Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many
solutions, using set notation to express their solution sets.
87)
y =13 –3x
12x +4y =52
87)
A)
{(5, –2)}
B)
{(0, 13)}
C)
{(x, y) 3x + y =13}
D)
Graph the solution set of the system of inequalities or indicate that the system has no solution.
88)
y 1
2 x
6
x – 2y –2
x + y 6
88)
A)
B)
41
C)
D)
Solve the system by the addition method.
89)
2x + 7y =12
2x + 2y =22
89)
A)
{(–2, 13)}
B)
{(–13, 7)}
C)
{(–13, 2)}
D)
{(13, –2)}
Find the maximum or minimum value of the given objective function of a linear programming problem. The figure
illustrates the graph of feasible points.
90)
Objective Function: z = x + 5y
Find maximum.
90)
A)
maximum: 14
B)
maximum: 22
C)
maximum: 27
D)
maximum: 18
42
Solve the problem.
91)
An office manager is buying used filing cabinets. Small file cabinets cost $7 each and large file
cabinets cost $10 each, and the manager cannot spend more than $113 on file cabinets. A small
cabinet takes up 7 square feet of floor space and a large cabinet takes up 8 square feet, and the
office has no more than 103 square feet of floor space available for file cabinets. The manager must
buy at least 7 file cabinets in order to get free delivery. Let x = the number of small file cabinets
bought and y = the number of large file cabinets bought. Write a system of inequalities that
describes these constraints.
91)
A)
7x +10y 113
7x +8y 103
x + y 7
B)
7x +10y 113
7x +8y 103
x + y 7
C)
7x +10y 113
8x +7y 103
x 7
D)
7x +10y 113
7x +8y 103
y 7
Write the partial fraction decomposition of the rational expression.
92)
13x + 2
(x – 1)(x2+ x + 1)
92)
A)
5
x – 1 +
–5
x + 1 +3
x – 1
B)
5
x – 1 +3x – 5
x2+ x + 1
C)
5
x – 1 +
–5x +3
x2+ x + 1
D)
–5
x – 1 +5x +3
x2+ x + 1
Solve the problem.
93)
Mrs. White wants to crochet hats and afghans for a church fundraising bazaar. She needs 7 hours to
make a hat and 2 hours to make an afghan, and she has no more than 35 hours available. She has
material for no more than 10 items, and she wants to make at least two afghans. Let x = the number
of hats she makes and y = the number of afghans she makes. Write a system of inequalities that
describes these constraints.
93)
A)
2x +7y 35
x + y 10
x 2
B)
7x +2y 35
x + y 10
y 2
C)
7x +2y 35
x + y 10
x 2
D)
7x +2y 35
x + y 10
y 2
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.
94)
Is there a profit when 956 binoculars are produced?
94)
A)
Yes
B)
No
Solve the system of equations by the substitution method.
95)
x + y =6
y=2x
95)
A)
{(–2, 4)}
B)
{(–2, –4)}
C)
{(2, 4)}
D)
{(2, –4)}
Write the partial fraction decomposition of the rational expression.
96)
6x3+ 5x2+ 32x + 26
(x2+ 6)3
96)
A)
x + 1
x2+ 6
+6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
B)
x
x2+ 6
+6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
C)
6x – 5
(x2+ 6)2+
–4x + 4
(x2+ 6)3
D)
6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
Graph the solution set of the system of inequalities or indicate that the system has no solution.
44
97)
y < – x +8
y >7x – 3
97)
A)
B)
C)
D)
Graph the inequality.
45
98)
x – y < – 6
98)
A)
B)
C)
D)
46
Solve the system of equations.
99)
x+ y = – 3
3x – 3y+ 2z=11
x– z = – 5
99)
A)
{(4, –1, –2)}
B)
{(–1, –2, 4)}
C)
{(4, –2, –1)}
D)
{(–1, 4, –2)}
Determine if the given ordered triple is a solution of the system.
100)
(–3, –2, 5)
x + 3y + 4z = – 13
2y + 5z = – 19
z = – 3
100)
A)
solution
B)
not a solution
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.
101)
Fewer than how many binoculars must be produced and sold for the company to have a profit loss?
101)
A)
750 binoculars
B)
1500 binoculars
C)
2700 binoculars
D)
2250 binoculars
Determine if the given ordered triple is a solution of the system.
102)
(3, 1, –2)
x – y + z =6
x + y + z =2
x + y – z = – 4
102)
A)
solution
B)
not a solution
Solve the problem.
103)
A ceramics workshop makes wreaths, trees, and sleighs for sale at Christmas. A wreath takes 3
hours to prepare, 2 hours to paint, and 9 hours to fire. A tree takes 16 hours to prepare, 3 hours to
paint, and 4 hours to fire. A sleigh takes 4 hours to prepare, 17 hours to paint, and 7 hours to fire. If
the workshop has 135 hours for prep time, 87 hours for painting, and 126 hours for firing, How
many of each can be made?
103)
A)
6 wreaths, 3 trees, 9 sleighs
B)
10 wreaths, 7 trees, 4 sleighs
C)
9 wreaths, 6 trees, 3 sleighs
D)
3 wreaths, 9 trees, 6 sleighs
104)
A system for tracking ships indicated that a ship lies on a hyperbolic path described by
5x2–y2=145. The process is repeated and the ship is found to lie on a hyperbolic path described
by y2–2x2=2. If it is known that the ship is located in the first quadrant of the coordinate system,
determine its exact location.
104)
A)
(7, 10)
B)
(–7, –10)
C)
(–10, –7)
D)
(10, 7)
Solve the system by the substitution method.
105)
y = x + 4
y2=16x
105)
A)
{(4, 8), (4, –8), (–4, 0)}
B)
{(4, 8), (–4, 0)}
C)
{(4, 8), (4, –8)}
D)
{(4, 8)}
48
Decide if the system of equations in two variables is linear or nonlinear.
106)
y =x2–2
x2+y2=3
106)
A)
Linear
B)
Nonlinear
Write the partial fraction decomposition of the rational expression.
107)
x2+6x – 2
(x2+2)2
107)
A)
1
x2+ 2
+
–x – 4
(x2+ 2)2
B)
x + 1
x2+ 2
+6x – 4
(x2+ 2)2
C)
–1
x2+ 2
+6x – 4
(x2+ 2)2
D)
1
x2+ 2
+6x – 4
(x2+ 2)2
Explanation:
Solve the system by the substitution method.
108)
x2+y2=85
x + y = – 13
108)
A)
{(6, 7), (7, 6)}
B)
{(–6, 7), (–7, 6)}
C)
{(6, –7), (7, –6)}
D)
{(–6, –7), (–7, –6)}
Explanation:
Solve the problem.
109)
A dietitian needs to purchase food for patients. She can purchase an ounce of chicken for $0.35 and
an ounce of potatoes for $0.03. Let x = the number of ounces of chicken and y = the number of
ounces of potatoes purchased per patient. Write the objective function that describes the total cost
per patient per meal.
109)
A)
z =35x +3y
B)
z = 0.35x + 0.03y
C)
z =3y +35y
D)
z = 0.03x + 0.35y
Explanation:
Explanation:
Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear
equations. Solve the system and find the numbers.
110)
The difference between the squares of two numbers is 51. Twice the square of the second number
subtracted from the square of the first number is 2. Find the numbers.
110)
A)
10 and 7; –10 and –7
B)
10 and 7; –10 and 7; 10 and –7
C)
10 and 7; –10 and 7; 10 and –7; –10 and –7
D)
10 and 7
Solve the system by the addition method.
111)
x2–y2=3
x2+y2=9
111)
A)
{( 6, 3), (–6, 3)}
B)
{(0, 0)}
C)
{( 6, 3), (–6, 3), ( 6, –3), (–6, –3)}
D)
{( 6, 3), (–6, –3)}
Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear
equations. Solve the system and find the numbers.
112)
The sum of the squares of two numbers is 40. The difference of the two numbers is 8. Find the two
numbers.
112)
A)
–6 and –2; 2 and 6
B)
–6 and 2
C)
–2 and 6
D)
–2 and 6; –6 and 2
Solve the problem.
113)
You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and
eventually strikes the ground. A mathematical model can be used to describe the relationship for
the ball’s height above the ground, y, after x seconds. Consider the following data:
x, seconds after
ball is thrown
y, ball’s height, in feet,
above the ground
1114
3166
590
Find the quadratic function y = ax2+bx + c whose graph passes through the given points.
113)
A)
y = – 16x2+ 80x + 50
B)
y = – 12x2+90x + 36
C)
y = – 10x2+ 100x + 24
D)
y = – 16x2+90x +40
Solve the system of equations by the substitution method.
114)
–5x + 3y= – 22
x –5y =0
114)
A)
{(5, –1)}
B)
{(5, 1)}
C)
{(1, 5)}
D)
{(6, 1)}
Write the partial fraction decomposition of the rational expression.
115)
5x3+ 19x – 2
(x2+ 3)2
115)
A)
5x
x2+3
+
–4x – 2
(x2+3)2
B)
5x
x2+3
+4x – 2
(x2+3)2
C)
5x + 1
x2+3
+4x – 2
(x2+3)2
D)
5x
x2+3
+4x + 2
(x2+3)2
Solve the system by the addition method.
116)
x2– 3y2= 1
2x2+ 3y2=11
116)
A)
{(2, 1), (–2, 1), (2, –1), (–2, –1)}
B)
{(2, 1), (–2, –1)}
C)
{(1, 2), (–1, –2)}
D)
{(1, 2), (–1, 2), (1, –2), (–1, –2)}
Solve the system by the substitution method.
117)
xy =6
x + y =5
117)
A)
{(–3, 2), (–2, 3)}
B)
{(3, 2), (2, 3)}
C)
{(3, –2), (2, –3)}
D)
{(–3, –2), (–2, –3)}
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
118)
3x + 4
(x + 4)(x2+ x – 4)2
118)
A)
A
x + 4 +B
x2+ x – 4
+C
(x2+ x – 4)2
B)
A
x + 4 +Bx + C
x2+ x – 4
C)
A
x + 4 +Bx + C
x2+ x – 4
+Dx + E
(x2+ x – 4)2
D)
A
x + 4 +B
x2+ x – 4
+Cx + D
(x2+ x – 4)2
Let x represent one number and let y represent the other number. Use the given conditions to write a system of nonlinear
equations. Solve the system and find the numbers.
119)
The sum of the squares of two numbers is 37. The sum of the two numbers is 7. Find the two
numbers.
119)
A)
–6 and 1; –1 and 6
B)
1 and 6
C)
–6 and –1
D)
1 and 6; –6 and –1
Solve the system by the addition method.
120)
x2+y2– 6x – 6y – 7 = 0
x2–y2– 6x + 6y – 25 = 0
120)
A)
{(3, 8), (3, –2)}
B)
{(8, 3), (2, 3)}
C)
{(3, –8), (3, 2)}
D)
{(8, 3), (–2, 3)}
Graph the inequality.
121)
y 6
121)
A)
B)
C)
D)
Solve the system by the addition method.
122)
x2– 3y2– 1 = 0
4x2+ 3y2– 19 = 0
122)
A)
{(2, 1), (–2, –1)}
B)
{(–1, 2), (1, –2)}
C)
{(2, 1), (2, –1), (–2, 1), (–2, –1)}
D)
{(1, 2), (–1, 2), (1, –2), (–1, –2)}
Determine whether the given ordered pair is a solution of the system.
123)
(5, 6)
4x + y =26
2x + 4y =34
123)
A)
not a solution
B)
solution
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.
124)
How many binoculars must be produced and sold for the company to break even?
124)
A)
1500 binoculars
B)
750 binoculars
C)
2250 binoculars
D)
2700 binoculars
Solve the problem.
125)
The Little Town Arts Center charges $24 for adults, $13 for senior citizens, and $9 for children
under 12 for their live performances on Sunday afternoon. This past Sunday, the paid revenue was
$12,903 for 854 tickets sold. There were 45 more children than adults. How many children
attended?
125)
A)
318 children
B)
283 children
C)
243 children
D)
328 children
126)
A vendor sells hot dogs and bags of potato chips. A customer buys 4 hot dogs and 4 bags of potato
chips for $11.00. Another customer buys 3 hot dogs and 2 bags of potato chips for $7.25. Find the
cost of each item.
126)
A)
$1.75 for a hot dog; $1.25 for a bag of potato chips
B)
$1.00 for a hot dog; $1.75 for a bag of potato chips
C)
$1.75 for a hot dog; $1.00 for a bag of potato chips
D)
$2.00 for a hot dog; $1.25 for a bag of potato chips
127)
In a 1–mile race, the winner crosses the finish line 11 feet ahead of the second–place runner and 29
feet ahead of the third–place runner. Assuming that each runner maintains a constant speed
throughout the race, by how many feet does the second–place runner beat the third–place runner?
(5280 feet in 1 mile.)
127)
A)
7.01 ft
B)
18.04 ft
C)
–11.06 ft
D)
–18.1 ft
Write the partial fraction decomposition of the rational expression.
128)
9x + 9
(x–8)2
128)
A)
9
x –8+144
(x–8)2
B)
9
x –8+x +81
(x–8)2
C)
1
x –8+x +81
(x–8)2
D)
9
x –8+81
(x–8)2
129)
11x2+ 21x + 40
(x + 4)(x2+ 6)
129)
A)
6
x + 4 +5x – 1
x2+ 6
B)
6
x + 4 +5x + 1
x2+ 6
C)
6
x + 4 +5
x2+ 6
D)
6
x + 4 +5
x + 6 +1
(x + 6)2
130)
2x2– x –19
x(x + 1)(x – 1)
130)
A)
19
x+8
x + 1 +
–9
x – 1
B)
19
x+
–9
x + 1 +8
x – 1
C)
19
x+
–8
x + 1 +
–9
x – 1
D)
19
x+
–8
x + 1 +9
x – 1
Solve the problem.
131)
Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food
intake to 120 g protein, 105 g fat, and 159 g carbohydrate. According to the health conscious
hostess, the marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy
meatballs have 14 g protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein,
15 g fat, and 6 g carbohydrate. How many of each snack can he eat to obtain his goal?
131)
A)
9 mushrooms; 6 meatballs; 3 eggs
B)
2 mushrooms; 8 meatballs; 5 eggs
C)
5 mushrooms; 2 meatballs; 8 eggs
D)
8 mushrooms; 5 meatballs; 2 eggs
132)
In 1985, in the town of Appleby, 19.4% of Hispanics were overweight, increasing by an average of
0.41% per year. In 1985, in the town of Appleby, 0.15% of whites were overweight, increasing by
an average of 32.0% per year. If these trends continue, in which year will the percentage of
Hispanics who are overweight be the same as the percentage of whites who are overweight?
Round to the nearest year. What percentage of Hispanics (to the nearest whole percent) will be
overweight at that time?
132)
A)
1984; 19%
B)
1982; 18%
C)
1986; 20%
D)
1988; 20%
Solve the system by the addition method.
133)
x + y = – 6
x – y =12
133)
A)
{(3, 9)}
B)
{(6, –9)}
C)
{(6, 3)}
D)
{(3, –9)}
Graph the solution set of the system of inequalities or indicate that the system has no solution.
134)
y 2x – 4
x + 2y
7
y –2
x 1
134)
57
A)
B)
C)
D)
Solve the problem.
135)
A bank teller has 46 $10 and $20 bills in her cash drawer. The value of the bills is $820. How many $
10 bills are there?
135)
A)
12 $10 bills
B)
36 $10 bills
C)
34 $10 bills
D)
10 $10 bills
136)
You invested $16,000 and started a business selling vases. Supplies cost $10 per vase and you are
selling each vase for $26. Let x represent the number of vases produced and sold and write the cost
function, C, and revenue function, R.
136)
A)
C(x) =10x +26
R(x) =16,000x
B)
C(x) =10x +16,000
R(x) =26
C)
C(x) =10x +16,000x
R(x) =26x
D)
C(x) =10x +16,000
R(x) =26x
137)
A bakery plans to market a mixed assortment of its two most popular cookies, Chocolate Chip and
Toffee Chunk. The marketing analyst proposes that the new assortment be constrained by the
inequality 3C + 4T
31, where C is the number of Chocolate Chip cookies and T is the number of
Toffee Chunk cookies. The sales analyst suggests that the assortment should be constrained by the
inequality 5C + 2T
33. The number of each type of cookie cannot be negative, so C 0 and T 0.
Graph the region satisfying all the requirements for the assortment using C as the horizontal axis
and T as the vertical axis. Does the combination of 6 Chocolate Chip cookies and 3 Toffee Chunk
cookies satisfy all of the requirements?
137)
A)
No
B)
Yes
59
C)
Yes
D)
No
138)
A person at the top of a 600 foot tall building drops a yellow ball. The height of the yellow ball is
given by the equation h = – 16t2+ 600 where h is measured in feet and t is the number of seconds
since the yellow ball was dropped. A second person, in the same building but on a lower floor that
is 516 feet from the ground, drops a white ball 1.5 seconds after the yellow ball was dropped. The
height of the white ball is given by the equation h = – 16(t –1.5)2+516 where h is measured in feet
and t is the number of seconds since the yellow ball was dropped. Find the time that the balls are
the same distance above the ground and find this distance.
138)
A)
2.5 seconds; 500 feet
B)
2 seconds; 536 feet
C)
1.5 seconds; 564 feet
D)
3 seconds; 456 feet
Graph the solution set of the system of inequalities or indicate that the system has no solution.
139)
y 2x – 4
x + 2y
7
x –2
y 1
139)
60