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.
92)
x
3+y
6= 1
x
4–y
8= 0
92)
A)
3
2, 3
B)
3, 3
2
C)
(x, y) x
3+y
6= 1
D)
Solve the problem.
93)
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.
93)
A)
2 seconds; 536 feet
B)
1.5 seconds; 564 feet
C)
3 seconds; 456 feet
D)
2.5 seconds; 500 feet
Solve the system of equations by the substitution method.
94)
x – 7y = – 40
4x – 8y = – 20
94)
A)
{(–9, 8)}
B)
{(9, 7)}
C)
{(8, 8)}
D)
Graph the solution set of the system of inequalities or indicate that the system has no solution.
41
95)
y 3x
y 8
95)
A)
B)
C)
D)
Write the partial fraction decomposition of the rational expression.
96)
5x3+ 19x – 2
(x2+ 3)2
96)
A)
5x
x2+3
+4x – 2
(x2+3)2
B)
5x
x2+3
+
–4x – 2
(x2+3)2
C)
5x
x2+3
+4x + 2
(x2+3)2
D)
5x + 1
x2+3
+4x – 2
(x2+3)2
97)
11x2– x –20
x3– x
97)
A)
20
x+
–4
x + 1 +5
x – 1
B)
20
x+
–5
x + 1 +4
x – 1
C)
20
x+
–4
x + 1 +
–5
x – 1
D)
20
x+4
x + 1 +
–5
x – 1
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.
98)
How many binoculars must be produced and sold for the company to break even?
98)
A)
2700 binoculars
B)
2250 binoculars
C)
750 binoculars
D)
1500 binoculars
Solve the problem.
99)
The sum of two numbers is 15. If one number is subtracted from the other, their difference is 3.
Find the numbers.
99)
A)
13, 2
B)
–9, 6
C)
9, –6
D)
9, 6
Decide if the system of equations in two variables is linear or nonlinear.
100)
x3+3x= y
x +7y =3
100)
A)
Linear
B)
Nonlinear
Explanation:
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)
At the break–even point both cost and revenue are what?
101)
A)
$2700
B)
$2250
C)
$1500
D)
$750
Explanation:
Graph the solution set of the system of inequalities or indicate that the system has no solution.
44
Explanation:
102)
y < x + 1
6x +6y >18
102)
A)
B)
C)
D)
Graph the inequality.
45
103)
(x – 4)2+(y – 4)2>9
103)
A)
B)
C)
D)
46
Write the partial fraction decomposition of the rational expression.
104)
4x2– 12x – 48
x(x – 4)(x – 6)
104)
A)
–2
x+5
x – 4 +1
x – 6
B)
–2
x+4
x – 4 +2
x – 6
C)
–2
x+3
x – 4 +3
x – 6
D)
x +4
x2– 4
+2
x – 6
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.
105)
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.
105)
A)
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; –10 and –7
Solve the problem.
106)
Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in.
The length of the garden is 5 feet more than 4 times the width. He needs 70 feet of fencing to do the
job. Find the length and width of the garden.
106)
A)
length: 25 feet; width: 5 feet
B)
length: 29 feet; width: 6 feet
C)
length: 33 feet; width: 7 feet
D)
length: 57 feet; width: 13 feet
Write the partial fraction decomposition of the rational expression.
107)
9x + 9
(x–8)2
107)
A)
9
x –8+x +81
(x–8)2
B)
9
x –8+144
(x–8)2
C)
1
x –8+x +81
(x–8)2
D)
9
x –8+81
(x–8)2
Graph the solution set of the system of inequalities or indicate that the system has no solution.
108)
y 1
2 x
6
x – 2y –2
x + y 6
108)
A)
B)
48
C)
D)
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 =3y +35y
C)
z = 0.35x + 0.03y
D)
z = 0.03x + 0.35y
Solve the system of equations by the substitution method.
110)
y =2x – 6
y =7x – 7
110)
A)
{(1, –4)}
B)
–28
5, 1
5
C)
1
5, –28
5
D)
An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.
Use these values to determine the maximum value of the objective function and the values of x and y for which the
maximum occurs.
111)
Objective Function z =17x – 20y
Constraints 0 x
5
0 y 8
4x + 5y 30
4x + 3y 20
111)
A)
Maximum: –78.75; at (1.25, 5)
B)
Maximum: 0; at (0, 0)
C)
Maximum: 85; at (5, 0)
D)
Maximum: –120; at (0, 6)
Graph the inequality.
112)
y –3x + 3
112)
A)
B)
50
C)
D)
Determine whether the given ordered pair is a solution of the system.
113)
(–2, 1)
2x =5– y
3x =8– 2y
113)
A)
solution
B)
not a solution
Solve by the method of your choice.
114)
x2+y2=68
(x –5)2+y2=73
114)
A)
{(2, 8), (2, –8)}
B)
{(8, 2), (8, –2), (–8, 2), (–8, –2)}
C)
{(8, 2), (8, –2)}
D)
{(2, 8), (2, –8), (–2, 8), (–2, –8)}
51
Solve the problem.
115)
A steel company produces two types of machine dies, part A and part B and is bound by the
following constraints:
· Part A requires 1 hour of casting time and 10 hours of firing time.
· Part B requires 4 hours of casting time and 3 hours of firing time.
· The maximum number of hours per week available for casting and firing are 100 and 70,
respectively.
· The cost to the company is $0.75 per part A and $3.00 per part B. Total weekly costs cannot exceed
$45.00.
Let x = the number of part A produced in a week and y = the number of part B produced in a week.
Write a system of three inequalities that describes these constraints.
115)
A)
x+4y 100
10x +3y 70
3x +0.75y 45
B)
x+10y 100
4x +3y 70
0.75x +3y 45
C)
x+10y 100
4x +3y 70
0.75x +3y 45
D)
x+4y 100
10x +3y 70
0.75x +3y 45
D)
Write the partial fraction decomposition of the rational expression.
116)
x +2
x3– 2x2+ x
116)
A)
2
x+
–2
x – 1 +5
(x – 1)2
B)
2
x+
–2
x – 1 +3
(x – 1)2
C)
2
x+3
x – 1 +
–2
(x – 1)2
D)
–2
x+2
x – 1 +3
(x – 1)2
D)
Graph the solution set of the system of inequalities or indicate that the system has no solution.
117)
2x + y >4
2x + y <1
117)
52
A)
B)
C)
no solution
D)
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.
118)
y = – 4x + 9
20x + 5y =45
118)
A)
B)
{(0, 0)}
C)
{(x, y) | 4x + y =9}
D)
{(0, 9)}
53
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
119)
3x + 5
(x – 3)(x + 1)
119)
A)
A
x – 3 +B
x + 1 +Cx + D
(x – 3)(x + 1)
B)
A
x – 3 +B
x + 1 +C
(x – 3)2(x + 1)2
C)
A
x – 3 +B
x + 1
D)
A
x – 3 +B
x + 1 +C
(x – 3)(x + 1)
Write the partial fraction decomposition of the rational expression.
120)
6x3+ 5x2+ 32x + 26
(x2+ 6)3
120)
A)
x + 1
x2+ 6
+6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
B)
6x – 5
(x2+ 6)2+
–4x + 4
(x2+ 6)3
C)
x
x2+ 6
+6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
D)
6x + 5
(x2+ 6)2+
–4x – 4
(x2+ 6)3
An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.
Use these values to determine the maximum value of the objective function and the values of x and y for which the
maximum occurs.
121)
Objective Function z =5x – 2y
Constraints x 0
0 y 4
x – y 7
x + 2y 10
121)
A)
Maximum: –8; at (0, 4)
B)
Maximum: 35; at (7, 0)
C)
Maximum: 2; at (2, 4)
D)
Maximum: 38; at (8, 1)
54
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.
122)
x + 3y = – 2
2x + 6y = – 4
122)
A)
{(0, 0)}
B)
{(–2, 0)}
C)
{(x, y) | x + 3y = – 2}
D)
Solve the system of equations by the substitution method.
123)
5x + 9y =41
3x – 4y = – 13
123)
A)
{(0, 5)}
B)
{(1, 5)}
C)
{(1, 4)}
D)
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
124)
3x + 4
(x + 4)(x2+ x – 4)2
124)
A)
A
x + 4 +Bx + C
x2+ x – 4
+Dx + E
(x2+ x – 4)2
B)
A
x + 4 +B
x2+ x – 4
+C
(x2+ x – 4)2
C)
A
x + 4 +Bx + C
x2+ x – 4
D)
A
x + 4 +B
x2+ x – 4
+Cx + D
(x2+ x – 4)2
Graph the inequality.
125)
–2x – 5y –10
125)
55
A)
B)
C)
D)
Determine whether the given ordered pair is a solution of the system.
126)
(6, –5)
4x – y =19
3x + 4y = – 2
126)
A)
solution
B)
not a solution
Graph the solution set of the system of inequalities or indicate that the system has no solution.
56
127)
3x + y <3
3x + y >1
127)
A)
B)
C)
D)
57
Determine if the given ordered triple is a solution of the system.
128)
(–4, –1, 5)
2x + 2y + z = – 5
5x – 4y – z = – 21
5x + y + 2z = – 11
128)
A)
solution
B)
not a solution
Find the maximum or minimum value of the given objective function of a linear programming problem. The figure
illustrates the graph of feasible points.
129)
Objective Function: z = x + 9y + 7
Find minimum.
129)
A)
minimum: 54
B)
minimum: 29
C)
minimum: 38
D)
no minimum
Solve the system of equations.
130)
x+ y = – 3
3x – 3y+ 2z=11
x– z = – 5
130)
A)
{(4, –1, –2)}
B)
{(4, –2, –1)}
C)
{(–1, 4, –2)}
D)
{(–1, –2, 4)}
An objective function and a system of linear inequalities representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of the objective function at each corner of the graphed region.
Use these values to determine the maximum value of the objective function and the values of x and y for which the
maximum occurs.
131)
Objective Function z =15x + 14y
Constraints 0
x
10
0
y
5
3x + 2y 6
131)
A)
Maximum: 70; at (0, 5)
B)
Maximum: 220; at (10, 5)
C)
Maximum: 150; at (10, 0)
D)
Maximum: 42; at (0, 3)
Solve the problem.
132)
The area of a rectangular piece of cardboard shown is 510 square inches. The cardboard is used to
make an open box by cutting a 5–inch square from each corner and turning up the sides. If the box
is to have a volume of 700 cubic inches, find the dimensions of the cardboard that must be used.
132)
A)
17 inches by 30 inches
B)
22 inches by 35 inches
C)
7 inches by 15 inches
D)
12 inches by 25 inches
133)
In Miguel’s home town, the percentage of women who smoke is increasing while the percentage of
men who smoke is decreasing. The function y =0.29x +16.2 models the percentage, y, of women in
this city who smoke x years after 1990. The function 0.21x + y =29.0 models the percentage, y, of
men in this city who smoke x years after 1990. Use these models to determine when the percentage
of women who smoke will be the same as the percentage of men who smoke. Round to the nearest
year. What percentage of women and what percentage of men (to the nearest whole percent) will
smoke at that time?
133)
A)
2015; 24%
B)
2016; 24%
C)
2017; 23%
D)
2018; 23%
Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the
constants.
134)
5x + 4
(x + 7)(x + 6)2
134)
A)
A
x + 7 +B
x + 6 +Cx + D
(x + 6)2
B)
A
x + 7 +B
x + 6 +C
(x + 6)2
C)
A
x + 7 +B
x + 6 +C
x + 6 +Dx + E
(x + 6)2
D)
A
x + 7 +B
x + 6 +C
x + 6 +D
(x + 6)2
Solve the system by the addition method.
135)
2x + 5y =2
–7x – 3y =22
135)
A)
{(–4, 2)}
B)
{(4, 2)}
C)
{(–4, –2)}
D)
{(4, –2)}
Graph the inequality.
136)
x – y < – 6
136)
A)
B)
60