Consider the following problem:

The flight of two planes is being tracked on a rectangular coordinate system. The flight of the first plane is defined by the equation 3x + 4y = 12 and the flight of the second plane is defined by the equation y = -(3/4)x + 10. If the planes continue along the same paths, is there any danger of them colliding? (Assume the planes are flying at the same altitude)

Solution:

To determine if the planes will collide, we need to find the point of intersection of the two lines.

Substitute -(3/4)x + 10 for y in the first equation and solve for y.

3x + 4[-(3/4)x + 10] = 12

3x - 3x + 40 = 12

40 = 12 is a false statement. Therefore, there is no solution. There is no chance of the two planes colliding if they keep flying along the same paths.

Another way to solve this is to get the slope of each line. If the slopes are the same without the same y – intercept, then the lines are parallel and therefore the planes cannot collide.

3x + 4y = 12

4y = 12 - 3x

y = 4 - (3/4)x. Notice the slope of both lines is -3/4. The y – intercepts are different. Therefore there is no chance of the planes colliding if they continue along their same path.