# What is the equation of the line passing through (91,-41) and (-25,7)?

Mar 15, 2017

$\left(y + \textcolor{red}{41}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x - \textcolor{red}{91}\right)$

Or

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x + \textcolor{red}{25}\right)$

#### Explanation:

First, we must determine the slope of the line passing through these 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:

$m = \frac{\textcolor{red}{7} - \textcolor{b l u e}{- 41}}{\textcolor{red}{- 25} - \textcolor{b l u e}{91}} = \frac{\textcolor{red}{7} + \textcolor{b l u e}{41}}{\textcolor{red}{- 25} - \textcolor{b l u e}{91}} = \frac{48}{- 116} = \frac{4 \times 12}{4 \times 29} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \times 12}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \times - 29}$

$m = - \frac{12}{29}$

Now, use the point-slope formula to find an equation for the line passing through the two points. The point-slope formula states: $\left(y - \textcolor{red}{{y}_{1}}\right) = \textcolor{b l u e}{m} \left(x - \textcolor{red}{{x}_{1}}\right)$

Where $\textcolor{b l u e}{m}$ is the slope and $\textcolor{red}{\left(\left({x}_{1} , {y}_{1}\right)\right)}$ is a point the line passes through.

Substituting the slope we calculated and the first point gives:

$\left(y - \textcolor{red}{- 41}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x - \textcolor{red}{91}\right)$

$\left(y + \textcolor{red}{41}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x - \textcolor{red}{91}\right)$

We can also substitute the slope we calculated and the second point giving:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x - \textcolor{red}{- 25}\right)$

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{12}{29}} \left(x + \textcolor{red}{25}\right)$