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
2) 3x + 5y = 4
18x + 30y = 24
1
A) - , 1
3
C) -
D) (1, 4)
B) No solution
5
4
y + , y for any real number y
3
3
D)
4
,0
3
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
4y - z
3y - 4z
5y - 5z
A)
x - 2y
4y
3y
12y
C)
x - 2y
4y
3y
-7y
- 6w =
- 4w =
+ 2w =
- 2w =
4
-5
-3
8
B)
+ 5z - 6w =
- z - 4w =
- 4z + 2w =
+ 3z - 14w =
4
-5
-3
-7
x - 2y
4y
3y
17y
+ 5z - 6w =
- z - 4w =
- 4z + 2w =
- 8z - 14w =
4
-5
-3
-7
x - 2y
12y
3y
5y
+ 5z - 6w = 4
- 3z - 12w = -15
- 4z + 2w = -3
- 5z - 2w = 8
D)
+ 5z - 6w =
- 4w =
- z
- 4z + 2w =
+ 8z + 10w =
4
-5
-3
23
1
Write the system of equations associated with the augmented matrix. Do not solve.
1 0 0 -8
6) 0 1 0 3
0 0 1 -3
A) x = 0
B) x = -5
C) x = 8
y = -5
y= 6
y = -3
z =-11
z= 0
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
0
1
0
0
0
0
1
0
0 13
0 0
0 -8
1 11/2
11
A) 13, -8,
,0
2
C) 13, 0, -8,
B) No solution
11
2
D) 13, w, -8,
11
for any real number w
2
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
7+z 5
A)
, z, z for any real number z
B) (1, 0, 0)
7
7
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
15) What is the size of the matrix?
12
1
2
A) 1 × 3
B) 3 × 0
Perform the indicated operation.
1
-1
16) Let C = -3 and D = 3 . Find C - 3D.
2
-2
A)
B)
4
-2
-12
6
-4
8
C) 6
D) 3
C) 3 × 1
D) 3 × 3
C)
D)
-4
12
-8
3
4
-6
4
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
20,000 15,000
M=
30,000
70,000
J =
5000
0 17,000
0 30,000
20,000 25,000 32,000
Able
Barnes
Able
Barnes
Find the matrix that would give total sales for the months of May and June.
A)
B)
140,000 40,000 84,000
90,000 15,000 35,000
50,000 25,000 49,000
C)
D)
90,000 15,000 35,000
90,000 0 35,000
50,000 25,000 32,000
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
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
x
8
-4
-6
-8
-10
A)
B)
10
y
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
y
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
C)
2
4
6
8
x
2
4
6
8
x
D)
10
y
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
5
y
19) 4y + x ≥ -2
y + 2x ≤ 10
4y ≤ 10x + 40
y≥0
10
y
8
6
4
2
-10 -8 -6 -4 -2
-2
2
4
6
x
8
-4
-6
-8
-10
A)
B)
10
y
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
y
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
C)
2
4
6
8
x
2
4
6
8
x
D)
10
y
10
8
8
6
6
4
4
2
2
-10 -8 -6 -4 -2
-2
2
4
6
8
x
-10 -8 -6 -4 -2
-2
-4
-4
-6
-6
-8
-8
-10
-10
6
y
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
y
10
-10
10
x
-10
A) Maximum of 52 when x = 4 and y = 4
C) Maximum of 32 when x = 3 and y = 2
B) Maximum of 32 when x = 2 and y = 3
D) Maximum of 24 when x = 4 and y = 0
7
z = 6x + 8y
2x + 4y ≥ 12
2x + y ≥ 8
x≥0
y≥0
22) Minimize
subject to:
y
10
-10
10
x
-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
10
4
when x =
and y =
3
3
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
2
3
4
C) x1
2
3
4
x2 s1 s2
5 1 0
3 0 1
1 0 0
x2 s1 s2
5 1 0
3 0 1
1 0 0
z
B) x1
0
0
1
z
19
11
0
0
0
1
11
19
0
x2 s1 s2
2
5 1 0
3
3 0 1
-4 -1 0 0
D) x1 x2 s1 s2
2
5 1 0
3
3 0 1
-4 -1 0 0
z
0
0
1
z
19
11
0
0
0
1
11
19
0
Find the pivot in the tableau.
27)
A) 4 in row 1, column 3
C) 2 in row 1, column 1
B) 4 in row 2, column 2
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
3
2
0 1 3 0 0 8
0 0 1 1 0 13
1 0 2 0 0 27
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
D) x1 = 0, x2 = 27, x3 = 8, x4 = 0, x5 = 13, z = 0
C) x1 = 0, x2 = 27, x3 = 8, x4 = 0, x5 = 13, z = 23
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
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
with
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)
B)
x1
3
1
2
- 62
x2 s1 s2 s3 s4
5 1 0 0 0 580
3 0 1 0 0 660
2 0 0 1 0 700
- 93 0 0 0 1
0
x1
3
1
2
- 93
C)
x2 s1 s2 s3 s4
5 1 0 0 0 580
3 0 1 0 0 660
2 0 0 1 0 700
62 0 0 0 1
0
D)
x1
3
1
2
93
x2 s1 s2 s3 s4
5 1 0 0 0 580
3 0 1 0 0 660
2 0 0 1 0 700
62 0 0 0 1
0
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 Departmental capacity
by product (hours)
(Total hours)
A B C
Assembling
2 3 2
30,000
Painting
1 2 2
38,000
Finishing
2 3 1
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
A 3 2
B 1 6
C 2 3
E3 E4
1 1
3 1
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 Key
Testname: 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