Question

Two trains leave a train station traveling different directions. The first train travels 12 miles west, then 6 miles north. The second train travels 20 miles east, then 35 miles north.
a. The train station is the origin. What is the coordinate of each train?
b. Using the city center and the stop point of the first train, what is the slope of the line? Is it horizontal, vertical or neither? Write the equation of the line in slope-intercept form and standard form.
c. The city wants to build a train station 2 miles directly north of the first train station. They are going to build a train track parallel to the path the first train would’ve traveled if it were a direct route from the city center. What would be the equation of the line in slope intercept form of the new track?

Answer 1

Draw a diagram to illustrate the problem as shown below.
The origin (0, 0) is the train station.

Part a.
The first train travels 12 miles west and 6 miles north to arrive at A.
It's coordinate is (-12, 6).

The second train travels 20 miles east and 35 miles north to arrive at B.
It's coordinate is (20, 35).

Part b.
The slope of the line from the origin to A is
m = (6 - 0)/(-12 - 0) = -1/2.
The slope is neither vertical nor horizontal.
The y-intercept is 0.

The line OA is given in slope-intercept form by
y = -(1/2)x
In standard form, the line is
y - 6 = -(1/2)*(x - (-12))
or
y - 6 = -(1/2)*(x + 12)

Part c.
The new train station is located at (0, 2) because it is 2 miles north of the old station.
If a new track is built parallel to OA, then it should have a slope of -1/2.
Because it's y-intercept is 2, its equation in slope-intercept form is
y = -(1/2)x + 2.