# What is the slope of any line perpendicular to the line passing through (-2,5) and (-3,1)?

Mar 14, 2018

See a solution process below:

#### Explanation:

First, we need to determine the slope of the line going through the two points in the problem. The slope can be found by using the formula: $m = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

Where $m$ is the slope and ($\textcolor{b l u e}{{x}_{1} , {y}_{1}}$) and ($\textcolor{red}{{x}_{2} , {y}_{2}}$) are the two points on the line.

Substituting the values from the points in the problem gives:

$m = \frac{\textcolor{red}{1} - \textcolor{b l u e}{5}}{\textcolor{red}{- 3} - \textcolor{b l u e}{- 2}} = \frac{\textcolor{red}{1} - \textcolor{b l u e}{5}}{\textcolor{red}{- 3} + \textcolor{b l u e}{2}} = \frac{- 4}{-} 1 = 4$

Now, Let's call the slope of a perpendicular line: $\textcolor{b l u e}{{m}_{p}}$

The slope of a line perpendicular to a line with slope $\textcolor{red}{m}$ is the negative inverse, or:

$\textcolor{b l u e}{{m}_{p}} = - \frac{1}{\textcolor{red}{m}}$

Substituting the slope for the line in the problem gives:

$\textcolor{b l u e}{{m}_{p}} = \frac{- 1}{\textcolor{red}{4}} = - \frac{1}{4}$