Choose the graph that matches the equation.
25)
y = – 2x + 3
25)
A)
B)
C)
D)
Find the equations of the system shown on the graphing calculator screen.
26)
10
–10 10
–10
26)
A)
x – 3y = 9
x + 2y = – 1
B)
x + y = 1
x – 3y = – 9
C)
5x + y = 7
x + 2y = – 1
D)
3x – y = 9
5x + y = 7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
27)
Can an equation of a vertical line be written in slope–intercept form? Explain.
27)
28)
Describe the three possible outcomes when graphing a system of equations, and relate each
to the type of solution(s) each system has.
28)
29)
Explain why the solution of a system of equations is the point of intersection of the graphs
of the equations.
29)
30)
2x + 3y = – 8
4x + 2y = – 1
What number would you multiply the top equation by to eliminate the x terms?
30)
22
31)
In solving a system of three equations in three variables, it is impossible to eliminate two
variables. Thus, the the final step yields the following equation:
x=y+ 1.
How many solutions does this system have? Explain geometrically.
31)
32)
The total number of reported cases of AIDS in the United States has risen from 372 in 1981
to 100,000 in 1989 and 200,000 in 1992. Does a linear equation fit this data? Explain.
32)
33)
Under what circumstances can a system of equations be solved more easily by graphing
than be substitution?
33)
34)
Without solving the following system of equations, tell what the solution is. Explain your
answer.
3x – 6y – 3z = 2
–18x + 36y + 18z = – 12
2x + 6y + 4z =2
34)
35)
Why could (–1, –10) not be a solution of the system that is graphed?
35)
36)
Describe a situation in which it would be more convenient to use the point–slope form of
the equation of a line than the slope–intercept form.
36)
37)
Solve the following system of equations by first solving Equation (1) for x and then
substituting for x in Equation (2). Then solve the system again by first solving Equation (1)
for 2y and substituting for 2y in Equation (2). Which procedure do you prefer? Explain
why you answered as you did.
x + 2y = 6 (1)
3x + 2y = 4 (2)
37)
38)
Without solving the following system of equations, tell what the solution is. Explain your
answer.
6x + 2y + 5z = – 5
18x + 6y + 15z = – 1
36x + 12y + 30z = – 5
38)
39)
While using the elimination method, if the following occurs, what can you say about the
graph of the system of equations?
6x – 8y = 5
– 6x + 8y = 7
0 = 12
39)
40)
A student tried to clear the fractions in
x
6+y
10 =10
x
2+y
5= – 9
by multiplying both fractions by zero. Is this an acceptable approach?
40)
41)
While using the elimination method, if the following occurs, what can you say about the
graph of the system of equations?
3x – 7y = 8
– 3x + 7y = – 8
0 = 0
41)
42)
Explain how the multiplication and addition principles are used in solving systems of
equations using the elimination method.
42)
43)
Write a system of equations that would be more easily solved using the elimination
method than the substitution method. Explain why elimination would be easier than
substitution.
43)
44)
In solving a system of three equations in three variables, the final step yields the following
equation:
0 = 0.
How many solutions does this system have? Explain geometrically.
44)
45)
–6x + 8y = A
3x – 10y = B
For what values of A or B can this system of equations have (0,0) as a solution?
45)
46)
Without solving the following system of equations, tell what the solution is. Explain your
answer.
5x – 4y + 6z = – 3
–30x + 24y – 36z = 18
20x – 16y + 24z = – 12
46)
47)
A student argued that it is impossible to use the substitution method to determine if a
system of equations has no solution or an infinite number of solutions. Is the student
correct?
47)
48)
Write a system of equations that would be more easily solved using the substitution
method than the elimination method. Explain why substitution would be easier than
elimination.
48)
49)
Describe two advantages of the substitution method over the graphing method for solving
systems of equations.
49)
50)
Can the point–slope form of the equation of a line be used to write an equation of a vertical
line? Explain.
50)
51)
A student solved the system of equations
x + 2y = 4
3x + 6y =12
for x in the first equation, and substituted into the second equation. The y’s also
disappeared in the process. The student claimed that the system of equations has no
solution. Is this correct?
51)
52)
In solving a system of three equations in three variables, the final step yields the following
equation:
0 =5.
How many solutions does this system have? Explain geometrically.
52)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the x– and y–intercepts. Then graph the equation.
26
53)
2y =3
5x
53)
A)
(0, 0); (0, 0)
B)
(0, 0); (0, 0)
C)
(0, 0); (0, 0)
D)
(0, 0); (0, 0)
Find the midpoint of the segment with the given endpoints.
54)
(1, –6) and (–8, –9)
54)
A)
–7
2, –15
2
B)
9, 3
C)
9
2, 3
2
D)
–7, –15
Solve the problem.
55)
It costs $42 per hour plus a flat fee of $23 for a plumber to make a house call. Find the number of
hours a plumber worked if the total cost was $233.
55)
A)
17
B)
5
C)
16
D)
4
B
56)
A cruise boat travels 60 miles downstream in 4 hours and returns to its starting point upstream in 6
hours. Find the speed of the stream.
56)
A)
27.5 mph
B)
12.5 mph
C)
2.5 mph
D)
15 mph
C
Find the slope of the line and sketch the graph.
57)
4x + 5y =28
57)
28
A
A)
Slope: 4
5
B)
Slope: 5
4
C)
Slope: –5
4
D)
Slope: –4
5
Solve the problem.
58)
Best Rentals charges a daily fee plus a mileage fee for renting its cars. Barney was charged $78.00
for 3 days and 300 miles, while Mary was charged $138.00 for 5 days and 600 miles. What does Best
Rentals charge per day and per mile?
58)
A)
$19 per day; 9¢ per mile
B)
$18 per day; 8¢ per mile
C)
$8 per day; 18¢ per mile
D)
$17 per day; 9 ¢ per mile
B
D
59)
A cabin cruiser travels 20 miles in the same time that a power boat travels 40 miles. The cruiser
travels 5 mph slower than the power boat. Find the speed of each boat.
59)
A)
Power boat: 15 mph; cabin cruiser: 10 mph
B)
Power boat: 10 mph; cabin cruiser: 5 mph
C)
Power boat: 20 mph; cabin cruiser: 15 mph
D)
Cannot be determined without more information
Find the slope of the line through the given pair of points, if possible. Based on the slope, indicate whether the line
through the points rises from left to right, falls from left to right, is horizontal, or is vertical.
60)
(9, –3) and (–1, 6)
60)
A)
10
9; rises
B)
–9
10 ; falls
C)
9
10 ; rises
D)
–10
9; falls
B
Solve the system of equations. If the system is inconsistent or has dependent equations, say so.
61)
x + 3y + 2z = 11
4y + 9z = – 12
x + 7y + 11z = – 11
61)
A)
{(0, 3, 1)}
B)
{(7, –1, –1)}
C)
; inconsistent system
D)
{(x, y, z)|x + 3y + 2z = 11}; dependent equations
C
Write the equation in slope–intercept form.
62)
3x – 10y = – 6
62)
A)
y =10
3x – 2
B)
y =3
10 x +3
5
C)
y = – 3
10 x +3
5
D)
y = 3x + 11
B
Find the x– and y–intercepts. Then graph the equation.
30
B
63)
–2x – 6y =6
63)
A)
(–3, 0); (0, –1)
B)
(–1, 0); (0, –3)
C)
(3, 0); (0, –1)
D)
(–1, 0); (0, 3)
Provide an appropriate response.
64)
When solving a system of equations, you get x + 2 = x + 4. How many solutions are there?
64)
A)
One solution
B)
No solutions
C)
Infinitely many
Plot the point on the rectangular coordinate system provided. Write the corresponding letter as your answer.
65)
(–5, –4)
65)
A)
L
B)
A
C)
G
D)
I
D
Provide an appropriate response.
66)
Decide whether the elimination method or the substitution method involves the least amount of
work. Do not solve.
x + 2y = 0
x – 2y =16
66)
A)
Substitution
B)
Elimination
B
Does the system have one solution, no solution, or an infinite number of solutions?
67)
3x – y =7
x + 6y =15
67)
A)
Infinite number
B)
One
C)
No Solution
B
B
Solve the system by substitution or elimination. If a system is inconsistent or has dependent equations, say so.
68)
3x + 5y + z =26
4x – 5y – z = – 5
3x + y + 5z =22
68)
A)
{(3, 2, 3)}
B)
; inconsistent system
C)
{(2, 3, 3)}
D)
{(3, 3, 2)}
Decide whether or not the ordered pair is a solution of the system.
69)
(–2, 2)
x + y =0
x – y = – 4
69)
A)
Yes
B)
No
Find the equation in slope–intercept form of the line satisfying the conditions.
70)
m = – 8, passes through (–8, 4)
70)
A)
y = – 8x +67
B)
8x + y =60
C)
y = – 8x –60
D)
y =8x –58
Plot the point on the rectangular coordinate system provided. Write the corresponding letter as your answer.
71)
(4, –5)
71)
A)
L
B)
A
C)
F
D)
K
Does the system have one solution, no solution, or an infinite number of solutions?
72)
4x – 16y =12
y =1
4x –3
4
72)
A)
No Solution
B)
One
C)
Infinite number
C
Decide whether or not the ordered pair is a solution of the system.
73)
(–2, 1)
2x = – 3– y
3x = – 4– 2y
73)
A)
No
B)
Yes
B
Find the midpoint of the segment with the given endpoints.
74)
(9, 2) and (2, 0)
74)
A)
7, 2
B)
11, 2
C)
11
2, 1
D)
7
2, 1
C
A
Solve the system by the elimination method.
75)
3x + 2y = – 5
–9x – 6y = – 15
75)
A)
{(15, 10)}
B)
{(–15, –10)}
C)
{(x, y)|3x + 2y = – 5}
D)
Solve the system by substitution or elimination. If a system is inconsistent or has dependent equations, say so.
76)
x –3y =1
x =5+3y
76)
A)
{(x, y) x –3y =1}; dependent equations
B)
{(1, 0)}
C)
{(1, 3)}
D)
; inconsistent system
Decide whether the pair of lines is parallel, perpendicular, or neither.
77)
The line through (–20, 5) and (–4, 7) and the line through (–5, 5) and (7, 4)
77)
A)
Parallel
B)
Perpendicular
C)
Neither
Graph the line described.
78)
m = 0; through (5, 3)
78)
35
A)
B)
C)
D)
Solve the system of equations. If the system is inconsistent or has dependent equations, say so.
79)
x + 4y – z = 3
–4x – 16y + 4z = – 12
3x + 12y – 3z = 9
79)
A)
{(0, 2, 5)}
B)
{(x, y, z)|x + 4y – z = 3}; dependent equations
C)
{(1, 1, 2)}
D)
; inconsistent system
36
Plot the point on the rectangular coordinate system provided. Write the corresponding letter as your answer.
80)
(4, 3)
80)
A)
K
B)
C
C)
E
D)
D
Provide an appropriate response.
81)
Solving for which variable in which equation involves the least amount of work?
2x =6
x – 8y =19
81)
A)
y in equation 2
B)
y in equation 1
C)
x in equation 1
D)
x in equation 2
Find an equation of the line that satisfies the conditions. Write the equation in standard form.
82)
Through (–1, 9); horizontal
82)
A)
y =1
B)
y =9
C)
x = – 1
D)
x = – 9
Find the equation in slope–intercept form of the line satisfying the conditions.
83)
Slope –1; y–intercept (0, 8)
83)
A)
y =8x
B)
y = – x +8
C)
–x + y =8
D)
y =8x – 1
Name the quadrant, if any, in which the point is located.
84)
(7, –19)
84)
A)
I
B)
IV
C)
II
D)
III
Clear fractions then solve by substitution.
85)
1
2x +1
2y =7
x – y =4
85)
A)
{(8, 6)}
B)
{(–9, 6)}
C)
{(9, 5)}
D)
C
Find an equation of the line, and write it in (a) slope–intercept form if possible and (b) standard form.
86)
Through (–3, –8); horizontal
86)
A)
(a) not possible
(b) x =8
B)
(a) y = – 8
(b) y = – 8
C)
(a) not possible
(b) x = – 3
D)
(a) y =3
(b) y =3
B
Decide whether or not the ordered pair is a solution of the system.
87)
(3, 3)
2x = – 3– y
3x = – 3– 2y
87)
A)
No
B)
Yes
A
Find the midpoint of the segment with the given endpoints.
88)
(–5, 0) and (3, 1)
88)
A)
– 4, –1
2
B)
–2, 1
C)
– 1, 1
2
D)
–8, –1
C
B
Solve the problem.
89)
An express train and a local train both leave Gray’s Lake at 3 P.M. and head for Chicago 50 miles
away. The express travels twice as fast as the local and arrives 1 hour ahead of the local. Find the
speed of each train.
89)
A)
Express: 25 mph; local: 12.5 mph
B)
Express: 50 mph; local: 25 mph
C)
Express: 60 mph; local: 30 mph
D)
Cannot be determined without more information
Solve the system by the elimination method.
90)
x + 2y = – 4
8x + 1y =58
90)
A)
{(–8, –5)}
B)
{(7, –5)}
C)
{(8, –6)}
D)
Solve the system. Express the solution in the form (x, y, z, w).
91)
4x + y –4z + w =1
–x +4y + z –4w = – 14
4x – y +4z +4w =29
x + y –4z – w = – 10
91)
A)
{(5, –10, 0, 2)}
B)
{(–5, 4, 4, 1)}
C)
{(–5, –3, 6, 2)}
D)
{(5, –5, 3, –2)}
Find an equation of the line that satisfies the conditions. Write the equation in standard form.
92)
Through (9, 2); undefined slope
92)
A)
x =9
B)
x =2
C)
y =2
D)
y =9
Decide if the graph of the system is a pair of parallel lines, intersecting lines, or one line.
93)
x + 6y =29
2x + 12y =63
93)
A)
Parallel
B)
Intersecting
C)
One Line
Solve the problem.
94)
During the 1998–1999 Junior Hockey League season, the Sharks played 64 games. Together, their
wins and losses totaled 57. They tied 15 fewer games than they lost. How many games did they tie
that season?
94)
A)
22 games
B)
7 games
C)
5 games
D)
35 games
95)
During the 1998–1999 Little League season, the Tigers played 49 games. They won 21 more games
than they lost. How many games did they lose that season?
95)
A)
12 games
B)
14 games
C)
35 games
D)
17 games
Find the equation in slope–intercept form of the line satisfying the conditions.
96)
m =7
5; through (0, –5)
96)
A)
y =7
5x – 5
B)
y = – 7
5x – 5
C)
y =7
5x + 5
D)
y = – 7
5x + 5
40