3.4 Applications of Linear Systems

Steps1) Read and underline important terms1) Read and underline important terms

2) Assign 2 variables x and y (use diagram or table as 2) Assign 2 variables x and y (use diagram or table as needed)needed)

3) Write a system of equations3) Write a system of equations

4) Solve for x and y4) Solve for x and y

5) Check answer5) Check answer

6) State answer6) State answer

Problem with quantities and costs

1) The average movie ticket (to US dollars) costs $10 in Tokyo and $8 in Paris. If a group of 36 people from these two cities paid $298 for tickets to see The Titanic, how many people from each city were there?

1) The average movie ticket (to US dollars) costs $10 in Tokyo and $8 in Paris. If a group of 36 people from these two cities paid $298 for tickets to see The Titanic, how many people from each city were there?

Let x: number of tickets (or people) in Tokyo

y: number of tickets (or people) in Paris

Equations: x + y = 36 -10x – 10y = -360

10x + 8y = 298 10x + 8y = 298

Use Elimination Method So -2y = -62

y = 31 & x = 5

Therefore, there were 31 people in Paris and 5 people

in Tokyo

Problem with Mixtures

2) How many liters of 25% alcohol solution must be mixed with 12% solution to get 13 liters of 15% solution?

2) How many liters of 25% alcohol solution must be mixed with 12% solution to get 13 liters of 15% solution?

Let x be the number of liters of the 25% alcohol solution

y be the number of liters of the 12% alcohol solution

Equations: x + y = 13 so x = 13 -y

0.25x + 0.12 y = 0.15 (13)

0.25 (13 –y) + 0.12y = 1.95 Use substitution method

3.25 - 0.25 y + 0.12 y = 1.95

- .13y = -1.3 so y = 10, x= 13-10=3

Therefore, there are 3 liters of the 25% alcohol solution and

10 liters of the 12% alcohol solution

Simple Interest Problems

PRINCIPAL * RATE * TIME = INTEREST

3)

Total Loan (Perkin and Federal) : $9600

Perkin Loan (P.L.): 5% simple interest

Federal Loan (F.L): 8% simple interest

Interest after 1 year: $633.

What was the original amount of each loan?

------ ------ ------

Let p: org. amount of P. L. and F: org. amount of F. L

p + f = 9600

.05p + .08f = 633

You can change the second equation to 5p + 8f = 63300 before you use either method to solve

Answer: Perkin Loan = $4500 and Federal Loan = $5100

Motion Problems

DISTANCE = RATE * TIME d = r * t

r = d / t

t = d / r

Problem with Distance, rate, time

4) In one hour Ann can row 2 mi against the current or 10 mi with the current. Find the speed of the current and Ann’s speed in still water

4) In one hour Ann can row 2 mi against the current or 10 mi with the current. Find the speed of the current and Ann’s speed in still water

Let x: Ann’s speed, and y: current’s speed

Equations: 1(x + y) = 10 x + y = 10

1(x – y) = 2 x - y = 2

2x = 12

so x = 6 and y = 4

Therefore, Ann’s speed is 6 mi/h and current’s speed is 4 mi/h

1x - y2Against Current

1x + y10With Current

TRD =

5) Two airplanes leave Boston at 12:00 noon and fly in opposite directions. If one flies at 410 mph and the other 120 mph faster, how long will it take them to be 3290 mi apart?

Let t be the number of hours and d be the distance of the slower plane

Boston

Faster Plane

Slower Plane

t410+120=5303290-d

t410d

TimeRate *Distance =

Equations: d = 410 t

3290-d = 530 t

Solve using sub. or eli. we should get t =3.5 hours

6) A bus leaves downtown terminal, traveling east at 35 mph. One hour later, a faster bus leaves downtown, also traveling east on a parallel tract at 40 mph. How far from the downtown will the faster bus catch up with the slower one.

6) A bus leaves downtown terminal, traveling east at 35 mph. One hour later, a faster bus leaves downtown, also traveling east on a parallel tract at 40 mph. How far from the downtown will the faster bus catch up with the slower one.

Let t and d be number of hours and distance for the slower bus

Equations: d = 35 t d = 40 (t-1)Solve this system using substitution, we should get t = 8

hours and d = 280mi

t-140dFaster bus

t35dSlower bus

timeRate *Distance =