7-4
Applications of Linear Systems
Example 1
Suppose you have just enough money, in coins, to pay for a loaf of bread
priced at $1.95. You have 12 coins, all quarters and dimes. Let Q equal the
number of quarters and D equal the number of dimes. Write a system of
equations to solve the problem. How many quarters do you have? Dimes?
# of Coins:
• 12 coins total, adding
quarters and dimes
together
Value of Coins:
• Have $1.95 total
• Quarters = $0.25
• Dimes = $0.10
.25Q + .10D = 1.95
Q + D = 12
System:
Q + D = 12
.25Q + .10D = 1.95
Example 2
Several students decide to start a T-shirt company. After initial expenses of
$280, they purchase each T-shirt wholesale for $3.99. They sell each T-shirt
for $10.99. How many must they sell to break even?
Let: T = T-shirts
M = money
Expenses:
Income:
• Spent $280 on
• Sell each shirt for $10.99
miscellaneous supplies
(only income!)
• Spent $3.99 per shirt
M = 3.99T + 280
M = 10.99T
To
System:
Solve:
M = 3.99T
M = M+ 280
3.99TM+ =280
10.99T
= 10.99T
Example 3
Suppose you are trying to decide whether to buy ski equipment. Typically, it
costs you $60 a day to rent ski equipment and buy a lift ticket (the ticket is
included in that rate). You can buy ski equipment for about $400. A lift ticket
alone costs $35 for one day. How many days must you ski for it to be worth it
to buy the equipment? (break-even point)
Let: D = days
Renting:
• Spend $60 per day
M = money
Buying:
• Spend $400 flat rate to buy
equipment
• Spend $35 per day
M = 60D
M = 35D + 400
To
System:
Solve:
M
M == 60D
M
60D
M ==35D
35D++400
400
Example 3 Solution:
To Solve:
M=M
60D = 35D + 400
60D = 35D + 400
-35D
-35D
25D = 400
25
25
D = 16
Example 4
You have 28 coins in your pocket, consisting of only quarters and dimes. If
the total amount of money in your pocket is $5.20, how many quarters and
dimes do you have?
# of Coins:
• 28 coins total, adding
quarters and dimes
together
Value of Coins:
• Have $5.20 total
• Quarters = $0.25
• Dimes = $0.10
.25Q + .10D = 5.20
Q + D = 28
System:
Q + D = 28
.25Q + .10D = 5.20
“Easy” variable to solve for is in
first equation. (D is “easy” too!)
Example 4 Solution:
System:
Q + D = 28
.25Q + .10D = 5.20
2. 25Q + 10D = 520
1.
Pattern: 1, 2, 1
1
Q + D = 28
-D
-D
Q = 28 - D
Get rid of
decimals
2
25(28 – D) + 10D = 520
700 – 25D + 10D = 520
700 – 15D = 520
-700
-700
– 15D = -180
-15
-15
D = 12
System:
Pattern: 1, 2, 1
Example 4 Solution:
D = 12
1
Q = 28 - D
Q = 28 – 12
Q = 16
Q = 28 - D
25Q + 10D = 520
Example 5
Suppose you want to combine two types of fruit to drink to create 24kg of a
drink that will be 5% sugar by weight. Fruit drink A is 4% sugar by weight and
fruit drink B is 8% sugar by weight.
Don’t forget to
convert
percents to
decimals!
Fruit Drink
(kg)
Sugar (kg)
Fruit Drink A Fruit Drink B Mixed Fruit
4% Sugar
8% Sugar
Drink
5% Sugar
A
B
24
.04A
.08B
.05(24)
Example 5 Solution:
System:
A + B = 24
.04A + .08B = 1.2
18 kg of fruit drink A
and 6 kg of fruit drink
B.
Example 6
A plane takes about 6 hours to fly you 2400 miles from NYC to Seattle. At the
same time, your friend flies from Seattle to NYC. His plane travels with the
same average airspeed, but his flight takes 5 hours. Find the average airspeed
of the planes. Find the average wind speed.
Let: A = airspeed
W = wind speed
So,
Airspeedwhich
is the speed ofplane
an aircraft! is faster?
Wind speed is the speed of the wind!
Rate = airspeed + wind speed (faster!)
Rate = airspeed – wind speed (slower!)
r=A+W
d = (A + W)(t)
2400 = (A + W)(5)
r=A–W
d = (A – W)(t)
2400 = (A – W)(6)
5
480 = A + W
5
6
400 = A – W
6
Example 6 Solution:
System:
A + W = 480
A – W = 400
A + W = 480
A – W = 400
2A
= 880
2
A
2
= 440
A + W = 480
440 + W = 480
-440
-440
W = 40
The average airspeed of
the planes is 440 mph
and the average wind
speed is 40 mph.