Week 1, Day 2

Today I learned how to solve for variables through elimination. The steps to perform this operation is as follows: First, obtain coefficients that differ only in sign. Second, add equations to eliminate a variable. Then back substitute to solve for the second equation. Finally, check your solution! Once you have determined the value of each variable, write your solution as an ordered pair. There are a number of possibilities in regard to solutions, three of which are intersecting lines (one solution), parallel lines (no solution), and the same line (infinitely many solutions). The following are example problems demonstrating how to solve for variable through elimination:

Example 1:

5x + 3y = 9 

2x - 4y = 14

Step 1:

 4(5x + 3y = 9) --> 20x + 12y = 36

 3(2x - 4y = 14) --> 6x - 12y = 42

Step 2: 

 20x + 12y = 36

   6x - 12y = 42

+ -----------------

26x = 78

x = 3

Step 3: 

2(3) - 4y = 14

6 - 4y = 14

-4y = 8

y = -2

Step 4:

2(3) - 4(-2) = 14

6 + 8 = 14

14 = 14

(3, -2) 

Example 2:

An airplane flying into a headwind travels the 2000-mile flying distance between two cities in 4 hours and 24 minutes. On the return flight, the same distance is traveled in 4 hours. Find the airspeed of the plane and the speed of the wind, assuming that both remain constant.

r(1) = speed of the plane

r(2) = speed of the wind

r(1) - r(2) = speed of the plane against the wind

r(1) + r(2) = speed of the plane with the wind

Use the formula: Distance = (rate)(time)

2000 = (r(1) - r(2)) (4 + 24/60) --> 5000 = 11r(1) - 11r(2)

2000 = (r(1) + r(2)) (4) --> 500 = r(1) + r(2)

By elimination, the solution is:

r(1) = 5250/11 = 477.27 miles per hour (speed of the plane)

r(2) = 250/11 = 22.73 miles per hour (speed of the wind)