Question

What is the equation of the line that passes through `(5,7)` and is perpendicular to the line that passes through the following points: `(1,3),(-2,8)`?

Answer 1

`(y - color(red)(7)) = color(blue)(3/5)(x - color(red)(5))`

Or

`y = 3/5x + 4`

Explanation:

First, we will find the slope of the perpendicular 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 two points from the problem gives:

`m = (color(red)(8) - color(blue)(3))/(color(red)(-2) - color(blue)(1))`

`m = 5/-3`

A perpendicular line will have a slope (let's call it `m_p`) which is the negative inverse of the line or `m_p = -1/m`

Substituting gives `m_p = - -3/5 = 3/5`

Now that we have the slope of the perpendicular line and one point we can use the point-slope formula to find the equation. The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

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

Substituting the perpendicular slope we calculated and using the point from the problem gives:

`(y - color(red)(7)) = color(blue)(3/5)(x - color(red)(5))`

Or, if we solve for `y`:

`y - color(red)(7) = (color(blue)(3/5) xx x) - (color(blue)(3/5) xx color(red)(5))`

`y - color(red)(7) = 3/5x - 3`

`y - color(red)(7) + 7 = 3/5x - 3 + 7`

`y - 0 = 3/5x + 4`

`y = 3/5x + 4`