Question

A line passes through `(6 ,2 )` and `(5 ,7 )`. A second line passes through `(3 ,8 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

The point `(1,18)` is one possible point on the second line.

(but there is literally an infinite number of possible points)

Explanation:

The lines are parallel so they will have the same gradient (slope). First we find the gradient of the first line:

`m=(y_2-y_1)/(x_2-x_1)=(7-2)/(5-6)=5/(-1)=-5`

To find the y-intercept we can write the equation of the line in gradient-intercept form:

`y=mx+c`

We rearrange this equation and plug in the `x` and `y` values of the point we have:

`c=y-mx=8-(-5)(3)=8+15=23`

The equation of the second line, then, is `y=-5x+23`. Any set of `x` and `y` that makes that equation true is a point on the second line.

Let's use `x=1`, then `y=-5(x)+23=18`, so the point `(1,18)` is on the second line.