Question

A line passes through `(3 ,4 )` and `(4 ,7 )`. A second line passes through `(7 ,9 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

Any point on the line `y = 3x-12` is a viable answer.

Explanation:

Slope formula from two points:

`m = (y_2-y_1)/(x_2-x_1)`

Slope of the given line:

`m = (7-4)/(4-3)=3/1=3`

The equation of this line using point-slope form:

`y-y_1=m(x-x_1)=>`

`y-7=3(x-4)`

`y-7=3x-12`

`y=3x-5`

Since the unknown line is parallel to this one, it has the same slope. Using the newfound slope, we can craft an equation for the new line:

`y-y_1=m(x-x_1)=>`

`y-9=3(x-7)`

`y-9=3x-21`

`y=3x-12`

This is the "second line" that the question mentions. Now, we just need to pick a point anywhere on this line by plugging in an `x`-value and get a `y`-value back. I'll pick 6 for an `x`-value:

`y=3x-12=>`

`y=3(6)-12`

`y=18-12`

`y=6`

A point on the line is `(6, 6)`.