7-4

Applications of Linear Systems

# of Coins:• 12 coins total, adding

quarters and dimes together

Q + D = 12

Value of Coins:• Have $1.95 total• Quarters = $0.25• Dimes = $0.10

.25Q + .10D = 1.95

Example 1Suppose 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?

System: Q + D = 12

.25Q + .10D = 1.95

Expenses:• Spent $280 on

miscellaneous supplies• Spent $3.99 per shirt

M = 3.99T + 280

Income:• Sell each shirt for $10.99

(only income!)

M = 10.99T

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?

System:M = 3.99T + 280

M = 10.99TBreak even means when

your expenses = income!

M = M

Let: T = T-shirts M = money

To Solve:M = M

3.99T + 280 = 10.99T

Renting:• Spend $60 per day

M = 60D

Buying:• Spend $400 flat rate to buy

equipment• Spend $35 per day

M = 35D + 400

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)

System: M = 60D

M = 35D + 400Break-even point is when

the cost for renting = the

cost for buying!

M = M

Let: D = days M = money

To Solve:M = M

60D = 35D + 400

Example 3 Solution:To Solve:

M = M60D = 35D + 400

60D = 35D + 400-35D -35D

25D = 400 25 25

D = 16

You would have to ski

for 16 days for the

price of purchasing

the ski equipment to

equal the price of

renting per day.

Setting them equal to

each other means th

at

the renting price

equals

the purchasin

g price!

# of Coins:• 28 coins total, adding

quarters and dimes together

Q + D = 28

Value of Coins:• Have $5.20 total• Quarters = $0.25• Dimes = $0.10

.25Q + .10D = 5.20

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?

System: Q + D = 28

.25Q + .10D = 5.20

Example 4 Solution:Usin

g substitution!

System: Q + D = 28

.25Q + .10D = 5.20

“Easy” variable to solve for is in first equation. (D is “easy” too!)

Pattern: 1, 2, 1

1.

1

Q + D = 28-D -D

Q = 28 - D

25(28 – D) + 10D = 5202

Get rid of decimals

25Q + 10D = 520

*Multiply equation #2 by 100!

2.

700 – 25D + 10D = 520700 – 15D = 520-700 -700

– 15D = -180-15 -15

D = 12

Example 4 Solution:D = 12

Pattern: 1, 2, 1

1

Q = 28 - D

System: Q = 28 - D

25Q + 10D = 520

Q = 28 – 12Q = 16 You have 16 quarters

and 12 dimes in your

pocket.

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.

Fruit Drink A4% Sugar

Fruit Drink B8% Sugar

Mixed Fruit Drink

5% Sugar

Fruit Drink (kg)

Sugar (kg)

A B 24

.04A .08B .05(24)

Don’t forget to convert percents to decimals!

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

Rate = airspeed + wind speed (faster!)

r = A + Wd = (A + W)(t)

2400 = (A + W)(5) 5 5

480 = A + W

Rate = airspeed – wind speed (slower!)

r = A – Wd = (A – W)(t)

2400 = (A – W)(6) 6 6

400 = A – W

Airspeed is the speed of an aircraft!

Wind speed is the speed of the wind!

So, which plane is faster?

Example 6 Solution:System:

A + W = 480A – W = 400

A + W = 480

Using elim

ination!

A – W = 4002A = 880

2 2A = 440

A + W = 480440 + W = 480-440 -440

W = 40The average airspeed of the planes is 440 mph and the average wind speed is 40 mph.