# How do you write an equation in point slope form given (2p, -q), (p, p-q)?

Apr 24, 2017

See the entire solution process below:

#### Explanation:

First, you must determine the slope of the line. 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:

$m = \frac{\textcolor{red}{p - q} - \textcolor{b l u e}{- q}}{\textcolor{red}{p} - \textcolor{b l u e}{2 p}} = \frac{\textcolor{red}{p - q} + \textcolor{b l u e}{q}}{\textcolor{red}{p} - \textcolor{b l u e}{2 p}} = \frac{p - 0}{-} p = \frac{p}{-} p = - 1$

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.

Now, substitute the slope you calculated and the values for the first point in the problem to give:

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

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

Or, substitute the slope you calculated and the values for the second point in the problem to give:

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