# What is the slope of any line perpendicular to the line passing through (7,23) and (1,2)?

Apr 2, 2017

See the entires solution process below.

#### Explanation:

First, we need to determine the slope of the line passing through the two points. 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:

$m = \frac{\textcolor{red}{2} - \textcolor{b l u e}{23}}{\textcolor{red}{1} - \textcolor{b l u e}{7}} = \frac{- 21}{-} 6 = \frac{- 3 \times 7}{- 3 \times 2} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 3}}} \times 7}{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 3}}} \times 2} = \frac{7}{2}$

So the slope of any line perpendicular to this line, let's call this slope ${m}_{p}$, will be the negative inverse of the slope of the line it is perpendicular to, or:

${m}_{p} = - \frac{1}{m}$

Therefore, for the problem:

${m}_{p} = - \frac{2}{7}$