3.4 Applications of Linear Systems

1. 3.4 Applications of Linear Systems

2. Steps 1) Read and underline important terms 2) Assign 2 variables x and y (use diagram or table as needed) 3) Write a system of equations 4) Solve for x and y 5) Check answer 6) State answer

3. 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?

4. 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

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

6. 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

7. Simple Interest Problems PRINCIPAL * RATE * TIME = INTEREST

8. 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

9. Motion Problems DISTANCE = RATE * TIME d = r * t t = d / r r = d / t

10. 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

11. 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 D = R T With Current 10 x + y 1 Against Current 2 x - y 1

12. 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 Distance = Rate * Time Slower Plane d 410 t Faster Plane 3290-d 410+120=530 t Equations: d = 410 t 3290-d = 530 t Solve using sub. or eli. we should get t =3.5 hours

13. 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.

14. 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 Distance = Rate * time Slower bus d 35 t Faster bus d 40 t-1