Olympic College - Topic 8 Applications of a Linear Equation
Topic 8 Applications of a Linear Equation
There are a number of practical problems that involve using linear equations. These applications
are varied and some of the most common forms are given below.
1. Questions where you are given the linear equation.
In these questions you are given the linear equation and are asked to use it to solve some
problems using this equation. The linear equations do not typically use the x and y variables such
as y = 3x – 8 but will more likely be in the form C = 25t + 50 or W = 0.12B etc..
Example 1: A rental company charges a fixed cost of $10 plus $25 per day. The cost C of hiring
a compressor for D days is C = 25D + 10.
(a) How much will it cost to hire the compressor for 5 days?
(b) How much will it cost to hire the compressor for 14 days?
(c) The rental company charged $210, how many days was the compressor rented
for?
(d) The rental company charged $85, how many days was the compressor rented
for?
Solution (a): Substitute D = 5 into the formula
C = 25D + 10 = 25(5) + 10 = $135
Solution (b): Substitute D = 14 into the formula C = 25D + 10 = 25(14) + 10 = $360
Solution (c): To solve this problem we need to solve the equation 25D + 10 = 210
25D + 10
25D
D
=
=
=
=
210
200
subtract 10 from both sides
divide both sides by 25
8
The person rented the compressor for 8 days.
Solution (d): To solve this problem we need to solve the equation 25D + 10 = 85
25D + 10
25D
D
=
=
=
=
85
75
subtract 10 from both sides
divide both sides by 25
3
The person rented the compressor for 3 days.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 2: A tennis club charges a yearly fee of $400 plus $15 per hour for renting a court.
The cost C of playing h hours of tennis in a year is a C = 15h + 400.
(a)
(b)
(c)
(d)
What will be your total cost if you play 200 hours of tennis in a year?
What will be your total cost if you play 52 hours of tennis in a year?
If you are charged $550, how many hours of tennis did you play?
If you are charged $625, how many hours of tennis did you play?
Solution (a): Substitute h = 200 into the formula C = 15h + 400
C
C
C
=
=
=
15h + 400
15(200) + 400
$3400
The total cost for playing 200 hours of tennis in a year is $3400
Solution (b): Substitute h = 52 into the formula C = 15h + 400
C
C
C
=
=
=
15h + 400
15(52) + 400
$1280
The total cost for playing 52 hours of tennis in a year is $1280
Solution (c): To solve this problem we need to solve the equation 15h + 400 = 550
15h + 400
15h
h
=
=
=
=
550
150
subtract 400 from both sides
divide both sides by 15
10
The person played 10 hours of tennis in a year.
Solution (d): To solve this problem we need to solve the equation 15h + 400 = 625
15h + 400
15h
h
=
=
=
=
625
225
subtract 400 from both sides
divide both sides by 15
15
The person played 15 hours of tennis in a year.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 3: A plumber charges a fixed call out charge of $120 plus $80 per hour to make
repairs. The total cost T he charges for a job that requires h hours to repair is.
T = 80h + 120
(a)
(b)
(c)
(d)
What will be your total cost if a repair takes 8 hours?
What will be your total cost if a repair takes 4.5 hours?
If you are charged $520 for a repair, how many hours did the plumber work?
If you are charged $420 for a repair, how many hours did the plumber work?
Solution (a): Substitute h = 8 into the formula
T
T
T
=
=
=
T = 80h + 120
80h + 120
80(8) + 120
$760
The total cost for an 8 hour repair is $760.
Solution (b): Substitute h = 4.5 into the formula
T
T
T
=
=
=
T = 80h + 120
80h + 120
80(4.5) + 120
$480
The total cost for an 4.5 hour repair is $480.
Solution (c): To solve this problem we need to solve the equation 80h + 120 = 520
80h + 120
80h
h
=
=
=
=
520
400
subtract 120 from both sides
divide both sides by 80
5
The plumber worked for 5 hours.
Solution (d): To solve this problem we need to solve the equation 80h + 120 = 420
80h + 120
80h
h
=
=
=
=
420
300
subtract 120 from both sides
divide both sides by 80
3.75
The plumber worked for 3.75 or
hours.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 4: The volume in litres of water V left in a leaking tank after m minutes is given by
the formula V = 240 – 12m
(a)
(b)
(c)
(d)
(e)
How much liquid was in the tank when it was full?
How much liquid will be left in the tank after 8 minutes?
How much liquid will be left in the tank after 14 minutes?
How long will it take until the tank has a volume of 168 litres?
How long before the tank is empty?
Solution (a): The tank will be full t = 0 so we substitute m = 0 into the formula V = 240 – 12m
V
=
240 – 12m
=
240 – 12(0) =
240 litres
The tank had 240 litres when it is full.
Solution (b): Substitute m = 8 into the formula V = 240 – 12m
V
=
240 – 12m
=
240 – 12(8) =
240 – 96 = 144 litres
The tank had 144 litres when it is full.
Solution (c): Substitute m = 14 into the formula V = 240 – 12m
V
=
240 – 12m
=
240 – 12(14) =
240 – 168 = 72 litres
The tank had 72 litres when it is full.
Solution (d): To solve this problem we need to solve the equation 240 – 12m = 168
240 – 12m
– 12m
h
= 168
= – 72
=
= 6
subtract 240 from both sides
divide both sides by – 12
It will take 6 hours for the volume to reach 168 litres.
Solution (e): To solve this problem we need to solve the equation 240 – 12m = 0
240 – 12m
– 12m
h
= 0
= – 240
=
= 20
subtract 240 from both sides
divide both sides by – 12
It will take 20 hours for the tank to become empty.
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Olympic College - Topic 8 Applications of a Linear Equation
Exercise 1:
1.
A fitness club charges a joining fee of $50 plus $30 per hour for using the facilities.
The cost C of using the facilities for h hours is given by the formula C = 30h + 50
(a)
(b)
(c)
(d)
2.
An electrician charges a fixed call out charge of $150 plus $90 per hour to make repairs.
The total cost T he charges for a job that requires h hours to repair is T = 90h + 150
(a)
(b)
(c)
(d)
3.
How much liquid was in the tank when it was full?
How much liquid will be left in the tank after 10 minutes?
How much liquid will be left in the tank after 15 minutes?
How long will it take until the tank has a volume of 100 litres?
How long before the tank is empty?
The profit P that a theater makes when it sells t tickets is given by the formula P = 12t – 240
(a)
(b)
(c)
(d)
(e)
(f)
5.
What will be your total cost if a repair takes 4 hours?
What will be your total cost if a repair takes 6.5 hours?
If you are charged $600 for a repair, how many hours did the electrician work?
If you are charged $465 for a repair, how many hours did the electrician work?
The volume in litres of water V left in a leaking tank after m minutes is given by the formula
V = 255 – 15m
(a)
(b)
(c)
(d)
(e)
4.
What will be your total cost if you use the facilities for 150 hours?
What will be your total cost if you use the facilities for 200 hours?
If you are charged $350, how many hours did you use the facilities?
If you are charged $650, how many hours did you use the facilities?
How much profit will the theater make if it sells 100 tickets?
How much profit will the theater make if it sells 250 tickets?
How much money will the theater lose if it only sells 15 tickets?
How many tickets must the theater sell in order to make a profit of $480
How many tickets must the theater sell in order to make a profit of $11,760
How many tickets must the theater sell in order to break even (Profit = $0)
The temperature T of a chemical after s seconds is given by the formula T = 12s + 10
(a)
(b)
(c)
(d)
(e)
(f)
What will be the temperature of the chemical be after 15 seconds?
What will be the temperature of the chemical be after 40 seconds?
What will be the temperature of the chemical be after 2 minutes?
How long before the temperature reaches 472oF?
Approximately how long before the temperature reaches 100oF?
The chemical will explode when it reaches a temperature of 1810oF, how long before it
will explode?
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Olympic College - Topic 8 Applications of a Linear Equation
2. Questions where you are asked to find the Linear Equation
In these questions you will be asked to create the linear equation first and then you will be asked
to use the equation to solve some problem.
Example 1: A truck rental company charges a fixed cost of $60 plus $45 per day.
(a)
(b)
(c)
(d)
(e)
Find a formula for the cost C of hiring a truck for d days?
How much will it cost to hire a truck for 3 days?
How much will it cost to hire a truck for 7 days?
The rental company charged $510, how many days was the truck rented for?
The rental company charged $240, how many days was the truck rented for?
Solution (a): The general formula will be of the form C = md + b where m is the rate per day
and b is the fixed cost. The rate used in this question was m = $45 per day and the
fixed cost was $60 we put this information together to get the formula C = 45d + 60
Solution (b): Substitute d = 3 into the formula
C
C
C
=
=
=
C = 45d + 60
45d + 60
45(3) + 60
$195
Solution (c): Substitute d = 7 into the formula
C = 45d + 60 = 45(7) + 60 = 315 + 60 = $375
Solution (d): To solve this problem we need to solve the equation 45d + 60 = 510
45d + 60 = 510
45d
= 450
=
d = 10
subtract 60 from both sides
divide both sides by 45
The person rented the truck for 10 days.
Solution (e): To solve this problem we need to solve the equation 45d + 60 = 240
45d + 60 = 240
45d
= 180
=
d = 4
subtract 60 from both sides
divide both sides by 45
The person rented the truck for 4 days.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 2: A plumber charges a fixed call out charge of $100 plus $95 per hour to make
repairs.
(a) What is the formula for the total cost T for a repair that takes h hours?
(b) What will be your total cost if a repair takes 6 hours?
(c) What will be your total cost if a repair takes 2.5 hours?
(d) If you are charged $480 for a repair, how many hours did the plumber work?
(e) If you are charged $575 for a repair, how many hours did the plumber work?
Solution (a): The general formula will be of the form T = mh + b where m is the rate per hour
and b is the fixed call out charge. The rate used in this question was m = $95 per
hour and the fixed cost was $100 we put this information together to get the
formula T = 95h + 100
Solution (b): Substitute h = 6 into the formula
T
T
T
=
=
=
T = 95h + 100
95h + 100
95(6) + 100
$670
The total cost for an 6 hour repair is $670.
Solution (c): Substitute h = 2.5 into the formula
T
T
T
=
=
=
T = 95h + 100
95h + 100
95(2.5) + 100
$337.50
The total cost for an 6 hour repair is $337.50.
Solution (d): To solve this problem we need to solve the equation 95h + 100 = 480
95h + 100
95h
h
=
=
=
=
480
380
subtract 100 from both sides
divide both sides by 95
4
The plumber worked for 4 hours.
Solution (e): To solve this problem we need to solve the equation 95h + 100 = 575
95h + 100
95h
= 575
= 475
subtract 100 from both sides
=
divide both sides by 95
h = 5
The plumber worked for 5 hours.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 3: A tank with 2000 gallons has a leak; the water is pouring out at the rate of 10
gallons per second.
(a)
(b)
(c)
(d)
(e)
What is the formula for volume of water left V after s seconds?
How much liquid will be left in the tank after 10 seconds?
How much liquid will be left in the tank after 1 minute?
How long will it take until the tank has 250 gallons left?
How long before the tank is empty?
Solution (a): The general formula will be of the form V = ms + b where m is the rate per second
and b is the amount in the tank at the start. The rate used in this question was
m = – 10 per second (the rate is negative since the volume is going down) and the
amount of water at the start was b = 2000 gallons. This gives us the formula
V = – 10s + 2000 which can also be written as V = 2000 – 10s
Solution (b): Substitute s = 10 into the formula V = 2000 – 10s
V
=
2000 – 10s
=
2000 – 10(10) = 2000 – 100 = 1900 gallon
The tank had 1900 gallons after 10 seconds
Solution (c): Substitute s = 60 into the formula V = 2000 – 10s
V
=
2000 – 10s
=
2000 – 10(60) = 2000 – 600 = 1400 gallon
The tank had 1400 gallons after 1 minute
Solution (d): To solve this problem we need to solve the equation 2000 – 10s = 250
2000 – 10s
– 10s
s
=
=
=
=
250
– 1750
subtract 2000 from both sides
divide both sides by – 10
175
It will take 175 seconds for the volume to reach 250 gallons.
Solution (e): To solve this problem we need to solve the equation 2000 – 10s = 0
2000 – 10s
– 10s
s
=
=
=
=
0
– 2000
subtract 2000 from both sides
divide both sides by – 10
200
It will take 200 seconds until the tank is empty.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 4: The height of an animal in inches and its age in months is given in this linear graph
h
(a) What is the slope and y-intercept of this line?
(b) Find a formula for the height h.
20
of this animal after m months.
Height
(c) What will be the height of the animal
in
after 8 months?
inches
10
(d) What will be the height of the animal
after 20 months?
4
(e) When will the animal reach a height
of 28 inches.
0
(f) When will the animal reach a height
of 40 inches.
m
8
4
12
Age in months
Solution (a):
Using the graph the y-intercept is (0,4)
and using the points (0,4) and (12,22) slope =
Solution (b):
The formula is h =
Solution (c):
Substitute m = 8 into the formula h =
=
The animal will be 16 inches tall after 8 months.
= 12 + 4 = 16 inches
Solution (d):
Substitute m = 20 into the formula h =
=
The animal will be 34 inches tall after 20 months.
= 30 + 4 = 34 in
Solution (e):
To solve this problem we need to solve the equation
m
=
28
=
24
=
=
= 28
subtract 4 from both sides
divide both sides by
16
It will take 16 months for the animal to reach a height of 28 inches.
Solution (f):
To solve this problem we need to solve the equation
=
40
=
36
= 48
subtract 4 from both sides
=
divide both sides by
m = 24
It will take 24 months for the animal to reach a height of 40 inches.
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Olympic College - Topic 8 Applications of a Linear Equation
Example 5: The temperature T after h hours is drawn in the linear graph below.
(a)
(b)
(c)
(d)
(e)
(f)
Solution (a):
What is the slope and y-intercept of this line?
Find a formula for the Temp T after h hours?
What will be the temperature after 3 hours?
What will be the temperature after 12 hours?
When will temperature reach 5oF?
When will temperature reach – 20oF?
Using the graph the y-intercept is (0,40)
and using the points (0,40) and (8,0)
slope =
T
40
20
5
0
10
– 20
– 40
Time in hours
or T = 40 – 5h
Solution (b):
The formula is T =
Solution (c):
Substitute h = 3 into the formula T = 40 – 5h = 40 – 5(3) = 40 – 15 = 25oF
T
T
T
T
=
=
=
=
15
40 – 5h
40 – 5(3)
40 – 15
25oF
The temperature will be 25oF after 3 hours.
Solution (d):
Substitute h = 12 into the formula T= 40 – 5h = 40 – 5(12) = 40 – 60 = – 20 oF
The temperature will be – 20 oF after 12 hours.
Solution (e):
To solve this problem we need to solve the equation
40 – 5h =
– 5h
=
=
h =
5
– 35
40 – 5h = 5
subtract 40 from both sides
divide both sides by
7
It will take 7 hours before the temperature reaches 5oF .
Solution (f):
To solve this problem we need to solve the equation 40 – 5h = – 20
40 – 5h =
– 5h
=
=
h =
– 20
– 60
subtract 40 from both sides
divide both sides by
12
It will take 12 hours before the temperature reaches – 20oF .
Page | 10
h
Olympic College - Topic 8 Applications of a Linear Equation
Exercise 2:
1.
A truck rental company charges a fixed cost of $20 plus $32 per day.
(a)
(b)
(c)
(d)
(e)
2.
A golf club has an annual fee of $400 plus they charge you $30 per round of golf.
(a)
(b)
(c)
(d)
(e)
3.
What is the formula for the total cost C for using u units of gas?
What will be your total cost if you use 200 units of gas?
What will be your total cost if you use 2500 units of gas?
If you gas bill was $25.50, how many units of gas did you use in a month?
If you gas bill was $120, how many units of gas did you use in a month?
A tank with 15000 cm3 has a leak, the water is pouring out at the rate of 15 cm3 per second.
(a)
(b)
(c)
(d)
(e)
6.
What is the formula for the total cost T for a repair that takes h hours?
What will be your total cost if a repair takes 10 hours?
What will be your total cost if a repair takes 6.5 hours?
If you are charged $420 for a repair, how many hours did the electrician work?
If you are charged $600 for a repair, how many hours did the electrician work?
An gas utility company has a standing monthly charge of $10.50 plus $0.15 per unit of gas
used.
(a)
(b)
(c)
(d)
(e)
5.
Find a formula for the cost C of playing g games of golf in a year.
How much will it cost if you play 100 games of golf in a year?
How much will it cost if you play 50 games of golf in a year?
The total charge in a year was $1000 how many games of golf would you play?
The total charge in a year was $3430 how many games of golf would you play?
An electrician charges a fixed call out charge of $150 plus $90 per hour to make repairs.
(a)
(b)
(c)
(d)
(e)
4.
Find a formula for the cost C of hiring a truck for d days?
How much will it cost to hire a truck for 4 days?
How much will it cost to hire a truck for 9 days?
The rental company charged $340, how many days was the truck rented for?
The rental company charged $170, how many days was the truck rented for?
What is the formula for volume of water left V after s seconds?
How much liquid will be left in the tank after 30 seconds?
How much liquid will be left in the tank after 2 minute?
How long will it take until the tank has 1200 cm3 left?
How long before the tank is empty?
A tank with 2000 gallons has a leak, the water is pouring out at the rate of 20 gallons per
second. What is the formula for volume of water left V after s seconds?
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Olympic College - Topic 8 Applications of a Linear Equation
7.
A tank with 400 litres has a leak, the water is pouring out at the rate of 4 litres per minute.
What is the formula for volume of water left V after m minutes?
8. A company purchased a car for $25,000, and expects its value to depreciate $2,500 per year.
Write an equation for the value of the car, y, in terms of years, x.
9. The cost, C, of playing tennis in the Downtown Tennis Club includes an annual $200
membership fee plus $10 per hour, h, of court time.
(a) Write an equation for the annual cost of playing tennis in terms of hours played.
(b) Graph the equation for up to and including 300 hours.
10. The weight of an animal in pounds and its age in months is given in this linear graph
W
(a) What is the slope and y-intercept of this line?
(b) Find a formula for the weight W of this animal
after m months.
(c) What will be the weight of the animal after 5
months?
10
Weight
in
pounds
5
3
(d) What will be the weight of the animal after a year?
(e) When will the animal reach a weight of 25 pounds.
m
0
4
2
(a) When will the animal reach a weight of 44 pounds.
6
Age in months
11. The temperature T after h hours is drawn in the linear graph below.
T
(a) What is the slope and y-intercept of this line?
10
(b) Find a formula for the Temp T after h hours?
5
(c) What will be the temperature after 4 hours?
0
(d) What will be the temperature after 10 hours?
–5
(e) When will temperature reach 5oF?
(f)
When will temperature reach – 20oF?
2
4
6
– 10
Time in hours
Page | 12
h
Olympic College - Topic 8 Applications of a Linear Equation
Solution:
Exercise 1:
1.(a) $4550
(b) $6,050
(c) 6 hours
(d) 20 hours
2.(a) $510
(b) $735
(c) 5 hours
(d) 3.5 hours
3.(a) 105 litres
(b) 30 litres
(c) 10 minutes
(d) 17 minutes
4.(a) $960
(c) lose = $60
(d) 60 tickets
(e) 1000 tickets
(f) 20 tickets
(c) 1450oF
(d) 38.5 sec
(e) 7.5 sec
(f) 150 sec
(b) $2760
5.(a) 190oF (b) 490oF
Exercise 2:
1.(a) C = 32d + 20
(b) $148
(c) $308
(d) 10 days
(e) 5 days
2.(a) C = 30g + 400 (b) $3400
(c) $1620
(d) 20 games
(e) 101 games
3.(a) T = 90h + 150 (b) $1050
(c) $735
(d) 3 hours
(e) 5 hours
4.(a) C = 0.15u + 10.50
(b) $40.50 (c) $385.50
5.(a) V = 15000 – 15s
(b) 14,550 cm3 (c) 13,200 cm3
6.
V = 2000 – 20s
7.
V = 400 – 4m
8.
y = 25000 – 2500x
9.(a) C = 10h + 200
(d) 100 units
(e) 730 units
(d) 920 sec
(e) 1000 sec
(b) graph
10.(a) slope = 1 y –intercept = 3
(b) W = m + 3
11.(a) slope = – 5 y –intercept = 10
(b) T = 10 – 5h (c) – 10oF
(c) 8
(d) 15
(e) 22
(d) – 40oF
(f) 41
(e) 1 (f) 6
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