Exam 2 Review Math1324

Solve the system of two equations in two variables.

1) 8x + 7y = 36

3x - 4y = -13

A) (1, 5) B) (0, 5) C) No solution D) (1, 4)

2) 3x + 5y = 4

18x + 30y = 24

A) - 1

3, 1 B) No solution

C) - 5

3y +

4

3, y for any real number y D)

4

3, 0

Solve the problem by writing and solving a suitable system of equations.

3) If 20 pounds of tomatoes and 10 pounds of bananas cost $17 and 40 pounds of tomatoes and 30 pounds of

bananas cost $39, what is the price per pound of tomatoes and bananas?

A) tomatoes: $0.70 per pound; bananas: $0.30 per pound

B) tomatoes: $0.60 per pound; bananas: $0.50 per pound

C) tomatoes: $0.70 per pound; bananas: $0.50 per pound

D) tomatoes: $0.50 per pound; bananas: $0.70 per pound

Solve the system by back substitution.

4) x + 4y+ 4z = -11

2y + 5z = -21

2z = - 10

A) (1, -5, 2) B) (1, 2, -5) C) (-6, 2, -5) D) No solution

Obtain an equivalent system by performing the stated elementary operation on the system.

5) Replace the fourth equation by the sum of itself and 3 times the second equation

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

4y - z - 4w = -5

3y - 4z + 2w = -3

5y - 5z - 2w = 8

A)

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

4y - z - 4w = -5

3y - 4z + 2w = -3

12y + 3z - 14w = -7

B)

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

4y - z - 4w = -5

3y - 4z + 2w = -3

17y - 8z - 14w = -7

C)

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

4y - z - 4w = -5

3y - 4z + 2w = -3

-7y + 8z + 10w = 23

D)

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

12y - 3z - 12w = -15

3y - 4z + 2w = -3

5y - 5z - 2w = 8

1

Write the system of equations associated with the augmented matrix. Do not solve.

6)1 0 0 -8

0 1 0 3

0 0 1 -3

A) x = 0

y = -5

z =-11

B) x = -5

y = 6

z = 0

C) x = 8

y = -3

z = 3

D) x = -8

y = 3

z = -3

The reduced row echelon form of the augmented matrix of a system of equations is given. Find the solutions of the

system.

7)

1 0 0 0 0 3

0 1 0 0 0 8

0 0 1 0 0 6

0 0 0 1 0 -6

0 0 0 0 0 1

A) (3, 8, 6, -6, w) for any real number w B) (3, 8, 6, -6)

C) (3, 8, 6, -6, 1) D) No solution

8)

1 0 0 0 13

0 1 0 0 0

0 0 1 0 -8

0 0 0 1 11/2

A) 13, -8, 11

2, 0 B) No solution

C) 13, 0, -8, 11

2D) 13, w, -8,

11

2 for any real number w

Perform row operations on the augmented matrix as far as necessary to determine whether the system is independent,

dependent, or inconsistent.

9) x + y - 2z = 8

3x + z = - 6

2x - y + 3z = -14

A) Independent B) Dependent C) Inconsistent

Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z.

10) 2x + y - z = 2

x - 3y + 2z = 1

7x - 7y + 4z = 7

A)7 + z

7,

5

7z, z for any real number z B) (1, 0, 0)

C) (2, 5, 7) D) No solution

2

Solve the problem by writing and solving a suitable system of equations.

11) Alan invests a total of $20,500 in three different ways. He invests one part in a mutual fund which in the first

year has a return of 11%. He invests the second part in a government bond at 7% per year. The third part he

puts in the bank at 5% per year. He invests twice as much in the mutual fund as in the bank. The first year

Alan's investments bring a total return of $1645. How much did he invest in each way?

A) mutual fund: $7000; bond: $11,000: bank: $3500 B) mutual fund: $7000; bond: $10,000: bank: $3500

C) mutual fund: $6400; bond: $10,900: bank: $3200 D) mutual fund: $7600; bond: $9100: bank: $3800

12) Linda invests $25,000 for one year. Part is invested at 5%, another part at 6%, and the rest at 8%. The total

income from all 3 investments is $1600. The income from the 5% and 6% investments is the same as the income

from the 8% investment. Find the amount invested at each rate.

A) $8000 at 5%, $10,000 at 6%, $7000 at 8% B) $10,000 at 5%, $5000 at 6%, $10,000 at 8%

C) $10,000 at 5%, $10,000 at 6%, $5000 at 8% D) $5000 at 5%, $10,000 at 6%, $10,000 at 8%

13) A store sells televisions for $360 and video cassette recorders for $270. At the beginning of the week its entire

stock is worth $56,430. During the week it sells three quarters of the televisions and one third of the video

cassette recorders for a total of $32,310. How many televisions and video cassette recorders did it have in its

stock at the beginning of the week?

A) 87 televisions; 93 video cassette recorders B) 88 televisions; 95 video cassette recorders

C) 89 televisions; 92 video cassette recorders D) 90 televisions; 89 video cassette recorders

Solve the problem.

14) What is the size of the matrix?

-5 5 -1

-4 -4 1

A) 2 × 3 B) 3 × 2 C) 6 D) 3

15) What is the size of the matrix?

12

1

2

A) 1 × 3 B) 3 × 0 C) 3 × 1 D) 3 × 3

Perform the indicated operation.

16) Let C = 1

-3

2

and D = -1

3

-2

. Find C - 3D.

A)

4

-12

8

B)

-2

6

-4

C)

-4

12

-8

D)

4

-6

4

3

Solve the problem.

17) Barnes and Able sell life, health, and auto insurance. Sales for May and June are given in the matrices.

Life Health Auto

M =

20,000 15,000 5000

30,000 0 17,000

Able

Barnes

J =

70,000 0 30,000

20,000 25,000 32,000

Able

Barnes

Find the matrix that would give total sales for the months of May and June.

A)

90,000 15,000 35,000

50,000 25,000 49,000

B)

140,000 40,000 84,000

C)

90,000 15,000 35,000

50,000 25,000 32,000

D)

90,000 0 35,000

50,000 0 49,000

4

Graph the feasible region for the system of inequalities.

18) 2y + x ≥ -2

y + 3x ≤ 9

y ≤ 0

x ≥ 0

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

5

19) 4y + x ≥ -2

y + 2x ≤ 10

4y ≤ 10x + 40

y ≥ 0

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

A)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

B)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

C)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

D)

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

x-10 -8 -6 -4 -2 2 4 6 8

y10

8

6

4

2

-2

-4

-6

-8

-10

6

Find the value(s) of the function on the given feasible region.

20) Find the maximum and minimum of z = 8x - 11y.

A) -45, -66 B) 40, 0 C) -66, 0 D) 40, -66

Use graphical methods to solve the linear programming problem.

21) Maximize z = 6x + 7y

subject to: 2x + 3y ≤ 12

2x + y ≤ 8

x ≥ 0

y ≥ 0

x-10 10

y

10

-10

x-10 10

y

10

-10

A) Maximum of 52 when x = 4 and y = 4 B) Maximum of 32 when x = 2 and y = 3

C) Maximum of 32 when x = 3 and y = 2 D) Maximum of 24 when x = 4 and y = 0

7

22) Minimize z = 6x + 8y

subject to: 2x + 4y ≥ 12

2x + y ≥ 8

x ≥ 0

y ≥ 0

x-10 10

y

10

-10

x-10 10

y

10

-10

A) Minimum of 36 when x = 6 and y = 0 B) Minimum of 26 when x = 3 and y = 1

C) Minimum of 0 when x = 0 and y = 0 D) Minimum of 92

3 when x =

10

3 and y =

4

3

Find the value(s) of the function, subject to the system of inequalities.

23) Find the minimum of P = 23x + 18y + 23 subject to:

x ≥0, y ≥0, x + y ≥ 1.

A) 64 B) 41 C) 46 D) 23

State the linear programming problem in mathematical terms, identifying the objective function and the constraints.

24) A firm makes products A and B. Product A takes 2 hours each on machine L and machine M; product B takes 4

hours on L and 2 hours on M. Machine L can be used for 8 hours and M for 7 hours. Profit on product A is $6

and $7 on B. Maximize profit.

A) Maximize 6A + 7B

Subject to: 2A + 2B ≥ 8

2A + 4B ≥ 7

A, B ≤ 0.

B) Maximize 7A + 6B

Subject to: 2A + 2B ≤ 8

2A + 4B ≤ 7

A, B ≥ 0.

C) Maximize 6A + 7B

Subject to: 2A + 2B ≤ 8

2A + 4B ≤ 7

A, B ≥ 0.

D) Maximize 6A + 7B

Subject to: 2A + 4B ≤ 8

2A + 2B ≤ 7

A, B ≥ 0.

The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24

rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring, versus 2 man-hours to

make one SST ring.

25) How many of each type of ring should be made daily to maximize the company's profit, if the profit on a VIP

ring is $30 and on an SST ring is $40?

A) 0 VIP and 24 SST B) 12 VIP and 12 SST C) 8 VIP and 16 SST D) 16 VIP and 8 SST

8

Introduce slack variables as necessary and write the initial simplex tableau for the problem.

26) Maximize z = 4x1 + x2

subject to: 2x1 + 5x2 ≤ 11

3x1 + 3x2 ≤ 19

x1 ≥ 0, x2 ≥ 0

A) x1 x2 s1 s2 z

2 5 1 0 0 19

3 3 0 1 0 11

4 1 0 0 1 0

B) x1 x2 s1 s2 z

2 5 1 0 0 19

3 3 0 1 0 11

-4 -1 0 0 1 0

C) x1 x2 s1 s2 z

2 5 1 0 0 11

3 3 0 1 0 19

4 1 0 0 1 0

D) x1 x2 s1 s2 z

2 5 1 0 0 11

3 3 0 1 0 19

-4 -1 0 0 1 0

Find the pivot in the tableau.

27)

A) 4 in row 1, column 3 B) 4 in row 2, column 2

C) 2 in row 1, column 1 D) 2 in row 2, column 1

Use the indicated entry as the pivot and perform the pivoting once.

28)

A) B)

C) D)

9

Write the basic solution for the simplex tableau determined by setting the nonbasic variables equal to 0.

29) x1 x2 x3 x4 x5 z

1

3

0

0

1

0

3

1

0

1

0

0

8

13

3 1 0 2 0 0 27

2 0 0 3 0 1 23

A) x1 = 8, x2 = 27, x3 = 8, x4 = 13, x5 = 13, z = 23 B) x1 = 2, x2 = 0, x3 = 0, x4 = 3, x5 = 0, z = 1

C) x1 = 0, x2 = 27, x3 = 8, x4 = 0, x5 = 13, z = 23 D) x1 = 0, x2 = 27, x3 = 8, x4 = 0, x5 = 13, z = 0

Use the simplex method to solve the linear programming problem.

30) Maximize z = 2x1 + 5x2 + 3x3

subject to: 2x1 + x2 + 3x3 ≤ 9

4x1 + 3x2 + 5x3 ≤ 12

with x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

A) Maximum is 20 when x1 = 0, x2 = 4, x3 = 0 B) Maximum is 45 when x1 = 0, x2 = 9, x3 = 0

C) Maximum is 60 when x1 = 0, x2 = 9, x3 = 5 D) Maximum is 12 when x1 = 6, x2 = 0, x3 = 0

A bakery makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, 1 dozen eggs, and 2 lb of sugar. A

batch of donuts requires 5 lb of flour, 3 dozen eggs, and 2 lb of sugar. Set up an initial simplex tableau to maximize profit.

31) The bakery has 580 lb of flour, 660 dozen eggs, 700 lb of sugar. The profit on a batch of sweet rolls is $93.00 and

on a batch of donuts is $62.00.

A)

x1 x2 s1 s2 s3 s4

3 5 1 0 0 0 580

1 3 0 1 0 0 660

2 2 0 0 1 0 700

- 62 - 93 0 0 0 1 0

B)

x1 x2 s1 s2 s3 s4

3 5 1 0 0 0 580

1 3 0 1 0 0 660

2 2 0 0 1 0 700

- 93 62 0 0 0 1 0

C)

x1 x2 s1 s2 s3 s4

3 5 1 0 0 0 580

1 3 0 1 0 0 660

2 2 0 0 1 0 700

93 62 0 0 0 1 0

D)

x1 x2 s1 s2 s3 s4

3 5 1 0 0 0 580

1 3 0 1 0 0 660

2 2 0 0 1 0 700

- 93 - 62 0 0 0 1 0

10

A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and

$15 for C. The production requirements and departmental capacities are as follows:

Department Production requirement

by product (hours)

Departmental capacity

(Total hours)

Assembling

Painting

Finishing

A B C

2 3 2

1 2 2

2 3 1

30,000

38,000

28,000

32) What are the coefficients of the objective function?

A) 2, 3, 2 B) 1, 2, 2 C) 3, 6, 15 D) 2, 3, 1

Solve the problem.

33) An agricultural research scientist is developing three new crop growth supplements -- A, B, and C. Each pound

of each supplement contains four enzymes -- E1, E2, E3, and E4 -- in the amounts (in milligrams) shown in the

table.

E1 E2 E3 E4

A 3 2 1 1

B 1 6 3 1

C 2 3 1 5

The cost of E1 is $20/mg, the cost of E2 is $40/mg, the cost of E3 is $10/mg, and the cost of E4 is also $10/mg. The

growth benefit for crops is expected to be proportional to 10 times the amount of A used, 25 times the amount of

B used, and 60 times the amount of C used. However, the total cost of the enzymes used in A, B, and C must be

less than $5000 for each treatment. How many pounds each of A, B, and C should be produced to maximize the

growth effect?

A) 3.0, 3.0, 4.0 B) 9.0, 0, 9.5 C) 0, 0, 15 D) 0, 9.3, 10.0

11

Answer KeyTestname: MATH1324 EXAM2 REVIEW SPRING 2016

1) D

2) C

3) B

4) B

5) B

6) D

7) D

8) C

9) B

10) A

11) B

12) B

13) D

14) A

15) C

16) A

17) A

18) B

19) B

20) D

21) C

22) D

23) B

24) D

25) A

26) D

27) B

28) B

29) C

30) A

31) D

32) C

33) D

12