Question

A line passes through `(2 ,3 )` and `( 4, 5 )`. A second line passes through `( 7, 4 )`. What is one other point that the second line may pass through if it is parallel to the first line?

Answer 1

See a solution process below:

Explanation:

First, we need to determine the slope of the first line. The slope can be found by using the formula: `m = (color(red)(y_2) - color(blue)(y_1))/(color(red)(x_2) - color(blue)(x_1))`

Where `m` is the slope and (`color(blue)(x_1, y_1)`) and (`color(red)(x_2, y_2)`) are the two points on the line.

Substituting the values from the points in the problem gives:

`m = (color(red)(5) - color(blue)(3))/(color(red)(4) - color(blue)(2)) = 2/2 = 1`

Because the problem states the two lines are parallel therefore the second line will have the same slope of `m = 1`

Because we have a slope and one point for the second line we can write an equation for the line using the point-slope formula. The point-slope form of a linear equation is: `(y - color(blue)(y_1)) = color(red)(m)(x - color(blue)(x_1))`

Where `(color(blue)(x_1), color(blue)(y_1))` is a point on the line and `color(red)(m)` is the slope.

Substituting the values from the point in the problem and slope we calculated gives:

`(y - color(blue)(4)) = color(red)(1)(x - color(blue)(7))`

`y - color(blue)(4) = x - color(blue)(7)`

`y - color(blue)(4) + 4 = x - color(blue)(7) + 4`

`y - 0 = x - 3`

`y = x - 3`

To find another point on this second line substitute any number you want for `x` (other than 7) and calculate `y`.

I am going to choose an easy number - `0`

`y = x - 3` becomes:

`y = 0 - 3`

`y = -3`

So the point would be: `(0, -3)`