Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations by graphing.
1)
2x + y = – 5
x= – 2
A)
(–2, –1)
B)
(–1, –2)
C)
(2, 1)
D)
no solution
Answer:
2)
3x +y=9
2x +y=7
A)
(3, 2)
B)
(–2, –3)
C)
(2, 3)
D)
no solution
Answer:
1
3)
y=x + 7
y=3x + 9
A)
(6, –1)
B)
(–1, 6)
C)
(1, –6)
D)
no solution
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
4)
Use a graphing utility to solve the system y = 2x + 7
y = – 5x + 1 . Give the answer to three decimal places.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the system of equations by substitution.
5)
x + 2y =6
6x + 3y =9
A)
(1, 2)
B)
(–3, 0)
C)
(0, 3)
D)
no solution
Answer:
6)
x –2y = – 4
y =3
A)
(–2, 3)
B)
(3, 2)
C)
(2, 3)
D)
(–10, 3)
Answer:
Solve the system of equations by elimination.
7)
3x + y =18
5x – y =6
A)
(3, 9)
B)
(9, 3)
C)
infinitely many solutions
D)
no solution
Answer:
8)
x – 5y = 35
5x – 4y = 28
A)
(7, 0 )
B)
(1 , – 7)
C)
(0, – 7)
D)
(1, – 8)
Answer:
2
9)
x + y =8
x – y =8
A)
(0, 8)
B)
(–8, 0)
C)
(0, –8)
D)
(8, 0)
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10)
8x – 4y = 10
12x – 6y = – 15
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
11)
3x – 5y= – 7
–7x + 3y=25
A)
(4, 1)
B)
(4, –1)
C)
(–4, –1)
D)
(–4, 1)
Answer:
Solve the system mentally, without the use of a calculator or pencil–and–paper calculation. Try to visualize the graphs of
both lines.
12)
x + 0y = 9
0x + y = 4
A)
x = 9; y = 2
B)
x = 2; y =9
2
C)
x = 9; y = 4
D)
x = 4; y = 9
Answer:
13)
x – 0y = 5
3x + y = 7
A)
x = – 5; y = 8
B)
x = 3, y = 7
C)
x = 5; y = – 8
D)
x = 5; y = 4
Answer:
Solve the problem.
14)
Sam and Chad are ticket–sellers at their class play. Sam is selling student tickets for $2.00 each, and Chad selling
adult tickets for $5.50 each. If their total income for 24 tickets was $83.00, how many tickets did Sam sell?
A)
16 tickets
B)
15 tickets
C)
10 tickets
D)
14 tickets
Answer:
15)
Daisy has a desk full of quarters and nickels. If she has a total of 23 coins with a total face value of $4.35, how
many of the coins are nickels?
A)
9 nickels
B)
21 nickels
C)
7 nickels
D)
16 nickels
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
16)
A company that manufactures laser printers for computers has monthly fixed costs of $177,000 and variable
costs of $650 per unit produced. The company sells the printers for $1,250 per unit. How many printers must be
sold each month for the company to break even?
Answer:
3
17)
Suppose that the supply and demand equations for a logo sweat shirt in a particular week are p = 55 – 0.10q, for
the demand equation; and p = 0.20q + 25, for the supply equation. Find the equilibrium price and quantity.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
18)
Given matrix A:
A = 5 7 – 6
–4 9 1
What is the size of A?
A)
2 × 3
B)
3
C)
3 × 3
D)
3 × 2
Answer:
19)
Given the matrix B:
B =
–3
–1
1
What is the size of B?
A)
1 × 1
B)
3
C)
1 × 3
D)
3 × 1
Answer:
20)
Write the augmented matrix for the system.
8x1+ 9x2= 117
4x1+ 6x2= 66
A)
117 9 8
66 4 6
B)
8 4 117
9 6 66
C)
8 9 66
6 4 117
D)
8 9 117
4 6 66
Answer:
21)
Write the augmented matrix for the system.
6x1+ 4x2= 30
8x2= 72
A)
6 4 30
0 8 72
B)
30 4 6
72 0 8
C)
6 4 30
8 72 0
D)
8 0 72
6 4 4
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
22)
Given matrix A:
A =
5 –3
1 7
0 –2
–91
2
What is the size of A? Find a32 and a11.
Answer:
4
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Perform the indicated row operations on the following matrix.
1–5 4
2 2 5
23)
(–2)R1+R2
R2
A)
012 –3
2 2 5
B)
–210 –8
2 12 –5
C)
1–5 4
012 –3
D)
–210 –8
0 12 –3
Answer:
24)
–3R1
R1
A)
–3–5 4
–6 2 5
B)
1 –5 4
–117 7
C)
–315 –12
2 2 5
D)
1 –5 4
–6–6–15
Answer:
Identify the row operation that produces the resulting matrix.
25)
1–3 4
2 3 1
1–3 4
–1–6 3
A)
(–1)R2
R2
B)
R1+ (–1)R2
R1
C)
R1+ (–1)R2
R2
D)
(–1)R1+R2
R2
Answer:
26)
1 0 2
–1 1 3 1 0 2
0 1 5
A)
–R2+R1
R2
B)
R1+R2
R2
C)
R1+R2
R1
D)
–R1+R2
R2
Answer:
27)
2 0 2
–2 2 12 2 0 2
0 1 7
A)
–1
2R1+ – 1
2R2
R1
B)
R1+R2
R2
C)
1
2R1+1
2R2
R2
D)
1
2R2
R1
Answer:
28)
3 0 12
–2 4 10 3 0 12
0 2 9
A)
1
3R1+R2
R1
B)
1
2R1+1
3R2
R1
C)
R1+1
2R2
R2
D)
1
3R1+1
2R2
R2
Answer:
5
Write a system of equations associated with the augmented matrix. Do not try to solve.
29)
–5 0
1 1
–9–5
x
y
=
2
5
–5
A)
–5x =2
x + y =5
–9x – 5y = – 5
B)
–5x + y =2
x + 5y =5
–9x – 5y = – 5
C)
–5x + y + z =2
x – 5y =5
Answer:
30)
3 3 5 – 2
5 0 7 4
3 6 0 2
A)
3x1– 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
B)
3x1+ 3x2+ 5x3= 2
5x1+ 7x3= 4
3x1+ 6x2= 2
C)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= – 4
3x1+ 6x2= – 2
D)
3x1+ 3x2+ 5x3= – 2
5x1+ 7x3= 4
3x1+ 6x2= 2
Answer:
Provide an appropriate response.
31)
Use the augmented matrix to solve the system:
0.4x1+ 0.9x2= – 4.9
x1– 0.3x2= 0.5
A)
(–0.1, –0.5)
B)
(0, 0)
C)
(–1, –5)
D)
(–5, –1)
Answer:
32)
Solve the linear system corresponding to the following augmented matrix:
3 6 24
2 3 11
A)
(–2, –5)
B)
(5, –2)
C)
(–2, 5)
D)
(0, 0)
Answer:
33)
Only one of the following augmented matrices of a linear system is in a reduced form. Choose the matrix that is
in reduced form.
A)
1 0 –2
0 0 0
0 1 –1
B)
0 1 –2
1 0 –3
C)
1 0 –2 2
0 0 1 3
D)
1 4 0 4
0 0 1 –4
0 0 0 0
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
34)
Use a graphing utility and augmented matrix methods to solve the system: 1.8x1– 12.17x2= 33
–3.75x1+ 5.73x2= 7
Express your answer accurate to three decimal places.
Answer:
6
35)
Solve the linear system corresponding to the following augmented matrix:
1 4 0 5
0 0 1 6
0 0 0 0
Answer:
36)
Solve the linear system corresponding to the following augmented matrix:
1 0 2 –5
0 1 –4 2
0 0 1 1
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The matrix is the final matrix form for a system of two linear equations in variables x1 and x2. Write the Solution of the
system.
37)
1 0 5
0 1 3
A)
x1= – 5
x2= – 3
B)
x1= 5
x2= 3
C)
x1= 3
x2= 5
D)
x1= 5
x2= t for any real number t
Answer:
38)
1–410
0 0 0
A)
x1= 4t + 10
x2= t for any real number t
B)
No solution
C)
x1= t – 4
x2= t for any real number t
D)
x1= t for any real number t
x2= 10
Answer:
State whether the matrix is in reduced form or not in reduced form.
39)
1 3 0 1
0 0 1 0
0 0 0 1
A)
Not Reduced Form
B)
Reduced Form
Answer:
40)
1 0 –1 5
0 4 1 1
0 0 1 1
A)
Not Reduced Form
B)
Reduced Form
Answer:
7
41)
1–5 0 0 1
0 0 120
A)
Reduced Form
B)
Not Reduced Form
Answer:
Write the linear system corresponding to the reduced augmented matrix.
42)
1 0 0 4
0 1 1 6
0 0 0 0
A)
x1= – 4, x2= t + 6, x3= t for any real number t
B)
x1= 4, x2= t + 6, x3= 0
C)
No Solution
D)
x1= 4, x2= – t + 6, x3= t for any real number t
Answer:
43)
1 0 4
0 1 0
0 0 0
A)
x1= 4, x2= t for any real number t
B)
x1= 4, x2= 0
C)
No Solution
D)
x1= – 4, x2= 0
Answer:
User row operations to change the matrix to reduced form.
44)
1 0 2
–1 1 3
A)
1 0 2
0 0 5
B)
1 0 2
–1 1 3
C)
1 0 2
0 1 5
D)
0 1 5
–1 1 3
Answer:
45)
1–1 0 1
0 4 8 4
0 0 0 0
A)
1 0 2 2
0 1 2 0
0 0 0 0
B)
1 0 2 0
0 1 2 1
0 0 0 0
C)
1 0 2 2
0 1 2 1
0 0 0 0
D)
1–1 0 1
0 1 2 1
0 0 0 0
Answer:
Solve using Gauss–Jordan elimination.
46)
4x + 6y = 6
4x + 7y = 9
A)
(3, –3)
B)
(–3, –3)
C)
(–3, 3)
D)
No solution
Answer:
47)
x1+x2= 0
x1– x 2 = 12
A)
(–5, –6)
B)
(–6, –5)
C)
(6, –6)
D)
(5, –5)
Answer:
8
48)
9x – y – 9z = 9
6x + 4y + 5z = 92
–6x – 5y + z = – 77
A)
(–6, 9, 12)
B)
(6, 4, 9)
C)
(6, 9, 4)
D)
No solution
Answer:
49)
x + y + z = 7
x – y + 3z =21
5x + y + z =23
A)
(4, –2, 5)
B)
(5, –2, 4)
C)
(5, 4, –2)
D)
No solution
Answer:
50)
2x – 5y = – 9
–2x + 5y = – 3
A)
5
9x –2
9y, y
B)
(–1, –1)
C)
(–9, –3)
D)
No solution
Answer:
Find the system of equations to model the problem. DO NOT SOLVE THIS SYSTEM.
51)
There were 35,000 people at a ball game in Atlanta. The day’s receipts were $290,000. How many people paid
$14 for reserved seats and how many paid $6 for general admission? Let x represent the number of reserved
seats and y represent the number of general admission seats.
A)
15,000x + 14y = 20,000
x + y = 6
B)
20,000x + 14y = 6
x + y = 15,000
C)
14x + 6y = 105
x + y = 35,000
D)
25,000x + 14y = 35,000
x + y = 290,000
Answer:
52)
Hurst’s Feed & Seed sold to one customer 5 bushels of wheat, 2 of corn, and 3 of rye, for $31.00. To another customer h
sold 2 bushels of wheat, 3 of corn, and 5 of rye, for $27.60. To a third customer he sold 3 bushels of wheat, 5 of corn, and
2 of rye for $32.70. What was the price per bushel for each of the different grains?
Let x represent the price per bushel for wheat, y the price per bushel for corn, and z the price per bushel for rye.
A)
5x + 2y + 3z = 31.00
2x – 3y + 5z = 27.60
3x + 5y + 2z = 32.70
B)
5x + 2y – 3z = 31.00
2x + 3y – 5z = 27.60
3x + 5y – 2z = 32.70
C)
5x + 2y + 3z = 31.00
2x + 3y – 5z = 27.60
3x + 5y – 2z = 32.70
D)
5x + 2y + 3z = 31.00
2x + 3y + 5z = 27.60
3x + 5y + 2z = 32.70
Answer:
53)
A $124,000 trust is to be invested in bonds paying 9%, CDs paying 8%, and mortgages paying 10%. The sum of th
amount invested in bonds and the amount invested in CDs must equal the mortgage investment. To earn an $11
annual income from the investments, how much should the bank invest in each?
Let x represent the amount invested in bonds, y the amount invested in CDs, and z the amount invested in mortgages.
A)
x + y – z = 11,400
x – y + 9z = 22
8x + y + z = 124,000
B)
x + y + z = 0
x + y – 9z = 124,000
0.1x + 0.08y – 0.09z = 11,400
C)
x + y – z = 0
x + y + z = 124,000
0.09x + 0.08y + 0.1z = 11,400
D)
x + y – z = 0
x + y + z = 124,000
9x + 8y + z = 11,400
Answer:
9
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
54)
If $9,000 is to be invested, part at 13% and the rest at 8% simple interest, how much should be invested at each
rate so that the total annual return will be the same as $9,000 invested at 9%? Set up a system of linear
equations, letting x1 be the amount invested at 13% and x2 be the amount invested at 8%.
Answer:
55)
Labor and material costs for manufacturing each of three types of products M, N, and P are given in the table:
Product
M N P
Labor $50 $40 $50
Materials $60 $40 $70
The weekly allocation for labor is $50,000 and for materials is $80,000. There are to be 3 times as many units of prod
M manufactured as units of product P. How many of each type of product would be manufactured each week to us
exactly each of the weekly allocations? Set up a system of linear equations, letting x1, x2, and x3 be the number
of units of products M, N, and P, respectively, manufactured in one week.
Answer:
56)
In producing three types of bricks: face bricks, common bricks, and refractory bricks, a factory incurs labor,
material, and utility costs. To produce one pallet of face bricks, the labor, material, and utility costs are $50, $75,
and $35, respectively. To produce one pallet of common bricks, the labor, material, and utility costs are $50, $60,
and $30, respectively, while the corresponding costs for refractory bricks are $75, $100, and $45. In a certain
month the company has allocated $12,000 for labor costs, $14,500 for material costs and $6,000 for utility costs.
How many pallets of each type of brick should be produced in that month to exactly utilize these allocations?
Set up a system of linear equations, letting x, y, and z be the number of pallets of face, common, and refractory
bricks, respectively, that must be produced in that month.
Answer:
10
57)
A paper company produces high, medium, and low grade paper. The number of tons of each grade that is produc
from one ton of pulp depends on the source of that pulp. The following table lists three sources and the amount of each
grade of paper that can be made for one ton of pulp from each source.
(Number of Tons)
High Grade Medium Grade Low grade
Brazilian Pulp 0.6 0.3 0.1
Domestic Pulp 0.5 0.3 0.2
Recycled Pulp 0.3 0.4 0.3
The paper company has orders for 11 tons of high grade, 15 tons of medium grade, and 14 tons of low grade pape
How many tons of each type of pulp should be used to fill these orders exactly? Set up a system of linear equation
letting x, y, and z be the number of tons of Brazilian pulp, domestic pulp, and recycled pulp, respectively, needed to f
the orders.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
58)
A chemistry department wants to make 3 liters of a 17.5% basic solution by mixing a 20% solution with a 15%
solution. How many liters of each type of basic solution should be used to produce the 17.5% solution?
A)
1 liter of 15% solution, 2 liters of 20% solution
B)
1.5 liters of 15% solution, 1.5 liters of 20% solution
C)
2 liters of 15% solution, 1 liter of 20% solution
D)
0.5 liter of 15% solution, 2.5 liters of 20% solution
Answer:
59)
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)
$10,000 at 5%, $10,000 at 6%, $5000 at 8%
B)
$10,000 at 5%, $5000 at 6%, $10,000 at 8%
C)
$8000 at 5%, $10,000 at 6%, $7000 at 8%
D)
$5000 at 5%, $10,000 at 6%, $10,000 at 8%
Answer:
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
60)
A hospital dietitian wants to insure that a certain meal consisting of rice, broccoli, and fish contains exactly
26,800 units of vitamin A, 840 units of vitamin E, and 11,160 units of vitamin C. One ounce of rice contains 400
units of vitamin A, 20 units of vitamin E, and 180 units of vitamin C. One ounce of broccoli contains 800 units of
vitamin A, 60 units of vitamin E, and 540 units of vitamin C. And one ounce of fish contains 2,400 units of
vitamin A, 40 units of vitamin E, and 810 units of vitamin C. How many ounces of each food should this meal
include? Set up a system of linear equations and solve using Gauss–Jordan elimination.
Answer:
11
61)
A trucking firm wants to purchase 10 trucks that will provide exactly 28 tons of additional shipping capacity. A
model A truck holds 2 tons, a model B truck holds 3 tons, and a model C truck holds 5 tons. How many trucks
of each model should the company purchase to provide the additional shipping capacity? Set up a system of
linear equations and solve using Gauss–Jordan elimination. There may be more than one solution.
Answer:
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Perform the operation, if possible.
62)
–9 1
2 5
+6 2
2–3
A)
–3 3
4 2
B)
–3–7
–7–12
C)
3 3
4 2
D)
–3–3
4 2
Answer:
63)
– 1 0
3 1
––1 3
3 1
A)
–2 3
6 2
B)
0–3
0 0
C)
–3
D)
0 3
0 0
Answer:
64)
5 4 +– 2
7
A)
3
11
B)
3 11
C)
5– 2
4 7
D)
Not defined
Answer:
65)
Let A =
–6 5
–2–8
9 3
and B =
–7 3
–4–2
–7 5
. Find A + B.
A)
–13 –8
–6–10
2 8
B)
1 2
2 2
16 –1
C)
–13 8
–6–10
2 8
D)
–13 8
6–8
2–8
Answer:
66)
Let B =–1 6 7 –3. Find –2B.
A)
2–12 –14 6
B)
–3 4 5 –5
C)
–212 14 –6
D)
2 6 7 –3
Answer:
12