1. Solving Systems of Equations
Classic Applications (Travel)

2. Objectives Define the variables in Travel application problems.
Use the variables to set up the system of equations.
Solve the system using the Elmination or Substitution Methods.

3. Travel Problems.
A plane leaves New York City and heads for Chicago, which is 750 miles away. The plane, flying against the wind, takes 2.5 hours to reach Chicago. After refueling the plane returns to New York, traveling with the wind, in 2 hours. Find the rate of the wind and the rate of the plane with no wind.
This is the first problem given to students to begin the classic application lessons. All I wanted students to do for this part is to be able to extract the important information from the problem and organize it into a table to help them find the system of equations. x
= rate of the plane without any wind y
= the rate of the wind

4. Travel Problems. x = rate of the plane without any wind y
= the rate of the wind
Two Scenarios
Rate against the wind:
NYC to Chicago
x - y
Time:
2.5 hours
Distance: 750mi
In this part we begin to set up the problem to find our systems of equations. To find the rate of going with and against the plane I decided to get a student volunteer to help me with the problem. Using the student, I asked him to pretend he was a plane and fly towards the head of the class (Chicago) demonstrating the rate of the plane without wind. I asked the student to try again only this time I acted like rate of the wind, pushing against the student slowing him down. Students were able t make this connection and discover the the rate of the plane going against the wind was x-y. I did a similar example with the plane goin with the wind, pushing on the back of the student again making the connection, x+y.
Rate with the wind:
Chicago to NYC
x + y
Time:
2 hours
Distance: 750mi
Distance = Rate x Time

5. Travel Problems. x = rate of the plane without any wind y
= the rate of the wind
Two Scenarios
Rate x Time = Distance
(x - y)
2.5 =
750 miles
NYC to Chicago:
(x + y)
2.0 =
750 miles
Chicago to NYC:
The final set up of the problem.
Distance = Rate x Time

6. Travel Problems.
A boat can travel 10 miles downstream in 2 hours and the same distance upstream in 3.5 hours. Find the rate of the boat in still water and the rate of the current. x
= rate of the boat in still water y
= the rate of the wind

7. Travel Problems. x = rate of the boat in still water y
= rate of the current
Two Scenarios
Rate of the boat:
going downstream
x + y
Time:
2 hours
Distance: 10mi
Rate of the boat:
Going upstream
x - y
Time:
3.5 hours
Distance: 10mi
Distance = Rate x Time

8. Travel Problems. x = rate of the boat in still water y
= rate of the current
Two Scenarios
Rate x Time = Distance
Downstream:
(x + y) 2 =
10 miles
Upstream:
(x - y)
3.5 =
10 miles
Distance = Rate x Time