Linear Equations
Name_________________________________
1
8x + 2y = 6
3x + 3y = 5
A.
B.
C.
D.
2. The following system of equations is graphed below.
-2x - 4y = 0
-2x + 2y = 0
Find the solution to the system.
A. x = 0, y = 0
B. x = 0, y = 1
C. x = 1, y = 0
D. x = 5, y = -1
3. Mike's Movers charges a flat rate of $100 plus $20 per hour. If one of their customers was charged a total of
$200, for how many hours were they charged?
A. 5
B. 7
C. 4
D. 8
4.
4x + y = 8
3x – 2y = 6
4
A.
B.
C.
D.
5.
8x + 4y = 52
6x – 9y = -69
A.
B.
C.
D.
6. Dilbert's TV and Appliance discounts all televisions 9% to customers paying in cash. Rebecca paid $1,281.12
in cash for a 52-inch widescreen TV. What was the original price of the TV?
A. $1,153.00
B. $1,165.82
C. $1,396.42
D. $1,407.82
10x + 3y = 12
3x + y = 3
7.
A.
B.
C.
D.
8. A rental car company charges a base fee of $53.10 plus $0.40 per mile driven. If x represents the number of
miles driven, which of the following equations could be used to find y, the total cost of the bill?
A. y = $0.70x + $53.10
B. y = $53.10x + $0.40
C. y = $0.40x + $53.10
D. y = $53.50x
9. The following system of equations is graphed below.
3x - 5y = 19
4x + 5y = 2
Find the solution to the system.
A. x = 3, y = 2
B. x = -3, y = 2
C. x = -3, y = -2
D. x = 3, y = -2
10.
A.
B.
C.
D.
11.
A.
B.
C.
D.
12. A company manufactures and sells video games. A survey of video game stores indicated that at a price of
$86 each, the demand would be 300 games, and at a price of $36 each, the demand would be 1,800 games. If a
linear relationship between price and demand exists, which of the following equations models the price-demand
relationship?
(Let x represent the price per video game and y represent the demand.)
A.
B.
C.
D.
13. Betsy's high school is putting on a production of a play as a fundraiser for the school's music programs. A
local bank has agreed to allow the school to use a line of credit from which they can withdraw money to pay for
the play. Then, any deposits they make at the bank will be applied to the negative balance of the credit account.
The play cost $2,200.00 to produce, and they intend to sell tickets for $6 apiece. After the play, Betsy will take
the ticket proceeds and deposit them with the bank. If 987 people attend the play's opening night, what will the
balance of the bank account be?
A. $8,122
B. $3,722
C. $5,922
D. $-2,035
18x – 3y = 72
36x + 3y = 360
14.
A.
B.
C.
D.
15. A grocery store is offering a promotion where a customer receives a $0.15 discount per item on selected
items, and Paulo has a coupon where he receives $0.10 off regularly priced items.
Before applying the discounts, Paulo spent $56.46 on sale items and $44.49 on regularly priced items. After
applying the discounts, Paulo spent $52.71 on sale items and $42.89 on regularly priced items.
If Paulo bought 9 more sale items than regular items, how many items did he buy in all?
A. 34
B. 41
C. 25
D. 16
16. Blade-Z manufactures roller blades. The production facility has fixed costs of $400 a day and total
production costs of $3,400 per day at an output of 100 pair of skates per day. Which of the following equations
represents the daily production cost for Blade-Z based on the number of skates manufactured?
(Let C(x) represent the daily production cost and x represent the number of pairs of skates manufactured.)
A. C(x) = 34x
B. C(x) = 30x - 400
C. C(x) = 34x + 400
D. C(x) = 30x + 400
17.
5x + 2y = -14
3x + 8y = -22
A.
B.
C.
D.
18. The Bridgeport water department has a monthly service charge of $10.30 and a volume charge of $2.24 for
every 100 cubic feet of water. Which of the following equations can be used to determine the Morgan family's
monthly water bill?
(Let x represent 100 cubic feet of water and y represent the monthly cost)
A. y = 12.54x
B. y = 0.0224x + 10.30
C. y = 2.24x + 10.30
D. y = 2.24x - 10.30
19. The senior class officers have figured out that the prom will cost $3,000 for location rental and decorations
and another $40 per person for food.
Write an equation in general form that expresses the cost of the prom in dollars, y, as a function of the number
of people who attend, x.
A. 40x - y = 3,000
B. 40x - y = -3,000
C. x + 40y = -3,000
D. x + 40y = 3,000
20. Cassidy is going to the county fair with her friends but can only stay at the fair for 3 hours. The fair offers
two different pricing structures. Cassidy can purchase an all-day pass with unlimited rides for $45.00 or she can
pay an admission fee of $9.00 and pay $4.00 for each ride. If Cassidy rides 12 rides, how much does she save
by buying the all-day pass?
A. $21.00
B. They both cost the same.
C. $12.00
D. $51.00
Answers
1. D
2. A
3. A
4. D
5. B
6. D
7. C
8. C
9. D
10. A
11. A
12. D
13. B
14. C
15. B
16. D
17. A
18. C
19. B
20. C
Explanations
1.
2. The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (0, 0).
So, x = 0 and y = 0 is the solution.
3. This situation can be represented by a linear equation where the total charged is the dependent variable and
the number of hours spent moving is the independent variable.
The flat rate is the y-intercept, and the hourly charge is the rate of change, or slope.
Use the given information to develop an equation and solve for the number of hours spent moving.
y = mx + b
$200 = $20x + $100
$100 = $20x
5=x
The customer was charged for 5 hours of moving.
4.
5.
6. Let x represent the original price of the TV and y represent the discounted price. The discounted price is equal
to the original price minus the discount.
Use the discounted price to find the original price. Rearrange the equation to solve for x.
7.
8. The total bill is equal to the cost per mile driven, $0.40, times the number of miles driven, x, plus the base
price, $53.10. Therefore, the linear equation which represents the total rental cost is shown below.
y = $0.40x + $53.10
9. The point of intersection of the two graphs gives the solution to the system of equations.
The two lines intersect at (3, -2).
So, x = 3 and y = -2 is the solution.
10.
11.
12. The problem gives two points and indicates that the price-demand function is linear. Use the two points to
find the slope and then the equation of the line passing through the points.
So, given the points (86, 300) and (36, 1,800), find the slope, m.
Next, use the point-slope form of the equation of a line to find the price-demand function.
13. The balance in the bank account can be modeled by the linear equation
B = $6p - $2,200,
where B represents the balance of the account and p represents the number of people in attendance at the play.
Evaluate the equation with the given attendance figure to determine the balance of the bank account.
B = $6(987) - $2,200
B = $5,922 - $2,200
B = $3,722
14.
15. First, set up equations to represent the amount of money Paulo spent on sale items and the number of items
he bought.
Let s represent the number of sale items he bought, and let r represent the number of regular items he bought.
Money spent:
$52.71 + $42.89 = $56.46 - ($0.15/item)(s items) + $44.49 - ($0.10/item)(r items)
$95.60 = $100.95 - $0.15s - $0.10r
$0.15s + $0.10r = $5.35
Number of items:
s=r+9
Next, substitute r + 9 in for s in the first equation, and solve for r.
$0.15s + $0.10r
$0.15(r + 9) + $0.10r
$0.15r + $1.35 + $0.10r
$0.25r + $1.35
$0.25r
r
=
=
=
=
=
=
$5.35
$5.35
$5.35
$5.35
$4.00
16
Then, substitute 16 in for r in the second equation, and solve for s.
s = r+9
= 16 + 9
= 25
Since Paulo bought 25 sale items and 16 regularly priced items, he bought
25 + 16 = 41 items in all.
16. First, find the cost to produce one pair of skates. When 100 pairs of skates are produced, the company has a
daily production cost of $3,400 with a fixed cost of $400.
Daily production cost = (cost per pair)(# of pairs) + fixed costs
$3,400 = (cost per pair) · 100 + $400
$3,000 = (cost per pair) · 100
$30 = (cost per pair)
Next, use the cost per pair of skates to find an equation that represents the daily production costs for any
number of skates.
Daily production cost = (cost per pair)(# of pairs) + fixed costs
C(x) = 30x + 400
17.
18. Let x represent 100 cubic feet of water and y represent the monthly cost. The monthly cost is equal to the
volume charge plus the service charge. So,
monthly cost = volume charge + service charge
y = 2.24x + 10.30
19. The cost of location rental and decorations will have to be paid even if 0 people attend the prom (when x =
0). So, 3,000 is the y-intercept.
The cost will go up by $40 with each person that attends the prom. Calculate the slope.
The slope of the function is 40. Substitute the slope and y-intercept into the slope-intercept formula.
20. Let x represent the number of rides and y represent the total cost. Then, paying an admission fee of $9.00
plus an additional charge of $4.00 for each ride can be modeled by the following equation.
y = 4x + 9
Find the amount Cassidy would spend by paying per ride.
y = ($4.00/ride)(12 rides) + $9.00 = $57.00
Subtract the cost of the all-day pass to find the amount of money Cassidy would save.
$57.00 - $45.00 = $12.00
So, Cassidy would save $12.00.