Question

Two airplanes are flying to the same airport. Their positions are shown in the graph. Write a system of linear equations that represents this situation. Solve the system by elimination to justify your answer. Airplane #1 is on (2, 4). Airplane #2 is on (15, 9). The airport is on (6, 12).
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Answer 1

To write the system we need the slope of each line and at least one point on the line. The two lines to consider will be the lines connecting the location of each plane to the airport they are flying to. It is also worth noting that the coordinates of the airport represent the point of intersection of the two lines and thus the solution to the system.

1. slope of the line connecting airplane one and the airport: m = 2 you can see this clearly if you graph the two points. From airplane 1 location we rise 8 units and move to the right 4 units to get to the airport. Slope is defined as rise over run: so 8 divided by 4 = 2(the slope) Now substitute the slope and the point (2,4) into point-slope form of a line:

y - 4 = 2(x -4) the standard form of this equation is 2x - y = 0

2. slope of the line connecting airplane 2 and the airport: m = -$\frac{1}{3}$ To find this slope, simply observe the vertical change of down 3 and a horizontal shift of right 9 from the airport to airplane 2. Now substitute this slope and and the point (15,9) into point-slope form of a line:

y - 9 = $- \frac{1}{3}$(x - 15) the standard form of this equation is:
x + 3y = 42

Let's write the system:
2x - y = 0
x + 3y = 42
Multiply the first equation by 3 to get the new system
6x - 3y = 0
x + 3y = 42 add these two equations to get an equation in terms of x

7x = 42 thus x = 6 and substituting this value into 2x - y = 0 we see y = 12
In other words, we have proven that the location of the airport is in fact the solution to our system.




PS: You just have to do a little algebra to get from point-slope form of the two equations to standard form. I did not show this process, but if you need it just let me know... thanks