# How do you write the point slope form of the equation given (-1,3) and (-2,5)?

Jan 26, 2017

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{1}\right)$

Or

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{2}\right)$

#### Explanation:

First, we need to 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 problem gives:

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

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

$m = \frac{2}{-} 1 = - 2$

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.

We can substitute the slope and the first point from the problem to give:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{- 1}\right)$

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{1}\right)$

We can also substitute the slope and the second point from the problem to give:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{- 2}\right)$

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{2}\right)$