 Chapter 7 Linear Systems and Matrices

74. (a) cA � dB � �54

�85

�10�

75.

f�A� � A2 � 5A � 2I � �2

4

0

5 ��2

4

0

5� � 5 �2

4

0

5� � 2 �1

0

0

1� � ��4

8

0

2�

A � �2

4

0

5�

76.

f�A� � A2 � 7A � 6 � �5

1

4

2� �5

1

4

2� � 7�5

1

4

2� � 6�1

0

0

1� � �0

0

0

0�

A � �5

1

4

2�

77. 1.20�7035

50100

2570� � �84

4260

1203084� 78. 1.10�100

40

90

20

70

60

30

60� � �110

44

99

22

77

66

33

66�

79.

The entries in the last matrix BA represent the profit for both crops at each of the three outlets.

� �1037.50 1400 1012.50�BA � �3.50 6.00� �125100

100175

75125�

80.

The entries represent the costs of the three models of the product at each of the two warehouses.

�39.50 44.50 56.50� �500060008000

400010,000

5000� � �916,500 885,500�

81.

The entries represent the wholesale and retail prices of the inventory at each outlet.

ST � �3

0

4

2

2

2

2

3

1

3

4

3

0

3

2� �

840

1200

1450

2650

3050

1100

1350

1650

3000

3200� � �

$15,770

$26,500

$21,260

$18,300

$29,250

$24,150�

82.

This represents the labor cost for each boat size at each plant.

ST � �1.0

1.6

2.5

0.5

1.0

2.0

0.2

0.2

0.4� �

12

9

6

10

8

5� � �

$17.70

$29.40

$50.40

$15.00

$25.00

$43.00�

83.

This product represents the changes in party affiliation after two elections.

P2 � �0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.40

0.28

0.32

0.15

0.53

0.32

0.15

0.17

0.68�

(b)

� �54

�85

�10�

� � 2�2

42

�40� � �3

6�12

33 0 �

cA � dB � 2� 1�1

21

�20� � ��3���1

�24

�1�1

0�

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Page 84: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.5 Operations with Matrices 631

84.

As is raised to higher and higher powers, the resulting matrices appear to be approaching the matrix

�0.2

0.3

0.5

0.2

0.3

0.5

0.2

0.3

0.5�.

P

P8 � �0.203

0.305

0.492

0.199

0.309

0.492

0.199

0.292

0.508�

P7 � �0.206

0.308

0.486

0.198

0.316

0.486

0.198

0.288

0.514�

P6 � �0.213

0.311

0.477

0.197

0.326

0.477

0.197

0.280

0.523�

P5 � P4P � �0.250

0.315

0.435

0.188

0.377

0.435

0.188

0.248

0.565� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.225

0.314

0.461

0.194

0.345

0.461

0.194

0.267

0.539�

P4 � P3P � �0.300

0.308

0.392

0.175

0.433

0.392

0.175

0.217

0.608� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.250

0.315

0.435

0.188

0.377

0.435

0.188

0.248

0.565�

P3 � P2P � �0.4

0.28

0.32

0.15

0.53

0.32

0.15

0.17

0.68� �

0.6

0.2

0.2

0.1

0.7

0.2

0.1

0.1

0.8� � �

0.300

0.308

0.392

0.175

0.433

0.392

0.175

0.217

0.608�

85. True 86. False.

�40

0�1��

�62

�2�6� � ��24

�2�8

6�

��62

�2�6��

40

0�1� � ��24

826�

For 87–93, A is of order B is of order C is of order and D is of order 2 � 2.3 � 22 � 3,2 � 3,

87. is not possible. and are not of thesame order.

CAA � 2C 88. Not possible

89. AB is not possible. The number of columns of Adoes not equal the number of rows of B.

90. Possible. Order 2 � 2

91. is possible. The resulting order is 2 � 2.BC � D 92. Not possible

93. is possible. The resulting order is 2 � 3.D�A � 3B� 94. Possible.Order 2 � 3

95.

A2 � 2AB � B2 � �03

02�

�A � B�2 � �12

01� 96.

A2 � 2AB � B2 � �87

�1622�

�A � B�2 � �78

�1623�

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632 Chapter 7 Linear Systems and Matrices

97.

A2 � B2 � �25

�24�

�A � B��A � B� � �34

�23� 98. �A � B�2 � �1

201� � A2 � AB � BA � B2

99.

AC � BC, but A � B.

BC � �1

1

0

0� �2

2

3

3� � �2

2

3

3�

AC � �0

0

1

1� �2

2

3

3� � �2

2

3

3� 100.

but and B � 0.A � 0AB � 0

AB � �3

4

3

4� � 1

�1

�1

1� � �0

0

0

0�

A � �3

4

3

4�, B � � 1

�1

�1

1�

101. (a)

(b)

The identity matrix

B2 � �0

i

�i

0� �0

i

�i

0� � �1

0

0

1�,

A4 � A3A � ��i

0

0

�i� � i

0

0

i� � �1

0

0

1� and i4 � 1

A3 � A2A � ��1

0

0

�1� � i

0

0

i� � ��i

0

0

�i� and i3 � �i

A2 � � i

0

0

i� � i

0

0

i� � ��1

0

0

�1� and i2 � �1

102. The product of two diagonal matrices of the same order is a diagonal matrix whoseentries are the products of the corresponding diagonal entries of A and B.

103. (a)

(b) and are both zero matrices.

(c) If A is then will be the zero matrix.

(d) If A is then is the zero matrix.Ann � n,

A44 � 4,

B3A2

A � �0

0

2

0�, B � �0

0

0

2

0

0

3

4

0� 104. Matrix multiplication can be used to find the

number of subscribers each company will haveone year later. Multiply a matrix containingthe current number of subscribers per company on the left by the given matrix.

3 � 1

105. 3 ln 4 �13

ln�x2 � 3� � ln 43 � ln�x2 � 3�13 � ln� 64�x2 � 3�13�

106.

� ln� x�x2 � 36�3�

� ln x � ln�x2 � 36�3

ln x � 3�ln�x � 6� � ln�x � 6�� � ln x � 3�ln�x � 6��x � 6��

107.

� ln��x � 5�xx � 8 �

12

�2 ln�x � 5� � ln x � ln�x � 8�� � ln�x � 5� � ln x12 � ln�x � 8�12

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Section 7.6 The Inverse of a Square Matrix 633

108.

� ln�73�2t6

t3 � � ln�73�2t3�

� ln�73�2t6� � ln t3

32

ln 7t4 �35

ln t5 � ln�7t4�3�2 � ln�t5�3�5

Section 7.6 The Inverse of a Square Matrix

■ You should be able to find the inverse, if it exists, of a square matrix.

(a) Write the matrix that consists of the given matrix on the left and the identity matrix on the right to obtain Note that we separate the matrices and by a dotted line. We callthis process adjoining the matrices and

(b) If possible, row reduce to using elementary row operations on the entire matrix The result will be the matrix If this is not possible, then is not invertible.

(c) Check your work by multiplying to see that

■ You should be able to use inverse matrices to solve systems of equation.

■ You should be able to find inverses using a graphing utility.

AA�1 � I � A�1A.

A�I � A�1�.�A � I�.IA

I.AIA�A � I�.

In � nAn � 2n

1.

BA � � 3

�5

�1

2� �2

5

1

3� � �3�2� � ��1��5��5�2� � 2�5�

3�1� � ��1��3��5�1� � 2�3�� � �1

0

0

1�

AB � �2

5

1

3� � 3

�5

�1

2� � �2�3� � 1��5�5�3� � 3��5�

2��1� � 1�2�5��1� � 3�2�� � �1

0

0

1�

2.

BA � �2

1

1

1� � 1

�1

�1

2� � �2 � 1

1 � 1

�2 � 2

�1 � 2� � �1

0

0

1�

AB � � 1

�1

�1

2� �2

1

1

1� � � 2 � 1

�2 � 2

1 � 1

�1 � 2 � � �1

0

0

1�

3.

BA � ��232

1

�12� �1

3

2

4� � ��2 � 332 �

32

�4 � 4

3 � 2� � �1

0

0

1�

AB � �1

3

2

4� ��232

1

�12� � ��2 � 3

�6 � 6

1 � 1

3 � 2� � �1

0

0

1�

4.

BA � �35

� 25

1515� �1

2

�1

3� � �35 �

25

�25 �

25

�35 �

35

25 �

35� � �1

0

0

1�

AB � �1

2

�1

3� �35

� 25

1515� � �

35 �

25

65 �

65

15 �

15

25 �

35� � �1

0

0

1�

Vocabulary Check

1. square 2. inverse 3. nonsingular, singular

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634 Chapter 7 Linear Systems and Matrices

5.

� �100

010

001�� �

2 � 14 � 46 � 6

�17 � 11 � 6�34 � 44 � 9

�51 � 66 � 15

11 � 7 � 422 � 28 � 6

33 � 42 � 10�BA � �123

146

2�3�5��

2�1

0

�17113

11�7�2�

� �100

010

001� � �

2 � 34 � 33�1 � 22 � 21

6 � 6

2 � 68 � 66�1 � 44 � 42

12 � 12

4 � 51 � 55�2 � 33 � 35

�9 � 10�

AB � �2

�10

�17113

11�7�2��

123

146

2�3�5�

6.

BA � �100

010

001�

AB � �1

�11

012

�100� 13 �

00

�3

�21

�2

111� � �

100

010

001�

7. BA � �10

01�AB � ��1

1�4

2��1

�12

2�

12� � �1

001�;

9. BA � �10

01�AB � � 1.6

�3.52

�4.5��22.5

�17.510

�8� � �10

01�;

8. BA � �10

01�AB � �11

2�12�2��

�1�1

6112� � �1

001�;

10.

BA � �100

010

001�

AB � �410

023

�2�4

1��0.28

�0.020.06

�0.120.08

�0.24

0.080.280.16� � �

100

010

001�

11.

A�1 � �12

0

013� �

16 �3

0

0

2�

12R1 →13R2 →�1

001

��

12

0013� � �I � A�1�

�A � I� � �20

03

��

10

01�

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Section 7.6 The Inverse of a Square Matrix 635

12.

A�1 � � 7

�3

�2

1�

�2R2 � R1→ �1

0

0

1��

7

�3

�2

1� � �I � A�1�

�3R1 � R2→ �1

0

2

1��

1

�3

0

1�

�A � I� � �1

3

2

7��

1

0

0

1�

14.

A�1 � ��19

�4

�33

�7�

�7R2 → �1

0

0

1

��

�19

�4

�33

�7�

�33R2 � R1→ �1

00

�17

��

�1947

�331�

�4R1 � R2 → �1

0

�337

�17

��

�1747

0

1�

17R1→ �1

4�

337

�19��

�17

001�

�A � I� � ��74

33�19

��

10

01� 15.

A has no inverse because it is not square.

A � � 2

�3

7

�9

1

2�

16.

A has no inverse because it is not square.

A � ��2

6

0

5

�15

1�

13.

A�1 � ��3

�2

2

1�

2R2 � R1 → �1

001

��

�3�2

21�

�2R1 � R2 → �1

0�2

1��

1�2

01�

�A � I� � �12

�2�3

��

10

01�

17.

A�1 � �1

�3

3

1

2

�3

�1

�1

2�

� �I � A�1�

�R3 � R1→�R3 � R2→

2R3→�1

0

0

0

1

0

0

0

1

1

�3

3

1

2

�3

�1

�1

2�

�R2 � R1→12R2→

�3R2 � R3→�1

0

0

0

1

0

12

12

12

52

�32

32

�12

12

�32

0

0

1�

�3R1 � R2→�3R1 � R3→

�1

0

0

1

2

3

1

1

2

1

�3

�3

0

1

0

0

0

1�

�A � I� � �1

3

3

1

5

6

1

4

5

���

1

0

0

0

1

0

0

0

1�

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636 Chapter 7 Linear Systems and Matrices

18.

A�1 � ��13

12

�5

6

�5

2

4

�3

1�

� �I � A�1�

4R3 � R1→�3R3 � R2→ �

1

0

0

0

1

0

0

0

1

�13

12

�5

6

�5

2

4

�3

1�

�2R2 � R1→

2R2 � R3→�1

0

0

0

1

0

�4

3

1

7

�3

�5

�2

1

2

0

0

1�

�3R1 � R2→R1 � R3→

�1

0

0

2

1

�2

2

3

�5

1

�3

1

0

1

0

0

0

1�

�A � I� � �1

3

�1

2

7

�4

2

9

�7

���

1

0

0

0

1

0

0

0

1�

20.

Since the first 3 entries of row 3 are all zeros, the inverse does not exist.

→ �100

050

000

���

1�3

1

01

�1

001�

→ �100

055

000

���

1�3�2

010

001�

�A � I� � �132

055

000

���

100

010

001� 21. Not invertible.

does not exist.A�1

22. A�1 � ��175

9514

37�20�3

�1371� 23.

A�1 � ��12�4�8

�5�2�4

�9�4�6�

A � ��

12

1

0

34

0

�1

14

�3212

� 24.

does not exist.A�1

A � ��56

0

1

1323

�12

116

2

�52�

25.

A�1 �511�

0

�22

22

�4

11

�6

2

11

�8�

A � �0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4� 26.

A�1 � �3.75

3.45834.1667

0�1

0

�1.25�1.375

�2.5�

A � �0.60.7

1

0�1

0

�0.30.2

�0.9�

19.

Not invertible (row of zeros)

does not exist.A�1

��1

5�R1 →��2�R1 � R2 → �

10

�1

005

007

���

�1525

0

010

001�

�A � I� � ��5

2�1

005

007

���

100

010

001�

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Section 7.6 The Inverse of a Square Matrix 637

29. � 5�2

1�2�

�1

�1

5��2� � ��2��1� ��2

2�1

5� �1

�8��2

2�1

5� � �14

�14

18

�58�

36. �23

�51�

�1

�1

2�1� � ��5��3� �1

�352� �

117�

1�3

52� � � 1

17

�317

517

217�

37. � 1�2

20�

�1

� �012

�1214� ⇒ k � 0 38. ��1

211�

�1

� ��1323

1313� ⇒ k �

23

40. ��10

�22�

�1

� ��10

�112� ⇒ k �

12

30. �180

��10�11

0�8���8

110

�10��1

�1

��8���10� � 11�0� ��10�11

0�8�

32. Inverse does not exist.⇒�10 �

89

�13

23

�14� ��

1413

�2389�

�1

� 1

��14��8

9� � �13���2

3� �

89

�13

23

�14�

33. � 2�1

35�

�1

�1

2�5� � �3���1� �51

�32� �

113�

51

�32� � �

513

113

�313

213�

34. � 1�3

�22�

�1

�1

1�2� � ��2���3� �23

21� �

�14 �2

321� � ��

12

�34

�12

�14�

27.

A�1 � �1

0

2

0

0

1

0

1

1

0

1

0

0

1

0

2�

A � ��1

0

2

0

0

2

0

�1

1

0

�1

0

0

�1

0

1� 28.

A�1 � ��24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1�

A � �1

3

2

�1

�2

�5

�5

4

�1

�2

�2

4

�2

�3

�5

11�

31. �2059

�45

�15

3472� �

159 �

16�4

1570� �

7215

�3445�

�1

� 1

�72��4

5� � ��34��1

5� �45

�15

3472�

35. ��13

0�2�

�1

�1

��1���2� � 0�3� ��2�3

0�1� �

12�

�2�3

0�1� � ��1

�32

0�

12�

39. ��1�3

21�

�1

� �1535

�25

�15� ⇒ k �

35

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638 Chapter 7 Linear Systems and Matrices

41.

Answer: �5, 0�

�x

y� � ��3

�2

2

1� � 5

10� � �5

0� 42.

Answer: �6, 3�

�xy� � ��3

�2

2

1� �0

3� � �6

3�

43.

Answer: ��8, �6�

�x

y� � ��3

�2

2

1��4

2���8

�6� 44.

Answer: ��7, �4�

�xy� � ��3

�2

2

1� � 1

�2� � ��7

�4�

45.

Answer: �3, 8, �11�

�x

y

z� � �

1

�3

3

1

2

�3

�1

�1

2� �

0

5

2� � �

3

8

�11� 46.

Answer: �1, 7, �9�

�x

y

z� � �

1

�3

3

1

2

�3

�1

�1

2� �

�1

2

0� � �

1

7

�9�

47.

Answer: �2, 1, 0, 0�

�x1

x2

x3

x4

� � ��24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1� �

0

1

�1

2� � �

2

1

0

0� 48.

Answer: ��32, �13, �37, 15�

�x

y

z

w� � �

�24

�10

�29

12

7

3

7

�3

1

0

3

�1

�2

�1

�2

1� �

1

�2

0

�3� � �

�32

�13

�37

15�

49.

Answer: �2, �2�

� � 2�2� �x

y� � �1

11 � 3

�5

�4

3� ��2

4� � �1

11 ��22

22�

A�1 �1

9 � 20 � 3

�5

�4

3�

A � �3

5

4

3�

50.

Answer: �12, 13�

�xy� � A�1b �

112�

4�5

�23��

1323� � �

1213�

A�1 �1

18�24� � 12�30� �24

�30�12

18� �112�

4�5

�23�

A � �1830

1224�

51.

does not exist.

[The system actually has no solution.]

A�1

A�1 �1

1.6 � 1.6 ��4

�2

�0.8

�0.4�

A � ��0.4

2

0.8

�4� 52.

Answer: �6, �2�

�xy� �

�18 �35

25155��

2.4�8.8� � � 6

�2�

A�1 ��18 �35

25155�

A � � 0.2�1

�0.61.4�

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Section 7.6 The Inverse of a Square Matrix 639

53.

Answer: ��4, �8�

�xy� � A�1b � ��1

2

1213�� �2

�12� � ��4�8�

A�1 � ��1

2

1213�

A � ��1432

3834�

54.

Answer: ��12, 10�

�xy� �

119�

4216

�12�10��

�20�51� � ��12

10�

�119�

4216

�12�10�A�1 �

1

�56��

72 � �4

3��1���

72

�43

156� �

�1219 ��

72

�43

156�

A � �5643

�1

�72�

56.

Answer: �5, 8, �2�

�x

y

z� �

1

82 �

�21

�44

26

19

32

�4

16

14

�12� �

�2

16

4� � �

5

8

�2�

A�1 �1

82 �

�21

�44

26

19

32

�4

16

14

�12�

A � �4

2

8

�2

2

�5

3

5

�2�

57.

does not exist.

The system actually has an infinite number of solutions of the form

where is any real number.t

z � t

y � 1.1875t � 0.6875

x � 0.3125t � 0.8125

A�1

A � �5

2

�1

�3

2

7

2

�3

�8� 58.

Answer: ��1, 2, 0�

�xyz� � A�1B � �

�120�

B � �47

13�A � �235

359

5�917�

55.

Answer: ��1, 3, 2�

� ��1

32� �

155

�55

� 165

110

�x

y

z� �

155 �

18

3

�14

4

19

3

�5

�10

10��

�5

10

1

A�1 �155�

18

3

�14

4

19

3

�5

�10

10�

A � �4

2

5

�1

2

�2

1

3

6�

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640 Chapter 7 Linear Systems and Matrices

59. row reduces to

Answer: �5, 0, �2, 3�

�1000

0100

0010

0001

����

50

�23��

7�2

4�1

�3101

0010

2�1�2�1

����

41�13

12�8

60.

Answer: �6.21, �0.77, �2.67, 2.40�

�0.338

0.042

�0.141

0.113

�0.352

0.164

0.230

�0.117

0.141

�0.066

0.108

0.047

0.394

�0.117

�0.164

�0.202� �

11

�7

3

�1� � �

6.21

�0.77

�2.67

2.40��

x

y

z

w� �

A�1 �0.338

0.042

�0.141

0.113

�0.352

0.164

0.230

�0.117

0.141

�0.066

0.108

0.047

0.394

�0.117

�0.164

�0.202�

A � �2

1

2

1

5

4

�2

0

0

2

5

0

1

�2

1

�3�

61. (a) (b)

(c) (d) (e)

The point has been translated back to ��3, 2�.��1, 1�

A�1B���3

21�A�1��

100

010

�211�B � AX � �

�111�

�x � h, y � k� � ��3 � 2, 2 � ��1�� � ��1, 1��x, y� � ��3, 2�

62. (a) (b)

(c) (d) (e)

The point has been translated back to �1, �2�.��1, 1�

A�1B��1

�21�A�1��

100

010

2�3

1�B � AX � ��1

11�

�x � h, y � k� � �1 � 2, �2 � 3� � ��1, 1��x, y� � �1, �2�

63. (a) (b)

(c) (d) (e)

The point has been translated back to �2, �4�.��1, 0�

A�1B��2

�41�A�1��

100

010

3�4

1�B � AX � ��1

01�

�x � h, y � k� � �2 � 3, �4 � 4� � ��1, 0��x, y� � �2, �4�

64. (a) (b)

(c) (d) (e)

The point has been translated back to �0, �3�.�3, �5�

A�1B��0

�31�A�1��

100

010

�321�B � AX � �

3�5

1��x � h, y � k� � �0 � 3, �3 � 2� � �3, �5��x, y� � �0, �3�

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Section 7.6 The Inverse of a Square Matrix 641

For 65–68 use Using the methods of this section, we have A�1 �111�

50�13�26

�600200400

�45

�1�.A � �

10.065

0

10.07

2

10.09�1

�.

66.

Answer: 4000 in AAA bonds, 2000 in A bonds, 4000 in B bonds.

X � A�1B �1

11 �

50

�13

�26

�600

200

400

�4

5

�1� �

10,000

760

0� � �

4000

2000

4000�

68.

Answer: $200,000 in AAA bonds, $100,000 in A bonds, and $200,000 in B bonds.

X � A�1B �1

11 �

50

�13

�26

�600

200

400

�4

5

�1� �

500,000

38,000

0� � �

200,000

100,000

200,000�

70.

Answer:I3 � 15�7 ampsI1 � 5�7 amps, I2 � 10�7 amps,

�I1

I2

I3� �

114 �

5

�4

1

�4

6

2

4

8

�2� �

10

10

0� � �

5�7

10�7

15�7�

A�1 �114 �

5

�4

1

�4

6

2

4

8

�2�

A � �2

0

1

0

1

1

4

4

�1�

65.

Answer: $10,000 in AAA bonds, $5000 in A-bonds and $10,000 in B-bonds.

X � A�1 B �111�

50�13�26

�600200400

�45

�1� �25,000

19000� � �

10,0005000

10,000�

67.

Answer: $20,000 in AAA bonds, $15,000 in A-bonds and $30,000 in B-bonds.

X � A�1 B �111�

50�13�26

�600200400

�45

�1� �65,000

50500� � �

20,00015,00030,000�

69.

Answer: I1 � �3 amps, I2 � 8 amps, I3 � 5 amps

�I1

I2

I3� �

114 �

5

�4

1

�4

6

2

4

8

�2� �

14

28

0� � �

�3

8

5�

A�1 �114 �

5

�4

1

�4

6

2

4

8

�2�

A � �2

0

1

0

1

1

4

4

�1�

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642 Chapter 7 Linear Systems and Matrices

For Exercises 71–74, let number of muffins, number of bones, number of cookies.

A X

A�1 � [ 1�5

2

1�2

0

�28

�2]

[232

111

21

1.5][xyz] � [ Beef

ChickenLiver]

z �y �x �

71.

300 units of bones

200 units of cookies

A�1�700500600� � �

0300200� 72.

5 units of muffins

415 units of bones

50 units of cookies

A�1�525480500� � �

541550�

73.

100 units of muffins

300 units of bones

150 units of cookies

A�1�800750725� � �

100300150� 74.

150 units of muffins

300 units of bones

200 units of cookies

A�1�1000950900� � �

150300200�

75. (a)

(b)

(c)

2 pounds French vanilla

4 pounds hazelnut

4 pounds Swiss chocolate

X � A�1B � �113

�43

�43

�432323

�1323

�13��10

260� � �

244�

B�XA

�120

12.5

1

13

�1��fhs � � �

10260�

f2f

h2.5h

h

s3ss

10260

76. (a) Let number of roses.

Let number of lilies.

Let number of irises.

(b)

(c)

80 roses, 10 lilies, 30 irises

X � A�1B � �23

�7632

012

�12

13

�112

�14��120

3000� � �

801030�

B�XA

�1

2.51

14

�2

12

�2��xyz � � �

120300

0�

x2.5x

x

y4y2y

z2z2z

120300

0

z �

y �

x �

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Section 7.6 The Inverse of a Square Matrix 643

77. (a)

(b)

(c)

(d) For 2005, and thousand.

For 2010, and thousand.

For 2015, and thousand.

(e) Answers will vary.

y � 45,513t � 15

y � 20,583t � 10

y � 7503t � 5

0 50

8,000

y � 237t2 � 939t � 6273

A�1�534355896309� � �

abc� � �

237�9396273�

A � �49

16

234

111�, A�1 � �

12

�72

6

�1

6

�8

12

�52

3�� 4a

9a16a

2b3b4b

ccc

534355896309

���

�2, 5343��3, 5589��4, 6309�

78. (a)

(b)

(c)

(d) For 2005, and millions.

For 2010, and millions.

For 2015, and millions.

(e) Answers will vary.

y � 87,929t � 15

y � 37,199t � 10

y � 11,744t � 5

0 50

15,000

y � 505.5t2 � 2491.5t � 11,564

A�1�860386399686� � �

abc� � �

505.5�2491.5

11,564�

A � �49

16

234

111�, A�1 � �

12

�72

6

�1

6

�8

12

�52

3�� 4a

9a16a

2b3b4b

ccc

860386399686

���

�2, 8603��3, 8639��4, 9686�

79. True. AA�1 � A�1A � I 80. True

81.

A�1A �1

ad � bc � d

�c

�b

a� �a

c

b

d� �1

ad � bc �ad � bc

0

0

ad � bc � � �1

0

0

1�

�1

ad � bc �ad � bc

0

0

ad � bc�� �1

0

0

1�

AA�1 � �a

c

b

d� � 1

ad � bc�d

�c

�b

a� �1

ad � bc �a

c

b

d� � d

�c

�b

a�

82. (a)

—CONTINUED—

Given A � �a11

0

0

0

a22

0

0

0

a33�, A�1 �

1 a11

0

0

0

1 a22

0

0

0

1 a33

, a11, a22, a33, � 0

, a11 � 0, a22 � 0Given A � �a11

0

0

a22�, A�1 � �

1a11

0

0

1a22

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644 Chapter 7 Linear Systems and Matrices

82. —CONTINUED—

(b) In general, the inverse of the diagonal matrix A is

�assuming aii � 0�

1 a11

0

0

0

0

1 a22

0

0

0

0

1 a33

0

. . .

. . .

. . .

�. . .

0

0

0

� 1 ann

83.�9

x�6

x� 2

��9

x�6 � 2x

x �

9x

�x

6 � 2x�

96 � 2x

, x � 0

84.1 �

2x

1 �4x

�x � 2x � 4

, x � 0 85.

�x2 � 2x � 13

x�x � 2� , x � ±3

�2x2 � 4x � 26

2x2 � 4x

�4�x � 2� � 2�x2 � 9�

�x � 3��x � 2� � �x � 3��x � 2�

4x2 � 9

�2

x � 21

x � 3�

1x � 3

��x2 � 9��x � 2��x2 � 9��x � 2�

86. �x2 � 4x � 3

3, x � �1�

�x � 3��x � 1�3

1x � 1

�12

32x2 � 4x � 2

�2 � x � 12�x � 1� �

2�x � 1�2

3

87. e2x � 2ex � 15 � �ex � 5��ex � 3� � 0 ⇒ ex � 3 ⇒ x � ln 3 1.099

88.

ex � 4 ⇒ x � ln 4 1.386

ex � 6 ⇒ x � ln 6 1.792

e2x � 10ex � 24 � �ex � 6��ex � 4� � 0 89.

x �13 e12�7 1.851

3x � e12�7

ln 3x �127

7 ln 3x � 12

90.

x � e2 � 9 �1.611

x � 9 � e2

ln�x � 9� � 2 91. Answers will vary.

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Section 7.7 The Determinant of a Square Matrix 645

Section 7.7 The Determinant of a Square Matrix

■ You should be able to determine the determinant of a matrix of order by using the products of thediagonals.

■ You should be able to use expansion by cofactors to find the determinant of a matrix of order 3 or greater.

■ The determinant of a triangular matrix equals the product of the entries on the main diagonal.

■ You should be able to calculate determinants using a graphing utility.

2 � 2

1. �4� � 4 2. det� ��10�� � �10

3. �82 43� � 8�3� � 4�2� � 24 � 8 � 16 4. ��9

602� � ��9��2� � 0 � �18

5. � 18 � 10 � 28� 6�5

23� � 6�3� � �2���5� 6. �34 �3

�8� � 3��8� � ��3��4� � �24 � 12 � �12

7. ��712

6

3� � �7�3� � 6�12� � �21 � 3 � �24 8. �40 �3

0� � �4��0� � �0���3� � 0

9. �244 �1

2

2

0

1

1� � 2�22 1

1� � 4��1

2

0

1� � 4��1

2

0

1� � 2�0� � 4��1� � 4��1� � 0

10. ��2

1

0

2

�1

1

3

0

4� � 0� 2

�1

3

0� � 1��2

1

3

0� � 4��2

1

2

�1� � 0�3� � 1��3� � 4�0� � 3

11. (Upper Triangular)��1

0

0

2

3

0

�5

4

3� � ��1��3��3� � �9

12. (Lower Triangular)� 1

�4

5

0

�1

1

0

0

5� � �1���1��5� � �5

13. � 0.3

0.2

�0.4

0.2

0.2

0.4

0.2

0.2

0.3� � �0.002 14. � 0.1

�0.3

0.5

0.2

0.2

0.4

0.3

0.2

0.4� � �0.022

Vocabulary Check

1. determinant 2. minor 3. cofactor

4. expanding by cofactors 5. triangular 6. diagonal

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646 Chapter 7 Linear Systems and Matrices

15.

(a) (b)

C22 � �M22 � �3M22 � �3

C21 � �M21 � �4M21 � �4

C12 � �M12 � �2M12 � �2

C11 � �M11 � �5M11 � �5

�3

2

4

�5� 16.

(a)

M22 � 11

M21 � 0

M12 � �3

M11 � 2

� 11

�3

0

2�(b)

C22 � M22 � 11

C21 � �M21 � 0

C12 � �M12 � 3

C11 � M11 � 2

17.

(a)

M22 � ��4

1

3

�5� � 17

M21 � �60 3

�5� � �30

M13 � �71 �2

0� � 2

M12 � �71 8

�5� � �43

M11 � ��2

0

8

�5� � 10

��4

7

1

6

�2

0

3

8

�5�

M33 � ��4

7

6

�2� � �34

M32 � ��4

7

3

8� � �53

M31 � � 6

�2

3

8� � 54

M23 � ��4

1

6

0� � �6 (b)

C33 � �34

C32 � 53

C31 � 54

C23 � 6

C22 � 17

C21 � 30

C13 � 2

C12 � 43

C11 � 10

18.

(a)

M22 � ��2

6

4

�6� � �12

M33 � ��2

7

9

�6� � �51M21 � �97 4

�6� � �82

M32 � ��2

7

4

0� � �28M13 � �76 �6

7� � 85

M31 � � 9

�6

4

0� � 24M12 � �76 0

�6� � �42

M23 � ��2

6

9

7� � �68M11 � ��6

7

0

�6� � 36

��2

7

6

9

�6

7

4

0

�6�

(b)

C33 � ��1�6M33 � �51

C32 � ��1�5M32 � 28

C31 � ��1�4M31 � 24

C23 � ��1�5M23 � 68

C22 � ��1�4M22 � �12

C21 � ��1�3M21 � 82

C13 � ��1�4M13 � 85

C12 � ��1�3M12 � 42

C11 � ��1�2M11 � 36

19. (a)

(b) ��3

4

2

2

5

�3

1

6

1� � �2�42 6

1� � 5��3

2

1

1� � 3��3

4

1

6� � �2��8� � 5��5� � 3��22� � �75

��3

4

2

2

5

�3

1

6

1� � �3� 5

�3

6

1� � 2�42 6

1� � �42 5

�3� � �3�23� � 2��8� � 22 � �75

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Section 7.7 The Determinant of a Square Matrix 647

20. (a)

(b) ��3

6

4

4

3

�7

2

1

�8� � 2�64 3

�7� � ��3

4

4

�7� � 8��3

6

4

3� � 2��54� � �5� � 8��33� � 151

��3

6

4

4

3

�7

2

1

�8� � �6� 4

�7

2

�8� � 3��3

4

2

�8� � 1��3

4

4

�7� � �6��18� � 3�16� � �5� � 151

21. (a)

(b)

� 0 � 13��298� � 0 � 6�674� � 170

� 6

4

�1

8

0

13

0

6

�3

6

7

0

5

�8

4

2� � 0� 4

�1

8

6

7

0

�8

4

2� � 13� 6

�1

8

�3

7

0

5

4

2� � 0�648 �3

6

0

5

�8

2� � 6� 6

4

�1

�3

6

7

5

�8

4� � �4��282� � 13��298� � 6��174� � 8��234� � 170

� 6

4

�1

8

0

13

0

6

�3

6

7

0

5

�8

4

2� � �4�006 �3

7

0

5

4

2� � 13� 6

�1

8

�3

7

0

5

4

2� � 6� 6

�1

8

0

0

6

5

4

2� � 8� 6

�1

8

0

0

6

�3

7

0�

22. (a)

(b)

� 10�24� � 4�245� � 0��64� � 1�427� � �1167

�10

4

0

1

8

0

3

0

3

5

2

�3

�7

�6

7

2� � 10�030 5

2

�3

�6

7

2� � 4�830 3

2

�3

�7

7

2� � 0�800 3

5

�3

�7

�6

2� � 1�803 3

5

2

�7

�6

7� � 0��64� � 3��3� � 2��112� � 7�136� � �1167

�10

4

0

1

8

0

3

0

3

5

2

�3

�7

�6

7

2� � 0�800 3

5

�3

�7

�6

2� � 3�10

4

1

3

5

�3

�7

�6

2� � 2�10

4

1

8

0

0

�7

�6

2� � 7�10

4

1

8

0

0

3

5

�3�

23. Expand by Column 3.

� 1

3

�1

4

2

4

�2

0

3� � �2� 3

�1

2

4� � 3�13 4

2� � �2�14� � 3��10� � �58

24. (Lower Triangular)��3

7

1

0

11

2

0

0

2� � �3�11

2

0

2� � �3�22� � �66

25. Expand by Column 3.

�2213 6

7

5

7

6

3

0

0

2

6

1

7� � 6�213 7

5

7

6

1

7� � 3�213 6

5

7

2

1

7� � 6��20� � 3�16� � �168©H

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648 Chapter 7 Linear Systems and Matrices

26. Expand by Row 2.

� 3

�2

1

0

6

0

1

3

�5

6

2

�1

4

0

2

�1� � ���2��613 �5

2

�1

4

2

�1� � 6�310 6

1

3

4

2

�1� � 2��63� � 6��3� � �108

27. Expand by Column 2.

�� 3

�2

1

6

3

2

0

0

0

0

4

1

0

2

5

�1

3

4

�1

1

5

2

0

0

0�� � �2��2

1

6

3

1

0

2

5

3

4

�1

1

2

0

0

0� � ��2���2��163 0

2

5

4

�1

1� � 4�103� � 412

28. Expand by Column 1.

5

0

0

0

0

2

1

0

0

0

0

4

2

3

0

0

3

6

4

0

�2

2

3

1

2� � 5�1000 4

2

3

0

3

6

4

0

2

3

1

2� � 5 � 1�230 6

4

0

3

1

2� � 5��20� � �100

29. (Lower Triangular)�4611 0�5

3�2

0017

0003� � �4���5��1��3� � �60

30.

(Upper Triangular)

�A� � �5��6���2���1� � 60 31.

(Upper Triangular)

det�A� � ��6���1���7���2���2� � �168

32.

(Lower Triangular)

�A� � ��2��4��1��10���3� � 240

33. �1220 �1

6

0

2

8

0

2

8

4

�4

6

0� � �336 34. � 0

8

�4

�7

�3

1

6

0

8

�1

0

0

2

6

9

14� � 7441

35. �� 3

�1

5

4

1

�2

0

�1

7

2

4

2

0

�8

3

3

1

3

0

0

1

0

2

0

2�� � 410 36. �

�2

0

0

0

0

0

3

0

0

0

0

0

�1

0

0

0

0

0

2

0

0

0

0

0

�4� � �48

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Section 7.7 The Determinant of a Square Matrix 649

37. (a)

(c) ��1

0

0

3� �2

0

0

�1� � ��2

0

0

�3���1

0

0

3� � �3 (b)

(d) [Note: �AB� � �A� �B����2

0

0

�3� � 6

�20 0

�1� � �2

38. (a)

(c) AB � ��41

4�1�

�A� � �8 (b)

(d) �AB� � 0

�B� � 0

39. (a) (b)

(c) (d) � 1

�1

0

4

0

2

3

3

0� � �12��1

1

0

2

0

1

1

1

0� �

�1

0

0

0

2

0

0

0

3� � �

1

�1

0

4

0

2

3

3

0�

��1

0

0

0

2

0

0

0

3� � �6��1

1

0

2

0

1

1

1

0� � 2

40. (a)

(c) AB � �2

1

3

0

�1

1

1

2

0� �

2

0

3

�1

1

�2

4

3

1� � �

7

8

6

�4

�6

�2

9

3

15�

�A� � �213 0

�1

1

1

2

0� � 0 (b)

(d) �AB� � �786 �4

�6

�2

9

3

15� � 0

�B� � �203 �1

1

�2

4

3

1� � �7

[Note:

�AB� � �A� �B��

41. (a)

(b)

(c)

(d) Note: �AB� � �A� �B����AB� � 5500

AB � ��7�413

�2

�16�14

43

�1�11

42

�288

�42�

�B� � �220

�A� � �25 42. (a)

(b)

(c)

(d) �AB� � �4094

AB � �53

�1�29

35

�102

1816

105

�6�1

221

�1312

��B� � 89

�A� � �46

43.

Thus, �wy x

z� � �� y

w

z

x�.�� y

w

z

x� � ��xy � wz� � wz � xy

�wy x

z� � wz � xy 44.

Thus, �wy cx

cz� � c�wy x

z�.c�wy x

z� � c�wz � xy�

�wy cx

cz� � cwz � cxy � c�wz � xy�

45.

Thus, �wy x

z� � �wy x � cw

z � cy�.�wy x � cw

z � cy� � w�z � cy� � y�x � cw� � wz � xy

�wy x

z� � wz � xy 46.

Thus, � w

cw

x

cx� � 0.

� w

cw

x

cx� � cxw � cxw � 0

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650 Chapter 7 Linear Systems and Matrices

47.

� �y � x��z � x��z � y�

� �y � x��z�z � x� � y�z � x��

� �y � x��z2 � zx � zy � xy�

� �y � x��z2 � zy � zx � xy�

� �y � x��z2 � z�y � x� � xy�

� z2�y � x� � z�y � x��y � x� � xy�y � x�

� z2�y � x� � z�y2 � x2� � xy�y � x�

� yz2 � xz2 � y2z � x2z � xy�y � x�

� �yz2 � y2z� � �xz2 � x2z� � �xy2 � x2y�

�111 x

y

z

x2

y2

z2� � �yz y2

z2� � �xz x2

z2� � �xy x2

y2�

48.

� 3ab2 � b3 � b2�3a � b�

� a3 � 3a2b � 3ab2 � b3 � 3a3 � 3a2b � 2a3

� �a � b�3 � 3a2�a � b� � 2a3

� �a � b�3 � a2�a � b� � a2�a � b� � a3 � a3 � a2�a � b�

� �a � b���a � b�2 � a2� � a�a�a � b� � a2� � a�a2 � a�a � b��

�a � b

a

a

a

a � b

a

a

a

a � b� � �a � b��a � b

a

a

a � b� � a�aa a

a � b� � a� a

a � b

a

a �

49.

x � ±2

x2 � 4

x2 � 2 � 2

�x1

2x� � 2 50.

x � ± 4

x2 � 16

x2 � 4 � 20

� x�1

4x� � 20 51.

x � ± 32

x2 �94

4x2 � 9

4x2 � 6 � 3

� 2x�2

�32x� � 3

52.

x � ± 43

x2 �169

9x2 � 16

9x2 � 8 � 8

�x4

29x� � 8 54.

x � �2, 1

�x � 2��x � 1� � 0

x2 � x � 2 � 0

x2 � x � 2 � 4

�x � 1�1

2x� � 453.

x � 1 ± �2

x �2 ± �4 � 4

2

x2 � 2x � 1 � 0

x2 � 2x � 2 � �1

�x2

1x � 2� � �1

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Section 7.7 The Determinant of a Square Matrix 651

55.

x � �4, �1

�x � 4��x � 1� � 0

x2 � 5x � 4 � 0

�x � 3��x � 2� � 2 � 0

�x � 31

2x � 2� � 0 56.

x � �1 or x � 3

�x � 1��x � 3� � 0

x2 � 2x � 3 � 0

x�x � 2� � ��3���1� � 0

�x � 2

�3

�1

x� � 0

57.

x � 1, 12

�x � 1��2x � 1� � 0

2x2 � 3x � 1 � 0

2x2 � 2x � 1 � x

� 2x�1

1x � 1� � x 58.

x � 3, �2

�x � 3��x � 2� � 0

x2 � x � 6 � 0

�x2 � x � 6 � 0

2x � 2 � x2 � x � �8

�x � 1x � 1

x2� � �8

60.

x � 8

2x � 16

�12 � ��2x � 4� � 0

1�32 3�2��1�x

2�2�2� � 0 �Expand first column�

�110

x32

�23

�2� � 0

61. � 4u

�1

�1

2v� � 8uv � 1 62. �3x2

1

�3y2

1 � � 3x2 � ��3y2� � 3x2 � 3y2

63. � e2x

2e2x

e3x

3e3x � � 3e5x � 2e5x � e5x 64.

� e�2x

� e�2x � xe�2x � xe�2x

� e�x

�e�x

xe�x

�1 � x�e�x � � �1 � x�e�2x � ��xe�2x�

65. �x

1

ln x1x� � 1 � ln x

66.

� x � x ln x � x ln x � x

�x

1

x ln x

1 � ln x� � x�1 � ln x� � x ln x

67. True. Expand along the row of zeros. 68. True

59.

x � 3

21 � 7x

7 � 2��7� � x��7� � 0

1� 3�2

21� � 2��1

321� � x��1

33

�2� � 0

� 1�1

3

23

�2

x21� � 0

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652 Chapter 7 Linear Systems and Matrices

69. Let

Thus, Your answer may differ, depending on how you choose and B.A�A � B� � �A� � �B�. A � B � ��3

1

3

9�, �A � B� � ��3

1

3

9� � �30

�A� � � 1

�2

3

4� � 10, �B� � ��4

3

0

5� � �20

A � � 1

�2

3

4� and B � ��4

3

0

5�.

70.

For an matrix with consecutive integer entries, the determinant appears to be 0.

� �3x � 6x � 6 � 3x � 6 � 0

� �3x � �x � 1���6� � �x � 2���3�

� �x2 � 11x � 30�� � �x � 2���x2 � 10x � 21� � �x2 � 10x � 24��

� x��x2 � 12x � 32� � �x2 � 12x � 35�� � �x � 1���x2 � 11x � 24�

� �x � 6��x � 5�� � �x � 2���x � 3��x � 7� � �x � 6��x � 4��

� x��x � 4��x � 8� � �x � 7 ��x � 5�� � �x � 1���x � 3��x � 8�

� x

x � 3

x � 6

x � 1

x � 4

x � 7

x � 2

x � 5

x � 8� � x�x � 4

x � 7

x � 5

x � 8� � �x � 1��x � 3

x � 6

x � 5

x � 8� � �x � 2��x � 3

x � 6

x � 4

x � 7��n > 2�n � n

�57

61

65

69

58

62

66

70

59

63

67

71

60

64

68

72� � 0�19

23

27

31

20

24

28

32

21

25

29

33

22

26

30

34� � 0

��5

�2

1

�4

�1

2

�3

0

3� � 0�33

36

39

34

37

40

35

38

41� � 0

�10

13

16

11

14

17

12

15

18� � 0� 4

7

10

5

8

11

6

9

12� � 0

71. (a)

(b)

(c)

(d) In general, det�A�1� �1

det A.

det�A�1� �16

A�1 � �1313

�1316�

�A� � 6 72. (a)

(b)

(c)

(d) In general, det�A�1� �1

det A.

det�A�1� � �13

A�1 � �1323

�13

�53�

�A� � �3

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Section 7.7 The Determinant of a Square Matrix 653

80. x2 � 5x � 6 � �x � 2��x � 3�

81. 4y2 � 12y � 9 � �2y � 3�2 82. 4y2 � 28y � 49 � �2y � 7�2

83.

Answer: �2, �4�

y � �2 � 2 � �4

x � 2

13x � 26

3x � 10��x � 2� � 46

y � �x � 2

x � y � �2

3x � 10y � 46

84.

Answer: ��1, 4�

x � �1, y � 2 � 2��1� � 4

�9x � 9

5x � 7�2 � 2x� � 23

5x�4x

7y2y

23�4 ⇒ 2x � y � 2 ⇒ y � 2 � 2x

78. Answers will vary.

79. x2 � 3x � 2 � �x � 2��x � 1�

73. (a)

(b)

(c)

(d) In general, det�A�1� �1

det A.

det�A�1� �12

A�1 � ��4�1�1

�5�1�1

1.50.5

0��A� � 2 74. (a)

(b)

(c)

(d) In general, det�A�1� �1

det A.

det�A�1� �14

A�1 � ��1.25

0.25�0.5

201

�2.250.25

�1.5��A� � 4

75. (a) Columns 2 and 3 are interchanged.

(b) Rows 1 and 3 are interchanged.

76. (a) times Row 1 is added to Row 2.

(b) times Row 2 is added to Row 1.��2�

��5�

77. (a) 5 is factored out of the first row of

(b) 4 and 3 are factored out of columns 2 and 3.

A.

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654 Chapter 7 Linear Systems and Matrices

Section 7.8 Applications of Matrices and Determinants

■ You should be able to find the area of a triangle with vertices

The symbol indicates that the appropriate sign should be chosen so that the area is positive.

■ You should be able to test to see if three points, are collinear.

if and only if they are collinear.

■ You should be able to use Cramer’s Rule to solve a system of linear equations.

■ Now you should be able to solve a system of linear equations by substitution, elimination, elementary rowoperations on an augmented matrix, using the inverse matrix, or Cramer’s Rule.

■ You should be able to encode and decode messages by using an invertible matrix.n � n

�x1

x2

x3

y1

y2

y3

1

1

1� � 0,

�x1, y1�, �x2, y2�, and �x3, y3�,±

Area � ±12�x1

x2

x3

y1

y2

y3

1

1

1��x1, y1�, �x2, y2�, and �x3, y3�.

1. Vertices:

Area �52 square units

� 12 ��2��2� � 4�3� � 13� �

12 �5� �

52

12��22

�1

435

111� �

12��2�35 1

1� � 4� 2�1

11� � � 2

�135��

��2, 4�, �2, 3�, ��1, 5�

2. Vertices:

Area square units� 28

12��3

23

56

�5

111� �

12 ���3�11 � 5��1� � 1��28�� �

12 ��56�

��3, 5�, �2, 6�, �3, �5�

3. Vertices:

Area �338 square units

12�0524 1

2

03

111� �

12 ��1

2��32� �

152 � �

12 �33

4 � �338 .

�0, 12�, �52, 0�, �4, 3�

Vocabulary Check

1. collinear 2. Cramer’s Rule 3. cryptogram 4. uncoded, coded

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Section 7.8 Applications of Matrices and Determinants 655

5.

Area rhombus square units� ��24� � 24

��3

1

�1

2

2

�4

1

1

1� � �3 �6� � 2�2� � 1��2� � �24

6.

Area rhombus square units� ��54� � 54

��4

6

2

4

8

1

1

1

1� � �4 �7� � 4�4� � 1��10� � �54

7.

or

or

x � 0, �165

x � �165x � 0

5x � 8 � �85x � 8 � 8

8 � ± �5x � 8�

8 � ± ���1���2� � 5��2 � x� � 1��4��

4 � ± 12��1

�2

x

5

0

2

1

1

1� 8.

or

or

x � 3, 19

x � 3x � 19

x � 11 � �8x � 11 � 8

8 � ± �x � 11�

8 � ± ��4�5 � x� � 2��2� � 1��3x � 5��

4 � ± 12��4

�3

�1

2

5

x

1

1

1�

9. Points:

The points are collinear.

� 3

0

12

�1

�3

5

1

1

1� � 3��8� � 12�2� � 0

�3, �1�, �0, �3�, �12, 5� 10. Points:

The points are not collinear.

� 8 � 26 � 33 � 15 � 0

�364 �5

1

2

1

1

1� � �64 1

2� � �34 �5

2� � �36 �5

1��3, �5�, �6, 1�, �4, 2�

11. Points:

The points are not collinear.

� �3 � 0

� 2

�4

6

�12

4

�3

1

1

1� � 2�7� �12 ��10� � 1��12�

�2, �12�, ��4, 4�, �6, �3� 12. Points:

The points are collinear.

� �3 � 3 � 0

� 02

�4

12

�172

111� � �

12 �2 � 4� � 1�7 � 4�

�0, 12�, �2, �1�, ��4, 72�

4. Vertices:

.

Area square units�1238

� 12 �30.75� � 15.375 �

1238

12� 9

2

20

06

�32

111� �

12 �9

2�6 �32� � 1��3��

�92, 0�, �2, 6�, �0, �3

2�©

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656 Chapter 7 Linear Systems and Matrices

15.

Answer: ��3, �2�

y ���7

3�1

9���7

311

�9� ��6030

� �2

x ���1

911

�9���7

311

�9� ��9030

� �3

��7x3x

11y9y

�19

16.

Answer: ��1, 2�

y ��46 �10

12��46 �3

9� �10854

� 2

x ���10

12�3

9��46 �3

9� ��5454

� �1

�4x6x

3y9y

�1012

17.

Cramer’s rule cannot be used.

(In fact, the system is inconsistent.)

�36 24� � 12 � 17 � 0

�3x6x

2y4y

�24

18.

Answer: �7, 5�

y �� 6�13

17�76�

� 6�13

�53� �

�235�47

� 5

x �� 17�76

�53�

� 6�13

�53� �

�329�47

� 7

� 6x�13x

5y3y

17�76

19.

Answer: 32

7,

30

7

y ���0.4

0.2

1.6

2.2��0.28

��1.20

�0.28�

30

7

x ��1.6

2.2

0.8

0.3��0.28

��1.28

�0.28�

32

7

D � ��0.4

0.2

0.8

0.3� � �0.28

��0.4x0.2x

0.8y0.3y

1.62.2

20.

Answer: �5, 1.5�

y ��2.44.6

10.824.8�

6.56�

9.846.56

� 1.5

x ��10.824.8

�0.81.2�

6.56�

32.86.56

� 5

D � �2.44.6

�0.81.2� � 6.56

�2.4x4.6x

0.8y1.2y

10.824.8

13.

x � 3

8x � 24 � 0

1��4� � 2�x � 5� � 1�6x � 10� � 0

�1x5 �2

2

6

1

1

1� � 0 14.

x � 3

�3x � �9

�25 � 3x � 24 � 6x � 10 � 0

��5

�3

x

5� � ��6

�3

2

5� � ��6

�5

2

x� � 0

��6

�5

�3

2

x

5

1

1

1� � 0

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Section 7.8 Applications of Matrices and Determinants 657

21.

Answer: ��1, 3, 2�

z ��425 �1

2

�2

�5

10

1�55

�110

55� 2y �

�425 �5

10

1

1

3

6�55

�165

55� 3,x �

��5

10

1

�1

2

�2

1

3

6�55

��55

55� �1,

D � �425 �1

2

�2

1

3

6� � 55�4x

2x

5x

y

2y

2y

z

3z

6z

�5

10

1

22.

Answer: �5, 8, �2�

z ��428 �2

2

�5

�2

16

4��82

�164

�82� �2 y �

�428 �2

16

4

3

5

�2��82

��656

�82� 8 x �

��2

16

4

�2

2

�5

3

5

�2��82

��410

�82� 5

D � �428 �2

2

�5

3

5

�2� � �82�4x2x8x

2y2y5y

3z5z2z

�2164

23. (a)

Answer: �0, �12, 12�

3x � 3��12� � 5�1

2� � 1 ⇒ x � 0

2y � 4�12� � 1 ⇒ y � �

12

z �12

�3x � 3y2y

5z4z23z

1113

�3x � 3y2y4y

5z4z

263 z

1173

�3x3x5x

3y5y9y

5z9z

17z

124

(b)

Answer: �0, �12, 12�

z ��335 3

5

9

1

2

4�4

�1

2

y ��335 1

2

4

5

9

17�4

� �1

2

x ��124 3

5

9

5

9

17�4

� 0

D � �335 3

5

9

5

9

17� � 4

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Page 111: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

658 Chapter 7 Linear Systems and Matrices

24. (a)

Answer: �1, �2, �1�

x � �1 � 5 � 6��2 � 1

y � �35 � 33 � �2

z � �1

�2x � 3yy

5z33z

�20z

1�35

20

�2x � 3y

12y32y

5z332 z592 z

1

�352

�652

�2x3x5x

3y5y9y

5z9z

17z

1�16�30

(b)

Answer: �1, �2, �1�

z ��235 3

5

9

1

�16

�30��20

�20

�20� �1

y ��235 1

�16

�30

�5

9

17��20

�40

�20� �2

x �� 1

�16

�30

3

5

9

�5

9

17��20

��20

�20� 1

D � �235 3

5

9

�5

9

17� � �20

25. (a)

Answer: �1, �1, 2�

x � 2��1� � 2 � 1 ⇒ x � 1

y � 2 � 1 ⇒ y � �1

z � 2

�x � 2yy

zz

�3z

11

�6

�x � 2y3y7y

z3z4z

131

�2xx

3x

y2yy

zzz

514

(b)

Answer: �1, �1, 2�

z ��213 �1

�2

1

5

1

4�9

� 2

y ��213 5

1

4

1

�1

1�9

� �1

x ��514 �1

�2

1

1

�1

1�9

� 1

D � �213 �1

�2

1

1

�1

1� � 9

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Page 112: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.8 Applications of Matrices and Determinants 659

(b)

z ��321 �1

1

1

1

�4

5��16

� �21

8

y ��321 1

�4

5

�3

2

�1��16

�7

2

x �� 1

�4

5

�1

1

1

�3

2

�1��16

� �9

8

D � �321 �1

1

1

�3

2

�1� � �1626. (a)

Answer: ��98, 72, �21

8 �

x �72

�218 � 5 ⇒ x � �

98

�72

� 4z � �14 ⇒ z � �218

y �72

x � y�y

�4y

z4z

5�14�14

�3x2xx

yyy

3z2zz

1�4

5

27. (a)

(b)

(c) No, because of the negative coefficient.t2

−1 50

1,500

y � �26.36t2 � 241.5t � 784

a �� 5

10

30

10

30

100

5543

12,447

38,333�700

��18,450

700� �26.36

b �� 5

10

30

5543

12,447

38,333

30

100

354�700

�169,070

700� 241.5

c �� 5543

12,447

38,333

10

30

100

30

100

354�700

�548,580

700� 784

D � � 5

10

30

10

30

100

30

100

354� � 700 28. (a)

(b)

(c) for t � 4.3, or 2004.y � 1500

−1 50

15,000

y � �689.21t2 � 2349.6t � 10,182

a �� 5

10

30

10

30

100

53,729

103,385

296,433�700

��482,450

700� �689.21

b �� 5

10

30

53,729

103,385

296,433

30

100

354�700

�1,644,690

700� 2349.6

c �� 53,729

103,385

296,433

10

30

100

30

100

354�700

�7,127,380

700� 10,182

D � � 5

10

30

10

30

100

30

100

354� � 700

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660 Chapter 7 Linear Systems and Matrices

31.

Cryptogram: 38 63 51 58 119 133 44 88 95�32�14�1

GEII

O

SN

NFHG

�7599

150

1914

14687�� 1

3�1

27

�4

29

�7� � �38

�15844

63�1411988

51�3213395

�32. H A P P Y _ B I R T H D A Y _

[8 1 16] [16 25 0] [2 9 18] [20 8 4] [1 25 0]

Cryptogram: �5 �41 �87 91 207 257 11 �5 �41 40 80 84 76 177 227

� 11 25 10�A � � 76 177 227�

� 20 28 14�A � � 40 80 84�

� 12 29 18�A � � 11 �5 2�41�

� 16 25 10�A � � 91 207 257�� 18 21 16�A � ��5 �41 �87�

33.

Message: HAPPY NEW YEAR �11642529234055

211125053467592

� ��5

3

2

�1� � �

816251423251

1160505

18

HPYNWYA

AP

E

ER

A�1 � �1

3

2

5��1

� ��5

3

2

�1�

29. The uncoded row matrices are the rows of the matrix on the left.

Answer:

��62, 15, 32�, ��54, 12, 27�, �23, �23, 0�

��68, 21, 35�, ��66, 14, 39�, ��115, 35, 60�

CLEOR

W

A

MR

LMTOO

3125�15

1823

100

13180

12132015150� �

11

�6

�102

0�1

3� � ��68�66

�115�62�54

23

2114351512

�23

35396032270�

6 � 3

30. The uncoded row matrices are the rows of the matrix on the left.

Answer: �43 6 9�, ��38 �45 �13�, ��42 �47 �14�, �44 16 10�, �49 9 12�, ��55 �65 �20�

43

�38

�42

44

49

�55

6

�45

�47

16

9

�65

9

�13

�14

10

12

�20

�4

�3

3

2

�3

2

1

�1

1� �

P

A

N

M

E

L

S

S

D

O

Y

E

E

E

N

16

1

0

14

13

5

12

19

19

4

15

25

5

5

5

0

14

0

6 � 3

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Page 114: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Section 7.8 Applications of Matrices and Determinants 661

34.

TO BE OR NOT TO BE

T—EO

—O

—OB

OB

—RNTT

—E

2005

150

150

152

1520

1814202005

856

10844290603019

1208

1511756

125804526

A � �23

34�, A�1 � ��4

33

�2�

��43

3�2� �

35.

YANKEES WIN WORLD SERIES

381142

�5268

3223

36171518282420

�1

�111

�27371720

�7�19

23�013

1�1

1

43

�123� �

2511199

2312199

150

1415455

145

230

180

1819

YKSI

WLSI

AE_

NODEE

NE

W_

R_

RS

� �23

013

1�1

1

43

�123�A�1 � �

1�1

1

20

�1

12

�2��1

36. Let A be the matrix needed to decode the message.

Message: MEET ME TONIGHT RON�

8

�15

�13

5

5

5

�1

20

�18

1

21

�10

�13

10

25

19

6

40

�18

16

� ��1

1

�2

1� � �

13

5

0

5

20

14

7

20

0

15

5

20

13

0

15

9

8

0

18

14

M

E

E

T

N

G

T

O

E

T

M

O

I

H

R

N

A � ��18

1

�18

16��1

� 0

15

18

14� � ��8

1351

270

�115115� � 0

15

18

14� � ��1

1

�2

1�

��18

1

�18

16�A � � 0

15

18

14�

O

R

N

2 � 2

37. True. Cramer’s Rule requires that the determinantof the coefficient matrix be nonzero.

38. False. The system has solutions,

yet det�12

12� � 0.

� x2x

y2y

12©

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Page 115: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

662 Chapter 7 Linear Systems and Matrices

39. Answers will vary. 40. Answers will vary.

41.

x � 4y � 19 � 0

4y � x � 19

4y � 20 � �x � 1

y � 5 �5 � 3

�1 � 7�x � 1� �

�14

�x � 1�

43.

2x � 7y � 27 � 0

7y � 2x � �27

7y � 21 � 2x � 6

y � 3 ��3 � 13 � 10

�x � 3� �27

�x � 3�

45.

Horizontal asymptote:

1

2 31−1−2−3

3

−1

−2

−3

x

y

y � 2

f �x� �2x2

x2 � 4.

42.

8x � y � 6 � 0

y � �8x � 6

y � 6 �10 � ��6�

�2 � 0�x � 0� � �8x

44.

5x � 4y � 28 � 0

4y � 5x � 28

4y � 48 � �5x � 20

y � 12 �12 � 2�4 � 4

�x � 4� � �54

�x � 4�

46.

Vertical asymptotes:

Horizontal asymptote:

4

6−4

2

−2 4

6

−6

−4

−8

8

x

y

y � 0

x � �6, 3

f �x� �2x

x2 � 3x � 18�

2x�x � 6��x � 3�

Review Exercises for Chapter 7

2.

Answer: ��3, �3�

x � �3 ⇒ y � �3

�x � 3

2x � 3�x� � 3

�2xx

3yy

3 0 ⇒ y � x

1.

Answer: �1, 1�

y � 2 � 1 � 1

x � 1

2x � 2 � 0

x � y � 0 ⇒ x � �2 � x� � 0

x � y � 2 ⇒ y � 2 � x�©

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Review Exercises for Chapter 7 663

4.

Answer: �13, 0�, �5, 12�

y � 12: x �13�39 � 2�12�� � 5

y � 0: x �13�39 � 2�0�� � 13

133 y�1

3y � 4� � 0 ⇒ y � 0, 12

139 y2 �

523 y � 0

169 �523 y �

49y2 � y2 � 169

19�1521 � 156y � 4y2� � y2 � 169

�13�39 � 2y��2

� y2 � 169

�x2

3x�

y2

2y�

169 39 ⇒ x �

13�39 � 2y�

3.

Answer: �5, 4�

x � 5

y � 4

2y � 1 � 9

�y � 1�2 � y2 � 9

x � y � 1 ⇒ x � y � 1

x2 � y2 � 9 �

5.

Answer: �0, 0�, ��2, 8�, �2, 8�

y � 0, y � 8, y � 8

x � 0, x � �2, x � 2

0 � x2�x � 2��x � 2� 0 � x2�x2 � 4� 0 � x4 � 4x2

y � x4 � 2x2 ⇒ 2x2 � x4 � 2x2

y � 2x2

� 6.

Answer: �5, 2�, �2, �1�

y � �1: x � �1 � 3 � 2

y � 2: x � 2 � 3 � 5

0 � � y � 2��y � 1� ⇒ y � 2, �1

0 � y2 � y � 2

y � 3 � y2 � 1

�xx

yy2

31

7.

Answer: �2, �12�

−9

−6

9

6

y2 � �x4

⇒ �x � 4y � 0

y1 �16

�7 � 5x� ⇒ 5x � 6y � 7

� 8.

Answer: �1.5, 5�

−9

−2

9

10

�8x2x

3y5y

�328

⇒ ⇒

y �

y �

13�8x � 3�

15�28 � 2x�

�83 x � 1

9.

Points of intersection: �0, 0�, �4, �4�

−6 12

−6

6

�y2

x�

4xy

00

⇒ ⇒

y2 �

y �

4x�x

⇒ y � ± 2�x 10.

Points of intersection: �2, 1�, �3.25, �1.5�

−2 10

−4

4

�y2

y�

x2x

�15

⇒ ⇒

y2 �

y �

x � 15 � 2x

⇒ y � ±�x � 1

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664 Chapter 7 Linear Systems and Matrices

14.

Point of intersection: �2, 10�

y � 16 � 3x

−2 70

12�3x � yy

161 � 3x

15.

Point of intersection: ��1, 1� −4 5

−3

3�y � ln�x � 2� � 1y � �x

16.

Point of intersection: �2, 3�−1 8

−1

5�y � ln�x � 1� � 3y � 4 �

12x

13.

Point of intersection: �4, 4�−2

−2

10

6�y � 2�6 � x�y � 2x�2

17. Revenue

Cost

Break even when Revenue Cost

x � 4762 units

2.10x � 10,000

4.95x � 2.85x � 10,000

� 2.85x � 10,000

� 4.95x 18.

Answer: More than $500,000

$500,000 � x

2500 � 0.005x

22,500 � 0.015x � 20,000 � 0.02x

�yy

22,50020,000

0.015x0.02x

19.

The dimensions are meters.96 � 144

l � 144

w � 96

5w � 480

2�1.50w� � 2w � 480

l � 1.50w

2l � 2w � 480�20.

The dimensions are 16 � 18 feet.

w � 16

l � 18

34l9

� 68

2l � 2�89

l � 68

w �89l

2l � 2w � 68

21.

Answer: �52, 3�

y � 3

x �5522 �

52

22x � 55

6x � 8y � 39 ⇒ 6x � 8y � 39

2x � y � 2 ⇒ 16x � 8y � 16� 22.

Answer: � 310, 25�

x �310

y �25

130y � 52

�40x20x

30y50y

24�14

⇒ ⇒

40x�40x

30y100y

2428

11.

Points of intersection:

�0.67, 2.56�, ��1, 2�−4 5

−2

4� y � 3 � x2

y � 2x2 � x � 112.

Points of intersection:

or

�1.41, �0.66�, ��1.41, 10.66�

��2, 5 � 4�2�, ���2, 5 � 4�2�

−2

−2

4

14�yy

2x2

x2

4x � 14x � 3

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Review Exercises for Chapter 7 665

24. Interchange the equations:

times Eq. 1 added to Eq. 2 produces:

Then

Answer: �35, �8�

�x �78��8� � �

385 ⇒ x �

35

�3796 y �

3712 ⇒ y � �85

12

Equation 1

Equation 2��x �78 y

512 x �

34 y

�385254

25.

Answer: �0, 0�

y � 0

x � 0

6x � 0

3x � 2�y � 5� � 10 ⇒ 3x � 2y � 0

3x � 2y � 0 ⇒ 3x � 2y � 0� 26.

Answer: ��3, 7�

y � 7

x � �3

�7x2x

12y3y

6315

⇒ ⇒

�7x8x

12y12y

�6360

28.

Inconsistent; no solution

0 � 5

�1.5x6x

2.5y10y

8.524

⇒ ⇒

3x�3x

5y5y

17�12

29.

Consistent.

Answer: �1.6, �2.4�

−4

−6

8

2

y � x � 4 x � y � 4 ⇒ y � �

32x 3x � 2y � 0 ⇒ � 30.

Consistent. Infinite number of solutions of formor All points on line

y � 6 � x.�6 � y, y�.�x, 6 � x�

−3

−1

9

7

�x � y�2y

6�12

⇒ � 2x ⇒

yy

66

xx

23.

Answer: ��12, 45� or ��0.5, 0.8�

x � �12

y � 45

5y � 4

25x � 12y � 15 ⇒ 4x � 5y � 2 ⇒ �20x � 25y � �10

15x � 310y � 750 ⇒ 20x � 30y � 14 ⇒ 20x � 30y � 14�

27.

Infinite number of solutions

Let then

Answer: �145 �

85a, a�5x � 8a � 14 ⇒ x �

145 �

85 a.y � a,

0 � 0

5x � 8y � 14 ⇒ �5x � 8y � �14

1.25x � 2y � 3.5 ⇒ 5x � 8y � 14�

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666 Chapter 7 Linear Systems and Matrices

33.

Consistent

Answer: ��4.6, �8.6�

4x � 1.5y � �5.5 ⇒ y � 83 x � 113

−11

−11

7

1 2x � 2y � 8 ⇒ y � x � 4

35.

Point of equilibrium: �500,0007

, 1597

x �500,000

7, p �

159

7

15 � 0.00021x

37 � 0.0002x � 22 � 0.00001x

Demand � Supply

34.

Consistent

Answer: ��7.52, 0.9�

�2x � 9.6y � 6.4 ⇒ y ��19.6

�2x � 6.4� −15

−7

6

7�x � 3.2y � 10.4 ⇒ y �1

3.2�x � 10.4�

36.

Points of equilibrium: �250,000, 95�

p � $95.00

x � 250,000 units

0.0003x � 75

45 � 0.0002x � 120 � 0.0001x

Supply � Demand

37. Let speed of the slower plane.

Let speed of the faster plane.

Then, distance of first plane distance of second plane 275 miles.

y � x � 25 � 218.75 mph

x � 193.75 mph

4x � 775

4x � 50 � 825

23x �

23�x � 25� � 275

y � x � 25

x�4060� � y�40

60� � 275

�rate of first plane��time� � �rate of second plane��time� � 275 miles

��

y �

x �

31.

Inconsistent; lines are parallel.

−7

−6

11

6

y �14�8 � 5x� �

54x � 2 �5x � 4y � 8 ⇒

y �54x � 10 14x � 15 y � 2 ⇒ � 32.

Inconsistent; lines are parallel.

−3

−1

3

3

�x � 2y � 4 ⇒ y � x2

� 2

72

x � 7y � �1 ⇒ y � �72�x � 1

7�

x2

�17�

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Review Exercises for Chapter 7 667

40.

Answer: �3, �1, �3�

x � 7y � 8z � �14 ⇒ x � �14 � 7��1� � 8��3� � 3

y � 9z � 26 ⇒ y � 9��3� � 26 � �1

z � �3

41.

Answer: �3817, 40

17, �6317�

x � 3�4017� � ��63

17� � 13 ⇒ x �3817

�6y � 3��6317� � �3 ⇒ y �

4017

172 z � �

632 ⇒ z � �

6317

�x �

3y6y

z3z

172 z

13�3�

632

�x �

3y6y

13y

z3z2z

13�3

�38

� x2x4x

3y

y

z5z2z

132314

42.

Answer: ��0.8, �2.4, 1.6�

x � 4 � 6�1.6� � 2��2.4� � �0.8

y � ��8 � 17�1.6����8� � �2.4

z �85 � 1.6

�x � 2y�8y

6z17z

�5z

4�8�8

�x � 2y�8y�8y

6z17z22z

4�8

�16

� x3x4x

2y2y

6zz

2z

440

38. Let amount invested in 6.75% bond.

Let amount invested in 7.25% bond.

Solving the system,

At most $18,000 can be invested in the 6.75% bond.

x � 18,000, y � 28,000.

�x � y � 46,0000.0675x � 0.0725y � 3245

y �

x � 39.

Answer: �4, 3, �2�

⇒ x � 4�3� � 3��2� � 14 � 4

x � 4y � 3z � �14 �y � z � �5 ⇒ �y � 2 � �5 ⇒ y � 3

z � �2

43.

Let then

Answer: where is any real number.

a�3a � 4, 2a � 5, a�

x � 3a � 4

x � 3a � 10 � �6

x � 2�2a � 5� � a � �6

y � 2a � 5.z � a,

�Eq. 2 � Eq. 3�x � 2y

y�

z2z0

�650

�2 Eq.1 � Eq. 2Eq. 1 � Eq. 3

�x � 2yyy

z2z2z

�655

� x2x

�x

2y3y3y

z

3z

�6�711

44.

Answer: �� 34, 0, � 5

4��x � 2�0� � 5�� 5

4� � 7 ⇒ x � �34

4�0� � 4z � 5 ⇒ z � �54

�3y � 0 ⇒ y � 0

�2 Eq. 2 � Eq. 3��x � 2y

4y�3y

5z4z

750

3 Eq. 1 � Eq. 23 Eq. 1 � Eq. 3

��x � 2y4y5y

5z4z8z

75

10

�Eq. 2 � Eq. 1��x3x3x

2y2yy

5z11z7z

7�16�11

Equation 1Equation 2Equation 3

�2x3x3x

2yy

6z11z7z

�9�16�11

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668 Chapter 7 Linear Systems and Matrices

45.

Answer: ��196 , 17

12, 13�x � 2�17

12� � 3�13� � 5 � �

196

4y � 5 � 2�13� �

173 ⇒ y �

1712

z �13

�x � 2y4y

3z2z3z

�551

�x � 2y8y4y

3zz

2z

�5115

� x2xx

2y4y2y

3z5zz

�510

46.

Adding the second and third equations,

Answer: �72, � 7

10, 110�

x � 2y � z � 5 � 2�� 710� �

110 � 5 �

72

z � 7y � 5 � 7�� 710� � 5 �

110

10y � �7 ⇒ y � �7

10

�x � 2y7y3y

zzz

5�5�2

� x2xx

2y3yy

zz

2z

553

47.

3 times Eq. 1 and times Eq. 2:

Adding,

Let then and

Answer: where is any real number.

a�a � 4, a � 3, a�

y � a � 3.x � a � 4z � a,

x � z � 4

5x � 5z � 20

5x � 5z � 36 � 16

5x � 12�z � 3� � 7z � 16

�y � z � 3 ⇒ y � z � 3.

� 15x�15x

36y35y

21z20z

48�45

��5�

Equation 1Equation 2�5x

3x�

12y7y

7z4z

169

48.

Let

Answer: �2 � 3a, 5a � 6, a�

x � �102 � 57a � 15�5a � 6��6 � �3a � 2

y � 5a � 6

z � a

�6x � 15yy

57z5z

1026

�6x6x

15y16y

57z62z

102108

�2x3x

5y8y

19z31z

3454

49.

�1, 0, 6��0, 0, 8�,�4, 0, 0�, �0, �2, 0�,

x y

22

64

64

8

z 50.

�0, 0, �9�, �0, 3, 0�, �3, 0, 0�, �1, 1, �3�

x y

z

−8

−10

6 64

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Review Exercises for Chapter 7 669

55.

x2 � 3x � 3x3 � 2x2 � x � 2

��1

x � 2�

2x � 1x2 � 1

C � �1, B � 2��1� � 4 � 2, A � 1 � 2 � �1

�A � BB � 2C

5C

14

�5

�A � B2B

�B�

C2C

13

�4

�A

A

� B2B �

C2C

13

�3

x2 � 3x � 3 � A�x2 � 1� � �Bx � C��x � 2�

�A

x � 2�

Bx � Cx2 � 1

x2 � 3x � 3x3 � 2x2 � x � 2

�x2 � 3x � 3

�x � 2��x2 � 1�

51.

4 � x

x2 � 6x � 8�

3

x � 2�

4

x � 4

⇒ A � 3, B � �4� A4A

B2B

�14

� �A � B�x � �4A � 2B�

4 � x � A�x � 4� � B�x � 2�

4 � x

x2 � 6x � 8�

A

x � 2�

B

x � 452.

Let :

Let :

�x

x2 � 3x � 2�

1

x � 1�

2

x � 2

2 � �B ⇒ B � �2x � �2

1 � Ax � �1

�x � A�x � 2� � B�x � 1�

�x

x2 � 3x � 2�

A

x � 1�

B

x � 2

53.

x2 � 2x

x3 � x2 � x � 1�

32

x � 1�

��12�x � 32

x2 � 1�

1

2�3

x � 1�

x � 3

x2 � 1

⇒ A �32, B � �

12, C �

32� A

�BA

BCC

120

x2 � 2x � A�x2 � 1� � �Bx � C��x � 1� � �A � B�x2 � �C � B�x � �A � C�

x2 � 2x

x3 � x2 � x � 1�

A

x � 1�

Bx � C

x2 � 1

54.

3x3 � 4xx4 � 2x2 � 1

�3x

x2 � 1�

x�x2 � 1�2

D � 0

C � 1

B � 0

A � 3

�A

AB

B� C

� D

3 0 4 0

3x3 � 4x � �Ax � B��x2 � 1� � Cx � D

3x3 � 4xx4 � 2x2 � 1

�3x3 � 4x�x2 � 1�2 �

Ax � Bx2 � 1

�Cx � D

�x2 � 1�2

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670 Chapter 7 Linear Systems and Matrices

56.

2x2 � x � 7x4 � 8x2 � 16

�2

x2 � 4�

�x � 1�x2 � 4�2

A � 0, B � 2, C � �1, D � �1

�A

4AB

4B� C

� D

0 2

�1 7

2x2 � x � 7 � �Ax � B��x2 � 4� � Cx � D

2x2 � x � 7x4 � 8x2 � 16

�2x2 � x � 7

�x2 � 4�2 �Ax � Bx2 � 4

�Cx � D

�x2 � 4�2

57.

Solving the system,

y � 2x2 � x � 5

a � 2, b � 1, c � �5.

��1, �4��1, �2�

�2, 5�

���

aa

4a

bb

2b

ccc

�4�2

5

y � ax2 � bx � c 58.

Solving the system,

y � �x2 � 2x � 3

a � �1, b � 2, c � 3.

��1, 0��1, 4��2, 3�

���

aa

4a

bb

2b

ccc

043

y � ax2 � bx � c

59. Let gallons of spray X

Let gallons of spray Y

Let gallons of spray Z

Subtracting gives

Then and

Answer: 10 gallons of spray X

5 gallons of spray Y

12 gallons of spray Z

y � 5.z � 12

15 x � 2 ⇒ x � 10.Eq. 2 � Eq. 1

�Chemical A: Chemical B: Cehmical C:

15x25x25x

� y �

13z13z13z

6

8

13

z �

y �

x � 60. Let be amount invested at 7%.

Let be amount invested at 9%.

Let be amount invested at 11%.

Solving the system,x � $8000, y � $5000, z � $7000.

�x

0.07xxx

y0.09y

y

z0.11z

z

20,000178030001000

z

y

x

62. Order 2 � 461. Order 3 � 1 63. Order 1 � 1 64. Order 1 � 5

65. �3

5

�10

4��

15

22� 66. ��110

1�4

��

12�90�

68. �3

�46

�501

1�2

0

���

25�14

15�

67. �8

3

5

�7

�5

3

4

2

�3

���

12

20

26�

69.

�5x � 2y � 7z � �9

4x � 2y � 7z � 10

9x � 4y � 2z � �3

�5

4

9

1

2

4

7

0

2

���

�9

10

3�

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Review Exercises for Chapter 7 671

75. �1.5

0.2

2.0

3.6

1.4

4.4

4.2

1.8

6.4� ⇒ �

1

0

0

0

1

0

0

0

1� 76. �

1

0

0

0

1

0

0

0

1�

78.

Answer: ��9, �4�

x � 2��4� � 1 � �9

y � �4

�3R1 � R2 → �10

�2�1

��

�14�

R2 � R1 → �13

�2�7

��

�11�

�23

�5�7

��

21�

80.

Answer: �0.6, 0.5�

x � 0.5�0.5� � 0.35 � 0.6

y � 0.5

5R1 →�2R1 � R2 → �1

0�0.5�0.3

��

0.35�0.15�

�0.20.4

�0.1�0.5

��

0.07�0.01�

70.

�13xx

4x

16y21y10y

7z8z4z

3w5w3w

212

�1

�13

1

4

16

21

10

7

8

�4

3

5

3

���

2

12

�1�

72.

Other answers possible

�31

�2

5�2

0

245� ⇒ �

100

�211

�4

4�10

13� ⇒ �100

�210

4�29

1�

73. �3

4

�2

�3

1

0

0

1� ⇒ �1

0

0

1

3

4

�2

�3� 74. �100

010

001

�611

�2

�46

�1

3�5

1�

71.

3R2 � R1→�R2→

�4R2 � R3→�1

0

0

0

1

0

1

1

�2�

R1 � R2→�R1 � R2→

�2R1 � R3→�1

0

0

3

�1

�4

4

�1

�6�

�0

1

2

1

2

2

1

3

2�

77.

Answer: �10, �12�

x � �8��12� � 86 � 10

y � �12

9y � �108

4R2 � R1→R1 � R2→

�1

0

8

9��

�86

�108�

� 5

�1

4

1��

2

�22�

79.

Answer: ��0.2, 0.7�

x � �0.2, y � 0.7

x � 2�0.7� � 1.6 � �0.2

5y � 3.5 ⇒ y � 0.7

�2R1 � R2→

�1

0

�2

5��

�1.6

3.5�

�1

2

�2

1��

�1.6

0.3�

�R1 � R2→

�2

1

1

�2��

0.3

�1.6�

�2

3

1

�1��

0.3

�1.3�

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672 Chapter 7 Linear Systems and Matrices

82.

Inconsistent, no solution

0 � 1

�2R2 � R1→

5R2 � R3→ �

1

0

0

0

1

0

0

3

0

���

�3

2

1�

�2R1 � R2→�3R1 � R3→

�1

0

0

2

1

�5

6

3

�15

���

1

2

�9�

�1

2

3

2

5

1

6

15

3

���

1

4

�6�

83.

Answer: a is a real number.�1 � 3a, �a, a�,

x � 1 � 3z

y � �z

�10

21

�11

��

10� ⇒ �1

001

�31

��

10�

�x � 2y � z � 1y � z � 0

84.

Answer: �1, �1, 0, �2�

�1102

�1310

4�2�1

1

�1111

����

4�4�3

0� ⇒ �

1000

0100

0010

0001

����

1�1

0�2

��

xx

2x

y3yy

4z2zzz

wwww

4�4�3

0

81.

Answer: �12, � 1

3, 1�x � 3�1� �

72 ⇒ x �

12

y � 1 � � 43 ⇒ y � � 1

3

z � 1

12R1→

� 13R2→

� 128R3→

�1

0

0

0

1

0

3

�1

1

��

72

� 43

1�

R2 � R1→

�3R2 � R3→ �2

0

0

0

�3

0

6

3

�28

���

7

4

�28�

�3R1 � R2→�6R1 � R3→ �

2

0

0

3

�3

�9

3

3

�19

���

3

4

�16�

�2

6

12

3

6

9

3

12

�1

���

3

13

2�

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Review Exercises for Chapter 7 673

88.

Answer: �15, �3

5, 110�

�1

2

�1

1

1

1

4

2

�2

���

0

0

�1� ⇒ �1

0

0

0

1

0

0

0

1

���

15

�35110�

� x2x

�x

yyy

4z2z2z

00

�1

90.

�x, y, z� � ��2, 1, 1�

�3

�2

0

0

1

1

6

0

2

���

0

5

3� ⇒ �

1

0

0

0

1

0

0

0

1

���

�2

1

1�

86.

Answer: �3142, 5

14, 1384�

x �3142, y �

514, z �

1384

8R3 � R1→12R3 � R2→

184R3→

�1

0

0

0

1

0

0

0

1

..

.

..

.

..

.

31425141384�

R2 � R1→12R2→

�8R2 � R3→ �

1

0

0

0

1

0

�8

�12

84

..

.

..

.

..

.

�12

�32

13�

R3 � R1→�4R1 � R2→�5R1 � R3→

�1

0

0

�1

2

8

4

�24

�12

..

.

..

.

..

.

1

�3

1�

�4

4

5

4

�2

3

4

�8

8

..

.

..

.

..

.

5

1

6� 85.

Answer: �2, �3, 3�

x � 2, y � �3, z � 3

R3 � R1→�R3 � R2→ �

1

0

0

0

1

0

0

0

1

���

2

�3

3�

13R3→

�1

0

0

0

1

0

�1

1

1

���

�1

0

3�

R2 � R1→

�9R2 � R3→�1

0

0

0

1

0

�1

1

3

���

�1

0

9�

15 R2→ �1

0

0

�1

1

9

�2

1

12

���

�1

0

9�

�R1→2R1 � R2→5R1 � R3→

�1

0

0

�1

5

9

�2

5

12

���

�1

0

9�

��1

2

5

1

3

4

2

1

2

���

1

�2

4�

87.

Answer: �1, 2, 12�

�1

1

2

1

�1

�1

2

4

2

���

4

1

1� ⇒ �

1

0

0

0

1

0

0

0

1

���

1

212�� x

x2x

yyy

2z4z2z

411

89. reduces to

Answer: �3, 0, �4�

�100

010

001

���

30

�4��104

2�1

0

�1�1�1

���

74

16�

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Page 127: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

674 Chapter 7 Linear Systems and Matrices

92.

Inconsistent. No solution �0 � 1�

�411

�2

1266

�10

241

�2

����

20128

�10� ⇒ �

1000

0100

0010

����

0001�

94.

y � 0

x � 8

96.

Answer: x � 12, y � �2

6 �12 x ⇒ x � 12

�4 � 2y ⇒ y � �2

2 � x � 10 ⇒ x � 12

93.

y � �7

x � 12

95.

Answer: x � 1, y � 11

6x � 6 ⇒ x � 1

y � 5 � 16 ⇒ y � 11

�4y � �44 ⇒ y � 11

x � 3 � 5x � 1 ⇒ x � 1

98. (a) not possible (b) not possible

(c) (d) not possibleA � 3B4A � ��44�28

64�8

764�

A � BA � B

97. (a)

(b) (c)

(d) A � 3B � � 7�1

35� � �30

42�60�9� � �37

41�57�4�

4A � � 28�4

1220�A � B � � �3

�15238�

A � B � � 7�1

35� � �10

14�20�3� � �17

13�17

2�

99. (a)

(b)

(c)

(d) A � 3B � �653

0�1

2

723� � �

0�12

6

1524

�3

3183� � �

6�7

9

1523

�1

10206�

4A � �242012

0�4

8

288

12�

A � B � �691

�5�9

3

6�4

2�

A � B � �653

0�1

2

723� � �

0�4

2

58

�1

161� � �

615

571

884�

91. reduces to

Answer: �2, 6, �10, �3�

�1000

0100

0010

0001

����

26

�10�3

��3150

�16

�14

541

�1

�2�1

3�8

����

�441

�1558

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Page 128: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 675

105. �32

6

�172

46

�32

33� 106. �4.1

�10.1�49

�10.260.95.1�

107. X � 3A � 2B � 3��4

1

�3

0

�5

2� � 2�

1

�2

4

2

1

4� � �

�14

7

�17

�4

�17

�2�

108. �1

6��13

�2

0

6

�17

20�X �

1

6�4A � 3B� �

1

6�4��4

1

�3

0

�5

2� � 3�

1

�2

4

2

1

4�

109. X �1

3 �B � 2A� �

1

3 ��

1

�2

4

2

1

4� � 2 �

�4

1

�3

0

�5

2� �

1

3 �

9

�4

10

2

11

0�

110. �1

3��13

12

�26

�10

�15

�16�X �

1

3�2A � 5B� �

1

3�2��4

1

�3

0

�5

2� � 5�

1

�2

4

2

1

4�

100. (a)

(c) 4A � �80

�1216

244�

A � B � ��11

25

112� (b)

(d) A � 3B � ��73

127

214�

A � B � � 5�1

�83

10�

101.

� ��13

0

�8

11

18

�19�

�2

0

1

5

0

�4� � 3�5

0

3

�2

�6

5� � �2

0

1

5

0

�4� � �15

0

9

�6

�18

15�

102. �2�1

5

6

2

�4

0� � 8�

7

1

1

1

2

4� � �

�2

�10

�12

�4

8

0� � �

56

8

8

8

16

32� � �

54

�2

�4

4

24

32�

103.

� � 9�9

�74�

� ��82

1�4� � ��10

150

�5� � �74

�83��� 8

�2�1

4� � 5 ��23

0�1� � �7

4�8

3�

104.

� ��3096

0�48

6090

�18�66�

� 6��516

0�8

1015

�3�11�6���4

2�1�5

�37

4�10� � ��1

141

�3138

�7�1�

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Page 129: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

676 Chapter 7 Linear Systems and Matrices

111. � �14

14

36

�2

�10

�12

8

40

48��

1

5

6

2

�4

0� �6

4

�2

0

8

0� � �1�6� � 2�4�

5�6� � ��4��4�6�6� � �0��4�

1��2� � 2�0�5��2� � ��4��0�

6��2� � �0��0�

1�8� � 2�0�5�8� � ��4��0�

6�8� � �0��0��112. is undefined.�1

2

5

�4

6

0� �7

0

5

1

2

0� 113. AB � �31

�24

09��

75

�1

033� � �11

18�639�

114. AB � �101

32

�1

2�4

3��400

�336

2�1

2� � �404

18�18

12

3�10

9�115. �

4

11

12

1

�7

3� �3

2

�5

�2

6

�2� � �14

19

42

�22

�41

�66

22

80

66� 116. ��2

4

3

�2

10

2� �1

�5

3

1

2

2� � �13

20

24

4�

117.

� � 1

12

17

36�

� �2�2� � 1��3�6�2� � 0

2�6� � 1�5�6�6� � 0�

�2

6

1

0� �� 4

�3

2

1� � ��2

0

4

4� � �2

6

1

0� � 2

�3

6

5�

118. �14

�12���

01

32��

15

0�3� � �1

4�1

2��1511

�9�6� � � 4

82�3

�48�

119. (a)

This is the sales and profit for milk on Friday,Saturday and Sunday.

(b) Profit � 47.4 � 66.2 � 77.2 � $190.80

� �438.2612.1

717.28

47.466.277.2�

AB � �406076

648296

527684��

2.652.812.93

0.250.300.35� 120. (a)

(b)

(c) This gives the total number of calories burnedby a 120 pound person and a 150 pound personwho perform the given exercises for the givenamount of time.

A � �2 12 3��

10912764

13615979� � �473.5 588.5�

A � �2 12 3�

121. BA � I2AB � ��47

�12��

�27

�14� � �1

001�;

122. BA � I3AB � �116

102

013��

�232

�334

1�1�1� � �

100

010

001�;

123. row reduces to

��6�5

54�

�1

� �45

�5�6�

�10

01

��

45

�5�6���6

�554

��

10

01�

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Page 130: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 677

129. �1

�10

21

�1

010�

�1

� �101

001

2�1

3� 130. �101

�112

�2�2�4�

�1

� � 0

�12

�14

�2

�12

�34

11214�

131. ��7�8

22�

�1

�1

��7��2� � �2���8� �28

�2�7� � �1

4�1�

72�

132. �107

43�

�1

�1

�10��3� � 4�7��3

�7�410� � �

32

�72

�2

5�133. ��1

21020�

�1

�1

��1��20� � 10�2��20

�2�10�1� �

1�40�

20

�2�10

�1� � ��12120

14140�

134. ��63

�53�

�1

�1

��6��3� � ��5��3��3

�35

�6� �1

�3�3

�35

�6� � ��11

�53

2�136.

Answer: ��12, �3�

�xy� � �

11427

314

�17���10

1� � ��12

�3��2

43

�1��1

� �11427

314

�17�

124. reduces to

��32

�53�

�1

� � 3�2

5�3�

�10

01

��

3�2

5�3���3

2�5

3��

10

01�

125. row reduces to

��1

31

�274

�297�

�1

� �13

�125

6�5

2

�43

�1��100

010

001

���

13�12

5

6�5

2

�43

�1���1

31

�274

�297

���

100

010

001�

126. reduces to

�0

�57

�2�2

3

1�3

4��1

� �1

�1�1

11�7

�14

8�5

�10��100

010

001

���

1�1�1

11�7

�14

8�5

�10��0

�57

�2�2

3

1�3

4

���

100

010

001�

127. �2

3

6

�6��1

� �151

10

15

�115� 128. �3

4

�10

2��1

� �123

� 223

523346�

135.

Answer: �1, �25�

�xy� � �

14320

14

�120���1

5� � � 1

�25�

�13

5�5�

�1� �

14320

14

�120�

� x3x

5y5y

�15

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Page 131: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

678 Chapter 7 Linear Systems and Matrices

138.

Answer: �2, �4, 3�

�xyz� � �

�11�3�1

�6�2�1

�2�1�1��

12�25

10� � �2

�43�

��1

2�1

4�9

5

�25

�4��1

� ��11�3�1

�6�2�1

�2�1�1�

140.

Answer: �1, 1, �2, 1�

�xyz

w� � �

232313

�23

�13

�131313

�1

0

0

1

23

�13

�2313

�� 1�3

22� � �

11

�21�

�1101

1�1

10

1200

1111��1

� �232313

�23

�13

�131313

�1

0

0

1

23

�13

�2313

��1

141.

Answer: ��3, 1�

x � �3, y � 1

�1

3

2

4��1

� ��232

1

�12� ⇒ �x

y� � ��232

1

�12� ��1

�5� � ��3

1�

� x3x

2y4y

�1�5

142.

Answer: �5, 6�

x � 5, y � 6

� 1

�6

3

2��1

� �0.1

0.3

�0.15

0.05� ⇒ �x

y� � �0.1

0.3

�0.15

0.05� � 23

�18� � �5

6�

� x�6x

3y2y

23�18

139.

Answer: �1, �2, 1, 0�

�xyz

w� � �

�1

2

�1

1

43

�53

1

�1

�2343

�1

1

�73

113

�2

3� ��2

101� � �

1�2

10�

�1210

21

�10

11

�31

�1101�

�1

� ��1

2

�1

1

43

�53

1

�1

�2343

�1

1

�73

113

�2

3�

�1

137.

Answer: �2, �1, �2�

�xyz� � �

�1

3

2

�18373

1

�73

�53

� � 6�1

7� � �2

�1�2�

�315

2�1

1

�121�

�1

� ��1

3

2

�18373

1

�73

�53

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Page 132: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 679

146.

� 36 � 77 � �41

��97

11�4� � ��9���4� � �11��7� 147. �50

10

�30

5� � 50�5� � ��30��10� � 550

148. �1412

�24�15� � 14��15� � ��24��12� � 78

149.

Minors:

Cofactors:

C22 � 2C12 � �7

C21 � 1C11 � 4

M22 � 2M12 � 7

M21 � �1M11 � 4

A � �27

�14� 150.

Minors:

Cofactors:

C22 � 3C21 � �6

C12 � �5C11 � �4

M22 � 3M21 � 6

M12 � 5M11 � �4

A � �35

6�4�

143.

Answer: �1, 1, �2�

x � 1, y � 1, z � �2

��3

0

4

�3

1

3

�4

1

4�

�1

� �1

4

4

0

4

�3

1

3

�3� ⇒ �

x

y

z� � �

1

4

�4

0

4

�3

1

3

�3� �

2

�1

�1� � �

1

1

�2�

��3x

4x

3yy

3y

4zz

4z

2�1�1

144.

does not exist.

Inconsistent; no solution.

�2

1

3

3

�1

7

�4

2

�10�

�1

�2xx

3x

3yy

7y

4z2z

10z

1�4

0

145. �82 5

�4� � 8��4� � 2�5� � �42

151.

Minors:

Cofactors:

C31 � 5, C32 � 2, C33 � 19

C21 � �20, C22 � 19, C23 � �22

C11 � 30, C12 � 12, C13 � �21

M31 � �25 �10� � 5, M32 � � 3

�2�1

0� � �2, M33 � � 3�2

25� � 19

M21 � �28 �16� � 20, M22 � �31 �1

6� � 19, M23 � �31 28� � 22

M11 � �58 06� � 30, M12 � ��2

106� � �12, M13 � ��2

158� � �21

A � �3

�21

258

�106�

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Page 133: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

680 Chapter 7 Linear Systems and Matrices

152. Minors:

Cofactors:

C31 � �47, C32 � 96, C33 � 22

C21 � �2, C22 � 32, C23 � �20

C11 � 19, C12 � 24, C13 � 26

M33 � �86 35� � 22M32 � �86 4

�9� � �96,M31 � �35 4�9� � �47,

M23 � � 8�4

31� � 20M22 � � 8

�442� � 32,M21 � �31 4

2� � 2,

M13 � � 6�4

51� � 26M12 � � 6

�4�9

2� � �24,M11 � �51 �92� � 19,

153. ��2�6

5

403

124� � 6�43 1

4� � 2��25

43� � 6�13� � 2��26� � 130

154. �4 � 126 � 13 � �117� 42

�5

7�3

1

�14

�1� � 4�3 � 4� � 7��2 � 20� � 1�2 � 15� �

155. � 1

0

�2

0

1

0

�2

0

1� � 1� 1

�2

�2

1� � 1�1� � ��2���2� � 1 � 4 � �3

156. � �12 � 32 � 20�051 3

�2

6

1

1

1� � �3�5 � 1� � 1�30 � 2�

157.

� 279

� 3�128 � 5 � 56� � 4��12�

� 3�8�8 � ��8�� � 1�1 � 6� � 2��4 � 24�� � 4�0 � 6�8 � 6� � 0�

�3060 0

8

1

3

�4

1

8

�4

0

2

2

1� � 3�813 1

8

�4

2

2

1� � ��4��060 8

1

3

2

2

1� �Expansion along Row 1�

158. (Expansion along Row 1.)

� �255

� �5�54 �3� � 6�9 �9�

� �5�6��1 � 10� � 3��5 � 4�� � 6���1 � 10� � 3�0 � 3��

��5

0

�3

1

6

1

4

6

0

�1

�5

0

0

2

1

3� � �5�146 �1

�5

0

2

1

3� � 6� 0

�3

1

�1

�5

0

2

1

3�159. (Upper Triangular)det�A� � 8��1��4��3� � �96 160. det (Lower Triangular)� ��5���2��2��14� � 280

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Page 134: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 681

167. The figure is a rhombus.

Area square units� 48

��24

�2

�19

�9

111� � �48

168. The figure is a rhombus.

Area square units� 64

��44

�4

800

111� � �64

169.

Not collinear

��124

751

111� � �8 � 0 170.

Collinear

�024 �517

111� � �2��12� � 4��6� � 0

171.

Answer: �1, 2�

y �� 1

�1

5

1�� 1

�1

2

1� �6

3� 2

x ��51 2

1�� 1

�1

2

1� �3

3� 1 172.

Answer: �x, y� � ��3, 4�

y ��23 �10

�1��23 �1

2� �28

7� 4

x ���10

�1

�1

2��23 �1

2� ��21

7� �3

162.

Area

Area � 24 square units

�12��4

4

0

0

0

6

1

1

1� �12 �48� � 24

��4, 0�, �4, 0�, �0, 6�161.

Area � 16 square units

12�155 0

0

8

1

1

1� �12 �32� � 16

�1, 0�, �5, 0�, �5, 8�

163.

Area �74 square units

12� 1

2

232

1�

52

1

111� �

12�7

2� �74 164.

Area square units�12� 3

2

4

4

1� 1

2

2

1

1

1� �12�25

4 � �258

�32, 1�, �4, �1

2�, �4, 2�

165.

Area square units�132

12�254 4

61

111� �

12��13� 166.

Area square units�352

12��3

2�4

2�3�4

111� �

12��35�

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Page 135: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

682 Chapter 7 Linear Systems and Matrices

175.

Answer: ��1, 4, 5�

z ���2

4�1

3�1�4

�11�315�

14�

7014

� 5

y ���2

4�1

�11�315

�516�

14�

5614

� 4

x ���11

�315

3�1�4

�516�

��24

�1

3�1�4

�516

� ��1414

� �1 176.

Answer: �6, 8, 1�

z � �532 �2�3�1

15�7�3�

65�

6565

� 1

y � �532 15�7�3

1�1�7�

65�

52065

� 8

x � � 15�7�3

�2�3�1

1�1�7�

�532 �2�3�1

1�1�7� �

39065

� 6

173.

Answer: �4, 7�

y �� 5�11

6�23�

� 5�11

�23� �

�49�7

� 7

x �� 6�23

�23�

� 5�11

�23� �

�28�7

� 4 174.

Answer: �3, �2�

y ��39 �7

37��39 8

�5� �174�87

� �2

x ���7

378

�5��39 8

�5� ��261�87

� 3

177.

Answer: �0, �2.4, �2.6�

z � �5220 � �2.6y � �

4820 � �2.4,x � 0,

�A3� � �52

�A2� � �48

�A1� � 0

�121 �32

�7

2�3

8� � 20 178.

Answer: �x, y, z� � 3

4,

25

28, �

73

28

z �� 14

�4

56

�21

2

�21

10

4

5�� 14

�4

56

�21

2

�21

�7

�2

7� ��4088

1568� �

73

28

y �� 14

�4

56

10

4

5

�7

�2

7�� 14

�4

56

�21

2

�21

�7

�2

7� �1400

1568�

25

28

x ��10

4

5

�21

2

�21

�7

�2

7�� 14

�4

56

�21

2

�21

�7

�2

7� �1176

1568�

3

4

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Page 136: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 683

(b)

Answer: �815

, �12

5,

�73

z � �124 2�1�2

�3�1

5�15

��3515

��73

y � �124 �3�1

5

�11

�1�15

��3615

��12

5

x � ��3�1

5

2�1�2

�11

�1�15

��815

D � �124 2�1�2

�11

�1� � 15180. (a)

Answer: �815

, �12

5,

�73

x � 2�125 � �

73 � �3 ⇒ x � �

815

�5y � 3�73 � 5 ⇒ y � �

125

z � �73

�x �

2y5y

z3z3z

�357

�x � 2y�5y

�10y

z3z3z

�35

17

� x2x4x

2yy

2y

zzz

�3�1

5

179. (a)

Answer: 5333

, �1733

, 6166

x � 3�1733 � 261

66 � 5 ⇒ x �5333

7y � 86166 � �11 ⇒ y � �

1733

z �6166

�x � 3y7y

2z8z

667 z

5�11

617

�x � 3y7y

10y

2z8z2z

5�11�7

� x2x2x

3yy

4y

2z4z2z

5�1

3

(b)

Answer: 5333

, �1733

, 6166

z � �122 �314

5�1

3�66

�6166

y � �122 5�1

3

2�4

2�66

��3466

��1733

x � � 5�1

3

�314

2�4

2�66

�10666

�5333

D � �122 �314

2�4

2� � 66

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684 Chapter 7 Linear Systems and Matrices

181. I _ H A V E _ A _ D R E A M

Cryptogram: 24 38 8 3 0

32 2 41 �39�2�39

�3�51�2�30

�1 13 0�A � �41 �2 �39�

�4 18 5�A � �32 2 �39�

�0 1 0�A � �3 0 �3�

�1 22 5�A � �38 8 �51�

�9 0 8�A � ��30 �2 24�

�1 13 0��4 18 5��0 1 0��1 22 5��9 0 8�

182. J U S T _ D O _ I T

Cryptogram: 52 28 4

57 33 40 20 09

�23�78�49

�20 0 0�A � �40 20 0�

�15 0 9�A � �57 33 9�

�20 0 4�A � �52 28 4�

�10 21 19�A � ��49 �78 �23�

�20 0 0��15 0 9��20 0 4��10 21 19�

183.

Message: I WILL BE BACK

II__

C

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LBBK

WLEA_

� �99003

01222

11

2312510��

12

�14

�12

�14

�38

�14

14

�1814��

3293

�1�8

�46�48�14�6

�22

3715102

�3�

A�1 � �12

�14

�12

�14

�38

�14

14

�1814�

184.

Message: CAN YOU HEAR ME NOW

�30

21500

23

12501

13140

14158

185

150

C_

UE__

W

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AMN_

NOHREO_

�1438

�116

�1418516

14

�18316� ��

305

3740

�31623

�71034

�78

�146

3080163836580

�A�1 � �

1438

�116

�1418516

14

�18316�

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Page 138: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

Review Exercises for Chapter 7 685

186.

Message: FLORIDA GATORS WIN NATIONAL TITLE

FRAAR

W_

TN_

T

LI_

TSI

NI

ATL

ODGO_

NAOLI

E

61811

18230

20140

20

1290

20199

1491

2012

1547

150

141

151295

�3525110

�1515

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15

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9138

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3139

�54�26�14�70�38�46�30�48�26�58�34

A�1 � �3525110

�1515

�15

15

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�310�

187. (a)

Using Cramer’s Rule, you obtain,

and

(c) when or 2010t � 20.15,y � 20

y � 0.41t � 11.74

b � 11.74a � 0.41

� 4b44b

44a504a

65 723.2

(b)

(d)

That is, year 2010.

t > 20.15

0.41t > 8.26

0.41t � 11.74 > 20

0 500

40

188. False

185.

Message: THAT IS MY FINAL ANSWER

TTSYI

LNE

H___

N_

SR

AI

MFAAW_

202019259

12145

8000

140

1918

19

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230

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1313

�16� �

212932311013375

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�196

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14�18�26�12

26�2�836

A�1 � �23

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Page 139: CHAPTER 7 Linear Systems and Matrices - Fraser HS Mathmrraysmathclass.weebly.com/uploads/3/9/0/5/3905844/chapter_7_s… · 552 Chapter 7 Linear Systems and Matrices 12. Substitute

686 Chapter 7 Linear Systems and Matrices

190. The row operations on matrices are equivalent tothe operations used in the method of elimination.

189. True. Expansion by Row 3 gives

Note: Expand each of these matrices by Row 3 to see the previous step.

� �a11

a21

a31

a12

a22

a32

a13

a23

a33� � �a11

a21

c1

a12

a22

c2

a13

a23

c3�� c1�a12

a22

a13

a23� � c2�a11

a21

a13

a23� � c3�a11

a21

a12

a22�� a32�a11

a21

a13

a23� � a33�a11

a21

a12

a22�� a31�a12

a22

a13

a23�� �a32 � c2��a11

a21

a13

a23� � �a33 � c3��a11

a21

a12

a22�� a11

a21

a31 � c1

a12

a22

a32 � c2

a13

a23

a33 � c3� � �a31 � c1��a12

a22

a13

a23�

191. A square matrix has an inverse if det�A� � 0.

n � n

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Practice Test for Chapter 7 687

Chapter 7 Practice Test

For Exercises 1–3, solve the given system by the method of substitution.

1.

3x � y � 15

3x � y � 51 2.

x2 � 6y � �5

2x � 3y � �3 3.

5x � 2y � z � �3

2x � y � 3z � 0

x � y � z � 6

4. Find the two numbers whose sum is 110 andproduct is 2800.

5. Find the dimensions of a rectangle if its perimeter is170 feet and its area is 2800 square feet.

For Exercises 6–7, solve the linear system by elimination.

6.

x � 3y � 23

2x � 15y � 24 7.

38x � 19y � 7

x � y � 2

8. Use a graphing utility to graph the two equations.Use the graph to approximate the solution of thesystem. Verify your answer analytically.

0.3x � 0.7y � �0.131

0.4x � 0.5y � �0.112

9. Herbert invests $17,000 in two funds that pay 11%and 13% simple interest, respectively. If he receives$2080 in yearly interest, how much is invested ineach fund?

10. Find the least squares regression line for the points and ��2, �1�.

�4, 3�, �1, 1�, ��1, �2�,

11.

x � 4y � 3z � �20

2x � y � z � �11

x � y � a � 1�2 12.

2x � 4y � 8z � 0

2x � y � z � 0

4x � y � 5z � 4 13.

6x � y � 5z � 2

3x � 2y � z � 5

For Exercises 11–13, solve the system of equations.

14. Find the equation of the parabola passing through the points �0, �1�, �1, 4� and �2, 13�.y � ax2 � bx � c

15. Find the position equation given that feet after 1 second, feetafter 2 seconds, and feet after 3 seconds.s � 4

s � 5s � 12s �12at2 � v0t � s0

16. Write the matrix in reduced row-echelon form.

�1

3

�2

�5

4

9�

For Exercises 17–19, use matrices to solve the system of equations.

17.

2x � 5y � �11

3x � 5y � �13 18.

3x � 2y � �1

3x � 2y � �8

2x � 3y � �3 19.

3x � y � 3z � �3

2x � y � 3z � �0

2x � y � 3z � �5

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688 Chapter 7 Linear Systems and Matrices

20. Multiply �1

2

4

0

5

�3� �1

0

�1

6

�7

2�

21. Given find 3A � 5B.A � � 9

�4

1

8� and B � �6

3

�2

5�,

22. Find

f�x� � x2 � 7x � 8, A � �3

7

0

1�f�A�: 23. True or false:

where andare matrices.

(Assume that exist.)A2, AB, and B2

BA�A � B��A � 3B� � A2 � 4AB � 3B2

For Exercises 24 and 25, find the inverse of the matrix, if it exists.

24. �1

3

2

5� 25. �1

3

6

1

6

10

1

5

8�

For Exercises 27 and 28, find the determinant of the matrix.

27. �6

3

�1

4� 28. �1

5

6

3

9

2

�1

0

�5�

29. Use a graphing utility to find the determinant of the matrix.

�1

0

3

2

4

1

5

0

2

�2

�1

6

3

0

1

1�

30. Evaluate ��60000 4

5

0

0

0

3

1

2

0

0

0

4

7

9

0

6

8

3

2

1�.

31. Use a determinant to find the area of the triangle with vertices and �3, 9�.�0, 7�, �5, 0�,

32. Use a determinant to find the equation of the line through �2, 7� and ��1, 4�.

For Exercises 33–35, use Cramer’s Rule to find the indicated value.

33. Find

2x � 5y � 11

6x � 7y � 14

x. 34. Find

3x � y � 4z � 2

3x � y � 4z � 3

3x � y � 4z � 1

z. 35. Find

745.9x � 105.6y � 19.85

721.4x � 129.1y � 33.77

y.

26. Use an inverse matrix to solve the systems.

(a) (b)

3x � 5y � �23x � 5y � 1

3x � 2y � �33x � 2y � 4

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