7.4: Application of Linear Systems Homework 52: p.421: 19-45, All

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1 7.4: Application of Linear Systems Homework 52: p.421: 19-45, All Objectives Choose the best method to solve a system of linear equations Use a system to model real life problems Determine which is the best method for solving each system. Choose from Graph/Table, Substitution, or Elimination. Do not solve. 1] y = 2x + 4 2x + y = 19 Substitution 4] 3x + 2y = 10 y = 2x 4 Substitution 2] 4x + 8y = 20 4x + 2y = 30 Elimination 5] y = 6x y = 2x + 1 Graph/Table 3] y = 4x 3 y = 3x + 1 Graph/Table 6] x y = 3 x + y = 3 Elimination

2 Example 1: Number Theory Find the value of two numbers if their sum is 12 and their difference is 4 x = first number y = second number x + y = 12 x y = 4 2x = 16 x = 8

3 Example 1: Number Theory Find the value of two numbers if their sum is 12 and their difference is 4 x = 8 x + y = 12 x y = y = 12 8 y = 4 y = 4 y = 4

4 Student Led Example 1: Number Theory The difference of two numbers is 3 and their sum is 13. Find the two numbers. x y = 3 x + y = 13 2x = 16 x = 8 y = 5

5 Example 2: Application A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Step 1: Define Variables Let: b = boat speed c = current speed

6 Example 2: Application A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Step 2: Separate Quantities Downstream: 210 miles in 10 hours (b + c) Upstream: 210 miles in 70 hours (b c)

7 Example 2: Application A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Step 3: Set Up Equations Downstream: 210 = 10(b + c) Upstream: 210 = 70(b c) Why not Downstream: 10 = 210(b + c) Upstream: 70 = 210(b c)

8 Example 2: Application A boat traveled 210 miles downstream and back. The trip This was not obvious. I played around downstream took 10 hours. The trip back took 70 hours. What is the with a calculator for a bit. My only other speed of the boat in still water? What is the speed of the current? option was to multiply the first equation by -7 but that made 210 turn into - Step 4: Solve for sumpin Um, no. Downstream: 210 = 10 b + c Upstream: 210 = 70(b c) 210 = 10b + 10c 210 = 70b 70c 210 = 10b + 10c 210 = 70b 70c [ 7] 210 = 10b + 10c 30 = 10b + 10c 180 = 20c 9 = c

9 Example 2: Application A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Step 5: Solve for d udder Downstream: 210 = 10 b + c Upstream: 210 = 70(b c) 210 = 10(b + 9) 210 = 10b = b = b 210 = 10b + 10c 210 = 70b 70c 210 = 70(b 9) 210 = 70b 63 3 = b 9 12 = b

10 Example 2: Application A boat traveled 210 miles downstream and back. The trip downstream took 10 hours. The trip back took 70 hours. What is the speed of the boat in still water? What is the speed of the current? Step 6: Explain The speed of the current is 9 mph. The speed of the boat (if it were in still water) was 12 mph. The boat travelled a combined speed of 21 mph downstream, but only 3 mph upstream.

11 Student Led Example 2: Application Us and a neighboring school combined resources to attend a state fair field trip. We ll call us School A for Awesome and they can be School B for blah. We rented and filled 4 vans and 1 bus with 54 students while they filled 8 vans and 8 busses for a total of 240 students. If each van held the same number of students and each bus did too, how many students can fit in a van and how many can fit in a bus?

12 Student Led Example 2: Application Us and a neighboring school combined resources to attend a state fair field trip. We ll call us School A for Awesome and they can be School B for blah. We rented and filled 4 vans and 1 bus with 54 students while they filled 8 vans and 8 busses for a total of 240 students. If each van held the same number of students and each bus did too, how many students can fit in a van and how many can fit in a bus? 4v + b = v 2b = 108 8v + 8b = 240 8v + 8b = 240 6b = 132 b = 22 4v + 22 = 54 4v = 32 v = 8 8v + 8(22) = 240 8v = 240 8v = 64

13 End of Lesson Your car s manual recommends that you use at least 89-octane gasoline. Your car s 16-gallon tank is almost empty. How much regular gasoline (87-octane) do you need to mix with premium gasoline (92-octane) to produce 16 gallons of 89-octane gasoline. An octane rating is the percent of isooctane in the gasoline, so 16 gallons of 89-octane gasoline contains 89% of 16, which is gallons of isooctane.

14 End of Lesson Your car s manual recommends that you use at least 89-octane gasoline. Your car s 16-gallon tank is almost empty. How much regular gasoline (87-octane) do you need to mix with premium gasoline (92-octane) to produce 16 gallons of 89-octane gasoline. x + y = 16.87x +.92y = 14.24

15 End of Lesson x + y = 16 y = 16 x.87x +.92y = x +.92(16 x) = x x = x =.48 x = = gallons of Premium 9.6 gallons of Regular

Just as in the previous lesson, all of these application problems should result in a system of equations with two equations and two variables:

Just as in the previous lesson, all of these application problems should result in a system of equations with two equations and two variables: The two methods we have used to solve systems of equations are substitution and elimination. Either method is acceptable for solving the systems of equations that we will be working with in this lesson.