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Transcript of 7-4
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.