Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1)
An electronics company kept comparative statistics on two products, A and B. For the years 2003 to
2011, the total number of Product A ever sold (in thousands) is given by the equation y =75x +240,
where x is the number of years since 2003. For that same period, the total number of Product B ever
sold (in thousands) is given by the equation y = – 30x + 434, where x is the number of years since
2003. Use the substitution method to solve the system and choose the statement that most
accurately describes the solution.
1)
A)
At about 1.8 years (to the nearest tenth) since 2003, the company sold the same number of
Product A as Product B.
B)
At some point between 2003 and 2011, the company had sold 1800 of each product.
C)
When 375,000 of Product A had been sold, 1.8 times as many of Product B had been sold.
D)
The company sold about 1.8 times as many of Product B as Product A.
Determine whether the system has exactly one ordered–pair solution, no solution, or infinitely many solutions.
2)
x
2+y
3= 4
x
4+y
6= 2
2)
A)
infinitely many solutions
B)
one ordered–pair solution
C)
no solution
Solve the system by the substitution method.
3)
x
6–y
8=1
x
9– y =8
3)
A)
(–8, 0)
B)
(8, 0)
C)
(0, –8)
D)
(0, 8)
Solve the system by graphing.
4)
4x +y= – 6
5x +6y =2
4)
A)
{(2, 2)}
B)
C)
{(x, y) 4x + y = – 6}
D)
{(–2, 2)}
Determine if the given ordered triple is a solution of the system.
5)
(5, –4, –3)
x + y + z = – 2
x – y + 2z =11
2x + y + z = – 5
5)
A)
not a solution
B)
solution
Solve the system by the addition method.
6)
5x –2y = – 1
7x +4y =53
6)
A)
{(3, 8)}
B)
C)
{(x, y) 5x – 2y = – 1}
D)
{(3, 9)}
The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars.
Use the information in the figure to answer the question.
7)
Use the revenue and cost functions to write the profit function from producing and selling x
binoculars.
7)
A)
P(x) = 2x – 1500
B)
P(x) = 2x + 1500
C)
P(x) = 4x – 1500
D)
P(x) = 4x + 1500
Solve the system by the addition method.
8)
7x +6y = – 5
–7x –11y=15
8)
A)
{(–1, 2)}
B)
{(x, y) 7x +6y = – 5}
C)
{(1, –2)}
D)
Use Cramer’s rule to solve the system.
9)
2x +6y =20
5x +y= – 20
9)
A)
{(5, –5)}
B)
{(–5, 5)}
C)
dependent equations; infinitely many solutions
D)
inconsistent; no solution or
Use Cramer’s rule to solve the linear system.
10)
–5x +8z =52
8x +6y –3z =17
4x –6y =4
10)
A)
{(4, –2, –9)}
B)
{(5, 0, 9)}
C)
{(2, 9, 2)}
D)
{(4, 2, 9)}
Solve the system by any method.
11)
–2x – 7y = – 31
–4x – 8y = – 32
11)
A)
(2, 6)
B)
C)
(–2, 5)
D)
(–3, 6)
Determine if the given ordered triple is a solution of the system.
12)
(–3, –1, 5)
x + 5y + 5z =17
3y + 4z =17
z =5
12)
A)
solution
B)
not a solution
Solve the problem.
13)
A company is planning to produce and sell a new video game console. The fixed cost will be
$450,000 and it will cost $150 to produce each console. Each video game console will be sold for
$400. Write the cost function, C, of producing x video game consoles.
13)
A)
C(x) =450,000 +150x
B)
C(x) =450,000 +400x
C)
C(x) =450,000 –150x
D)
C(x) =450,000x +150
Determine whether the system has exactly one ordered–pair solution, no solution, or infinitely many solutions.
14)
6x +y=2
–18x–3y = – 6
14)
A)
one ordered–pair solution
B)
infinitely many solutions
C)
no solution
Solve the system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations
are dependent, so state.
15)
3x –2y =4
–12x+8y = – 12
15)
A)
9
8, –3
4
B)
no solution or
C)
infinitely many solutions; dependent equations
D)
{(4, 3)}
Solve the problem.
16)
A chemist needs 180 milliliters of a 31% solution but has only 9% and 45% solutions available. Find
how many milliliters of each that should be mixed to get the desired solution.
16)
A)
80 milliliters of 9%; 110 milliliters of 45%
B)
70 milliliters of 9%; 110 milliliters of 45%
C)
110 milliliters of 9%; 70 milliliters of 45%
D)
80 milliliters of 9%; 120 milliliters of 45%
Solve the system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations
are dependent, so state.
17)
x–y+2z = – 2
3x +z=0
x+3y +z=6
17)
A)
no solution or
B)
infinitely many solutions; dependent equations
C)
{(0, 0, 2)}
D)
{(0, 2, 0)}
18)
x+y+z=1
x–y+2z =13
3x +y+z=3
18)
A)
{(4, –4, 1)}
B)
no solution or
C)
{(1, –4, 4)}
D)
infinitely many solutions; dependent equations
Solve the system. If there is no solution or if the system’s equations are dependent, so state.
19)
x–y+3z = – 5
4x +z=0
–x+y–3z =20
19)
A)
{(0, 0, 5)}
B)
infinitely many solutions; dependent equations
C)
no solution or
D)
{(3, 5, 0)}
20)
2x +4y +z=2
4x –3y –z= – 30
5x +y+2z =4
20)
A)
{(–5, 4, 2)}
B)
infinitely many solutions; dependent equations
C)
no solution or
D)
{(–5, 2, 4)}
Solve the problem.
21)
A deli sells three sizes of chicken sandwiches: the small chicken sandwich contains 6 ounces of
meat and sells for $3.50; the regular chicken sandwich contains 10 ounces of meat and sells for
$4.00; and the large chicken sandwich contains 13 ounces of meat and sells for $4.50. A customer
requests a selection of each size for a reception. She and the manager agree on a combination of 50
sandwiches made from 28 pounds 8 ounces of chicken for a total cost of $194. How many of each
size sandwich will be in this combination? (Note: 1 pound = 16 ounces)
21)
A)
22 small sandwiches, 20 medium sandwiches, 8 large sandwiches.
B)
24 small sandwiches, 14 medium sandwiches, 12 large sandwiches.
C)
26 small sandwiches, 8 medium sandwiches, 16 large sandwiches.
D)
20 small sandwiches, 10 medium sandwiches, 20 large sandwiches.
Solve the system by any method.
22)
–7x = – 42 – 9y
–4x – 2y =26
22)
A)
(–3, –6)
B)
(–4, –6)
C)
D)
(–3, –7)
Determine if the given ordered triple is a solution of the system.
23)
(3, –5, 1)
x + y + z = – 1
x – y + 5z =13
5x + y + z =11
23)
A)
solution
B)
not a solution
Solve the problem.
24)
Devon purchased tickets to an air show for 8 adults and 2 children. The total cost was $128. The
cost of a child‘s ticket was $6 less than the cost of an adult’s ticket. Find the price of an adult’s ticket
and a child’s ticket.
24)
A)
adult’s ticket: $13; child’s ticket: $7
B)
adult’s ticket: $16; child’s ticket: $10
C)
adult’s ticket: $15; child’s ticket: $9
D)
adult’s ticket: $14; child’s ticket: $8
Use Cramer’s rule to solve the system or to determine that the system is inconsistent or contains dependent equations.
25)
2x +y=10
6x +3y =30
25)
A)
{(0, 10)}
B)
dependent equations; infinitely many solutions
C)
{(5, 0)}
D)
inconsistent; no solution or
The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars.
Use the information in the figure to answer the question.
26)
Is there a profit when 808 binoculars are produced?
26)
A)
Yes
B)
No
Write the augmented matrix for the system of equations.
27)
–3x – 3y = 2
4x + 9y = 4
27)
A)
4 –3 2
–3 9 4
B)
–3–3 2
4 9 4
C)
4 –3–2
–3 9 –4
D)
–3–3–2
4 9 –4
Explanation:
Evaluate the determinant.
28)
1
10 –1
5
510
28)
A)
0
B)
2
C)
–2
D)
5
2
Solve the problem.
29)
The Little Town Arts Center charges $23 for adults, $13 for senior citizens, and $6 for children
under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was
$11,338 for 834 tickets sold. There were 41 more children than adults. How many children
attended?
29)
A)
302 children
B)
271 children
C)
261 children
D)
292 children
Determine whether the given ordered pair is a solution to the system.
30)
(5, –4)
x+ y = – 9
x– y = – 1
30)
A)
solution
B)
not a solution
Solve the problem.
31)
Jamil always throws loose change into a pencil holder on his desk and takes it out every two weeks.
This time it is all nickels and dimes. There are 7 times as many dimes as nickels, and the value of
the dimes is $3.25 more than the value of the nickels. How many nickels and dimes does Jamil
have?
31)
A)
6 nickels and 42 dimes
B)
5 nickels and 35 dimes
C)
4 nickels and 28 dimes
D)
35 nickels and 5 dimes
Write the augmented matrix for the system of equations.
32)
14x + 10y = 10
2x + 17y = 5
32)
A)
217 –10
14 10 –5
B)
210 10
14 17 5
C)
14 10 –10
217 –5
D)
14 10 10
217 5
Determine whether the system is inconsistent, dependent, or neither.
33)
x – y + 4z =3
2x + z =0
–x + y – 4z = – 9
33)
A)
neither
B)
inconsistent
C)
dependent
Solve the system using matrices.
34)
4x +5y =52
4x +2y =64
34)
A)
{(–18, 4)}
B)
{(18, –4)}
C)
infinitely many solutions; dependent equations
D)
no solution or
Use Cramer’s rule to solve the system or to determine that the system is inconsistent or contains dependent equations.
35)
4x +5y =18
3x +y= – 3
35)
A)
{(6, –3)}
B)
inconsistent; no solution or
C)
{(–3, 6)}
D)
dependent equations; infinitely many solutions
Solve the system. If there is no solution or if the system’s equations are dependent, so state.
36)
x + y + z = – 10
x – y +2z = – 3
4x + y + z = – 25
36)
A)
no solution or
B)
{(–5, –4, –1)}
C)
infinitely many solutions; dependent equations
D)
{(–1, –4, –5)}
The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars.
Use the information in the figure to answer the question.
37)
Fewer than how many binoculars must be produced and sold for the company to have a profit loss?
37)
A)
1500 binoculars
B)
2700 binoculars
C)
2250 binoculars
D)
750 binoculars
Solve the system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations
are dependent, so state.
38)
x+2y = – 6
5x +10y= – 30
38)
A)
{(0, 0)}
B)
{(–6, 0)}
C)
no solution or
D)
infinitely many solutions; dependent equations
Perform the matrix row operation and write the new matrix.
39)
–7–5 3
2–6 6
–1R1
39)
A)
7–5 3
–2–6 6
B)
7 5 –3
2–6 6
C)
–8–5 3
1–6 6
D)
–8–6 2
2–6 6
Solve the system using matrices. If there is no solution or if there are infinitely many solutions and a system’s equations
are dependent, so state.
40)
x+y= – 3
x–y=19
40)
A)
{(8, 11)}
B)
no solution or
C)
infinitely many solutions; dependent equations
D)
{(8, –11)}
Solve the problem.
41)
Find the values of a, b, and c such that the graph of the quadratic equation y = ax2+ bx + c passes
through the points (–2, –1), (3, 29), and (4, 47).
41)
A)
a =2; b = – 1; c =4
B)
a = – 2; b = – 1; c =4
C)
a =2; b =4; c = – 1
D)
a = – 2; b =4; c = – 1
42)
Given the cost function, C(x), and the revenue function, R(x), write the profit function from
producing and selling x units of the product.
C(x) =1.4x +1260
R(x) =2.3x
42)
A)
P(x) = – 0.9x –1260
B)
P(x) = – 0.9x +1260
C)
P(x) =0.9x +1260
D)
P(x) =0.9x –1260
Write a system of linear equations in three variables, and then use matrices to solve the system.
43)
The table below shows the number of birds for three selected years after an endangered species
protection program was started.
x (Number of years after 2002) 1 2 3
y (Number of birds) 40 58 82
Use the quadratic function y = ax2+ bx + c to model the data. Solve the system of linear equations
involving a, b, and c using matrices. Find the equation that models the data.
43)
A)
y =6x2– 27x +24
B)
y =4x2+18x +23
C)
y =3x2+9x +28
D)
y =5x2– 9x +31
Solve the system by any method.
44)
9x + 30 = – 4y
7x – 2y = – 54
44)
A)
{(x, y) 9x + 30 = – 4y}
B)
(–7, 7)
C)
(–6, 7)
D)
(–6, 6)
D
Use Cramer’s rule to solve the system or to determine that the system is inconsistent or contains dependent equations.
45)
11x+3y =1
5x +2y =4
45)
A)
inconsistent; no solution or
B)
–10
7, 39
7
C)
39
7, –10
7
D)
dependent equations; infinitely many solutions
B
C
Determine whether the system has exactly one ordered–pair solution, no solution, or infinitely many solutions.
46)
6x – 6y =7
18x – 18y =14
46)
A)
infinitely many solutions
B)
no solution
C)
one ordered–pair solution
Solve the problem.
47)
A company’s expenses included many factors. In 2012, travel costs were 2.52% of the expense
budget , increasing by 0.26% of the total expense budget per year. In 2012, office supplies were
5.73% of the expense budget, increasing by 0.02% of the total expense budget per year. In which
year will the cost of travel expenses and office supplies be the same? For that year, what will be the
cost of each expense as a percentage of the total expense budget?
47)
A)
2025; 5.9
B)
2027; 6.4
C)
2024; 5.6
D)
2023; 5.4
48)
Given the cost function, C(x), and the revenue function, R(x), find the number of units x that must
be sold to break even.
C(x) =0.9x +600
R(x) =1.5x
48)
A)
286 units
B)
1000 units
C)
1020 units
D)
1010 units
Solve the system by graphing.
49)
3x +y= – 9
4x +2y = – 10
49)
A)
{(–4, 3)}
B)
C)
{(x, y) 3x + y = – 9}
D)
{(4, 3)}
Determine whether the system has exactly one ordered–pair solution, no solution, or infinitely many solutions.
50)
2x – 8y =4
–6x + 24y = – 16
50)
A)
no solution
B)
one ordered–pair solution
C)
infinitely many solutions
The figure shows the graphs of the cost and revenue functions for a company that manufactures and sells binoculars.
Use the information in the figure to answer the question.
51)
What is the profit when 995 binoculars are produced?
51)
A)
$5480
B)
$3490
C)
$490
D)
$2480
Solve the system. If there is no solution or if the system’s equations are dependent, so state.
52)
x–6y –z=9
4x –24y–4z =36
5x –30y–5z =45
52)
A)
no solution or
B)
{(5, 4, –28)}
C)
{(4, 5, –35)}
D)
infinitely many solutions; dependent equations
Evaluate the determinant.
53)
9 8
5 5
53)
A)
–5
B)
5
C)
47
D)
85
Determine whether the system is inconsistent, dependent, or neither.
54)
3x + y = – 15
–3y + 4z =21
3x + 7y – 8z = – 57
54)
A)
neither
B)
inconsistent
C)
dependent
Solve the system by the addition method.
55)
6x +3y =51
2x –6y =38
55)
A)
{(–3, 10)}
B)
{(x, y) 2x – 6y = 38}
C)
{(10, –3)}
D)
56)
4x +y=14
4x +y=26
56)
A)
{(x, y) 4x + y =14}
B)
{(12, 2)}
C)
{(9, 10)}
D)
Solve the problem.
57)
The cost and revenue functions for producing and selling x units of a new kitchen appliance are
C(x) =70x +7000 and R(x) =150x.
Write the profit function, P, from producing and selling x kitchen appliances.
57)
A)
P(x) =220x –7000
B)
P(x) =80x –7000
C)
P(x) =7000 –150x
D)
P(x) =80x +7000
58)
A company is planning to produce and sell a new video game console. The fixed cost will be
$720,000 and it will cost $150 to produce each console. Each video game console will be sold for
$550. Determine the break–even point. Describe what this means.
58)
A)
(1800, 990,000); When 1800 video game consoles are sold, both cost and revenue are $990,000.
B)
(1800, 990,000); When 1800 video game consoles are sold, profit is $990,000.
C)
(18,000, 9,900,000); When 18,000 video game consoles are sold, profit is $9,900,000.
D)
(18,000, 9,900,000); When 18,000 video game consoles are sold, both cost and revenue are $
9,900,000.
59)
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by Area = ± 1
2
x1y11
x2y21
x3y31
,
where the symbol ± indicates that the appropriate sign should be chosen to yield a positive area.
Use this formula to find the area of a triangle whose vertices are (2, 9), (8, –8), and (–3, –9).
59)
A)
1 square units
B)
193
2 square units
C)
1
2 square units
D)
193 square units
B
Determine whether the given ordered pair is a solution to the system.
60)
(3, –2)
3x +y=11
2x + 3y=12
60)
A)
solution
B)
not a solution
B
Determine if the given ordered triple is a solution of the system.
61)
(–1, 5, 2)
x + 4y + 2z =20
2y + 2z =8
z = – 1
61)
A)
not a solution
B)
solution
A
A
Solve the system by any method.
62)
3x –4
5y = – 5
4x –2
3y = – 26
3
62)
A)
B)
(–2, –6)
C)
(3, –6)
D)
(–3, –5)
Solve the system. If there is no solution or if the system’s equations are dependent, so state.
63)
x+3y +3z = – 26
2y +2z = – 16
z= – 5
63)
A)
infinitely many solutions; dependent equations
B)
no solution or
C)
{(–2, –5, –3)}
D)
{(–2, –3, –5)}
Use matrices to determine whether the system is inconsistent, dependent, or neither.
64)
2x + y =0
4y – 2z = – 18
4x + 10y – 4z = – 36
64)
A)
neither
B)
dependent
C)
inconsistent
Determine whether the given ordered pair is a solution to the system.
65)
(–1, –2)
x+ y = – 3
x– y =1
65)
A)
not a solution
B)
solution
Solve the system by any method.
66)
2x – 6y =5
–6x + 18y = – 20
66)
A)
(2, 5)
B)
C)
4
15 , –4
5
D)
(3, 4)
Determine whether the system is inconsistent, dependent, or neither.
67)
x + y + z = – 6
x – y + 5z = – 20
4x +4y + 4z =0
67)
A)
dependent
B)
inconsistent
C)
neither
Solve the system by the substitution method.
68)
x= – 5y +5
9x –2y = – 2
68)
A)
{(1, 0)}
B)
{(x, y) x = – 5y +5}
C)
D)
{(0, 1)}
Determine whether the system has exactly one ordered–pair solution, no solution, or infinitely many solutions.
69)
x + y = – 6
x + y = – 8
69)
A)
infinitely many solutions
B)
one ordered–pair solution
C)
no solution