754 CHAPTER 10 Systems of Equations and Inequalities

Figure 9

rref < [ft][ [1[0[0

01e

<001

1.

2

Fil]

The augmented matrix of this system is:

30 4 9A = 35 33 13

200 10 300

6595

905

Then, using the RREF command on a TI-84 Plus graphing calculator, we obtain thematrix in Figure 9.

To meet the dietary requirements, the patient should receive 1.5 servings of

chicken a la king, - of a baked potato, and 2 glasses of milk. •<

10.2 Assess Your Understanding

Concepts and Vocabulary1. An m by n rectangular array of numbers is called a(n) 2. The matrix used to represent a system of linear equations is

called a(n) matrix.3. True or False: The augmented matrix of a system of two

equations containing three variables has two rows and fourcolumns.

4. True or False: The matrixform.

1 30 10 0

-2is in row echelon

Skill Building

In Problems 5-16, write the augmented matrix of the given system of equations.

5.x - 5y = 5

4x + 3y = 66.

. - O.Q3y = 0.06' \0.l3x + Q.Wy = 0.20

3x + 4y = 14x — 2y = 5

4 3 3

1 1 2

x + y - z = 213. < 3x - 2y '= 2

5x + 3y - 1 = 114.

2x + 3y - 4z = 0x - 5z + 2 = 0x + 2y - 3z = -2

11.

15.

| 2x + 3y - 6 = 0(4x - 6y + 2 = 0

x - y + z = 103x + 3y = 5

x + y + 2z = 2

x - y - z = 102x + y + 2z = -1

-3x + 4y =54x - 5y + z = 0

8.9x - y = 0

3x - y - 4 = 0

x— y + 2z — w = 516. \x + 3y - 4z + 2w = 2

3x — y ~ 5z — w = -1

In Problems 17-24, perform each row operation on the given augmented matrix.

\.

19.

21.

23.

1 -3[2 -5

1 -32 -5

_-3 3

1 -32 -5

_-3 -6

1 -32 -5

.-3 1

-2] ^?2 = -2/i

5j

464

234

164

3~| (a) /6 (b-U6_

-6"-4

6_

-2"-2

6_

(a) /( b ) /

(a) /(b ) /

+ r2

i2 = -2nF3'= -3rj H

J2 = -2rt

f3 = 3r, 4

;2 = -2rj!3 = 3r, -f

r2

18.

20.

22.

24.

1 -32 -5

1 -32 -5

_-3 -2

1 -32 -5

_-3 1

1 -39 CZ. J

.-3 -6

-3~| R2 = -2r, + r2

-4]

3-3

4

-464

-124

-5-5

6_

-6"-6

6_

(a) R2 = -2r,(b) R3 = 3rt 4

(a) R2 = -2r,(b) R3 = 3r, 4

2] (a) R2 = -2/-J6 (b) *3 = 3rt 4

6_

+ r2

r3

+ r2

r3

+ r2

r3

SECTION 10.2 Systems of Linear Equations: Matrices 755

In Problems 25-36, the reduced row echelon form of a system of linear equations is given. Write the system of equations corresponding tothe given matrix. Use x, y; or x, y, z; or xl, x2, x3, X4 as variables. Determine whether the system is consistent or inconsistent. If it is consis-tent, give the solution.

28.

31.

34.

[1_o

"l0

_0

"l0

-°

"l0

_0

0 51

1 -lj

0 0 0 ~1 0 00 0 2 _

0 0 01 0 10 1 2

0 0 01 0 00 1 2

123_

r23_

«29.

32.

35-

1 0 -40 1 0

" 1 0 2 - l "0 1 - 4 - 2

_0 0 0 0_

"l 0 0 00 1 0 2

_0 0 1 3

120_

"l 0 0 1 -2"0 1 0 2 20 0 1 - 1 00 0 0 0 0 _

V 27.

30.

33.

"l0

_0

"l0

_0

"l0

_0

"l00

_0

010

010

010

0100

000

430

r3̂_

4"20_

0 41 30 0

0 00 01 00 1

)

2"30_

r230

In Problems 37-72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

37.

46.

49.

x-y =

x - 2y + 3z = 72x + y + z = 4

-3* + 2y - 2z = -10

<2x - 3y - z = 0-x + 2y + z = 53x - 4y - z = 1

2x - 2y + 3z = 655. < 4x - 3y + 2z = 0

. -2x + 3y - lz = 1

x - y + z = -458. < 2x - 3y + 4z = -15

k 5x + y - 2z = 12

'

X 47.

X 53.

= 3

+ 2y = 4+ 4 = 8

-jc + y = -22

- > = 62x - 3z = 16•2y+ z= 4

2z + y - 3z = 0-2x + 2y + z = -1

3x - 4y -3z= 1

x+ y+ z-x + 2y - 3z3x - 2y - lz

3x - 2y + 2z56. { Ix - 3y + 2z

, 2x - 3 y + 4z

•- -4

= 0

6-1

0

x + 2y - z = -32x - 4y + z = -7

-2x + 2y -3z= 4

39•

42.

45.

48.

51.

3x + 2y = 3

3x — y = 19x — 3y =

I5x + 5y = 21

2x + y = -4-2y + 4z = 0

3x - 2z = -11

' 2x - 2y - 2z = 22x + 3y + z = 2

3x + 2y = 0

r2x - 3y - z = 03x + 2y + 2z = 2x + 5y + 3z = 2

x + y - z = 63x - 2y + z = -5

« + 3y - 2z = 14

x + 4y - 3z = -83*'- y + 3z = 12x + y + 6z = 1

756 CHAPTER 10 Systems of Equations and Inequalities

61. <

3x + y - z = -

2x - y + z = I

4x

62.o

2y = -

64.

67.

70.

x+y+z+w=

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

-2x + y ~ 2z + 2w =

x — y + z = 53x + 2y - 2z = 0

x - 3y + z = 12x - y - 4z = 0z - 3y + 2z = 1

x - 2y = 5

406

-1

65.

68.

71.

x + y = 1

2x - y + z = 18

x + 2y + z = -

x + 2y + z = 12* - >> + 2z = 2

. 3x + y + 3z = 3

-x + y + 3z = 1

63.

66.

2x - y + z3x + 2y + z - w- 2y - 2z + 2w

x + 2y - z = 32x - y + 2z = 6

- 3 + 3z = 4

406

-1

Applications and Extensions73. Curve Fitting Find the function y = ax2 + bx + c whose

graph contains the points (1, 2), (-2, —7), and (2, —3).

74. Curve Fitting Find the function y = ax2 + bx + c whosegraph contains the points (1, -1), (3, -1), and (—2,14).

75. Curve Fitting Find the function f ( x ) = ax3 + bx2 +ex + d for which /(-3) = -112,/(-l) = -2,/(l) = 4,and /(2) = 13.

76. Curve Fitting Find the function f ( x ) = ax3 + bx2 +ex + d for which /(-2) = -10, /(-I) = 3,/(I) = 5, and/(3) = 15.

77. Nutrition A dietitian at Palos Community Hospital wants apatient to have a meal that has 78 grams of protein, 59 gramsof carbohydrates, and 75 milligrams of vitamin A. The hospi-tal food service tells the dietitian that the dinner for today issalmon steak, baked eggs, and acorn squash. Each serving ofsalmon steak has 30 grams of protein, 20 grams of carbohy-drates, and 2 milligrams of vitamin A. Each serving of bakedeggs contains 15 grams of protein, 2 grams of carbohydrates,and 20 milligrams of vitamin A. Each serving of acorn squashcontains 3 grams of protein, 25 grams of carbohydrates, and32 milligrams of vitamin A. How many servings of each foodshould the dietitian provide for the patient?

78. Nutrition A dietitian at General Hospital wants a patientto have a meal that has 47 grams of protein, 58 grams of car-bohydrates, and 630 milligrams of calcium. The hospitalfood service tells the dietitian that the dinner for today ispork chops, corn on the cob, and 2% milk. Each serving ofpork chops has 23 grams of protein, 0 grams of carbohy-drates, and 10 milligrams of calcium. Each serving of cornon the cob contains 3 grams of protein, 16 grams of carbo-hydrates, and 10 milligrams of calcium. Each glass of 2%milk contains 9 grams of protein, 13 grams of carbohydrates,and 300 milligrams of calcium. How many servings of eachfood should the dietitian provide for the patient?

69.-x +

y - z = 0y + z = 0y + 3z = 5

= 472.

-4x + y = 52x - y + z - w = 5

z + w = 4

79. Financial Planning Carletta has $10,000 to invest. As herfinancial consultant, you recommend that she invest in Trea-sury bills that yield 6%, Treasury bonds that yield 7%, andcorporate bonds that yield 8%. Carletta wants to have anannual income of $680, and the amount invested in corpo-rate bonds must be half that invested in Treasury bills. Findthe amount in each investment.

80. Financial Planning John has $20,000 to invest. As his fi-nancial consultant, you recommend that he invest in Trea-sury bills that yield 5%, Treasury bonds that yield 7%, andcorporate bonds that yield 9%. John wants to have an annu-al income of $1280, and the amount invested in Treasurybills must be two times the amount invested in corporatebonds. Find the amount in each investment.

81. Production To manufacture an automobile requires paint-ing, drying, and polishing. Epsilon Motor Company pro-duces three types of cars: the Delta, the Beta, and the Sigma.Each Delta requires 10 hours for painting, 3 hours for dry-ing, and 2 hours for polishing. A Beta requires 16 horfrs forpainting, 5 hours for drying, and 3 hours for polishing, whilea Sigma requires 8 hours for painting, 2 hours for drying,and 1 hour for polishing. If the company has 240 hours forpainting, 69 hours for drying, and 41 hours for polishing permonth, how many of each type of car are produced?

82. Production A Florida juice company completes thepreparation of its products by sterilizing, filling, and labelingbottles. Each case of orange juice requires 9 minutes forsterilizing, 6 minutes for filling, and 1 minute for labeling.Each case of grapefruit juice requires 10 minutes for steriliz-ing, 4 minutes for filling, and 2 minutes for labeling. Eachcase of tomato juice requires 12 minutes for sterilizing, 4minutes for filling, and 1 minute for labeling.-If the companyruns the sterilizing machine for 398 minutes, the filling ma-chine for 164 minutes, and the labeling machine for 58 min-utes, how many cases of each type of juice are prepared?

SECTION 10.2 Systems of Linear Equations: Matrices 757

83. Electricity: Kirchhoff's Rules An application of Kirch-hoff's Rules to the circuit shown results in the following sys-tem of equations:

-4 + 8 -2 /2 = 08 = 5/4 + /,4 = 3 /3+ /!

Find the currents /j, /2, /?, and /4.

3ft

I,

5(1

'2n

8V

SOURCE: Based on Raymond Serway, Physics, 3rd ed.(Philadelphia: Saunders, 1990), Prob. 34, p. 790.

84. Electricity: Kirchhoff's Rules An application of Kirch-hoff's Rules to the circuit shown results in the following sys-tem of equations:

/I = /3 + h

24 - 6A -3/3 = 012 + 24 - 6/1 -6/2 = 0

Find the currents /], /2, and /3.

SOURCE: Ibid., Prob. 38, p. 791.

85. Financial Planning Three retired couples each require anadditional annual income of $2000 per year. As their finan-cial consultant, you recommend that they invest somemoney in Treasury bills that yield 7%, some money in cor-porate bonds that yield 9%, and some money in junk bondsthat yield 11%. Prepare a table for each couple showing thevarious ways that their goals can be achieved:(a) If the first couple has $20,000 to invest.(b) If the second couple has $25,000 to invest.(c) If the third couple has $30,000 to invest.(d) What advice would you give each couple regarding the

amount to invest and the choices available?

[Hint: Higher yields generally carry more risk.]

86. Financial Planning A young couple has $25,000 to invest.As their financial consultant, you recommend that they in-vest some money in Treasury bills that yield 7%, somemoney in corporate bonds that yield 9%, and some moneyin junk bonds that yield 11%. Prepare a table showing thevarious ways that this couple can achieve the followinggoals:(a) The couple wants $1500 per year in income.(b) The couple wants $2000 per year in income.(c) The couple wants $2500 per year in income.(d) What advice would you give this couple regarding the

income that they require and the choices available?

[Hint: Higher yields generally carry more risk.]

87. Pharmacy A doctor's prescription calls for a daily intakeof a supplement containing 40 mg of vitamin C and 30 mgof vitamin D. Your pharmacy stocks three supplementsthat can be used: one contains 20% vitamin C and 30% vi-tamin D; a second, 40% vitamin C and 20% vitamin D; anda third, 30% vitamin C and 50% vitamin D. Create a tableshowing the possible combinations that could be used tofill the prescription.

88. Pharmacy A doctor's prescription calls for the creation ofpills that contain 12 units of vitamin B12 and 12 units of vita-min E. Your pharmacy stocks three powders that can beused to make these pills: one contains 20% vitamin B12 and30% vitamin E; a second, 40% vitamin B12 and 20% vitaminE; and a third, 30% vitamin B12 and 40% vitamin E. Createa table showing the possible combinations of each powderthat could be mixed in each pill.

Discussion and Writing89. Write a brief paragraph or two that outlines your strategy

for solving a system of linear equations using matrices.90. When solving a system of linear equations using matrices,

do you prefer to place the augmented matrix in row echelonform or in reduced row echelon form? Give reasons foryour choice.

91. Make up a system of three linear equations containing threevariables that has:(a) No solution(b) Exactly one solution(c) Infinitely many solutionsGive the three systems to a friend to solve and critique.