# How do you write an equation in point slope and slope intercept form given (3,-6) and (-3,2)?

Apr 29, 2017

See the entire solution process below:

#### Explanation:

First, 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:

$m = \frac{\textcolor{red}{2} - \textcolor{b l u e}{- 6}}{\textcolor{red}{- 3} - \textcolor{b l u e}{3}} = \frac{\textcolor{red}{2} + \textcolor{b l u e}{6}}{\textcolor{red}{- 3} - \textcolor{b l u e}{3}} = \frac{8}{-} 6 = - \frac{4}{3}$

We can now use the point-slope formula to write an equation. 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 values from the first point in the problem gives:

$\left(y - \textcolor{red}{- 6}\right) = \textcolor{b l u e}{- \frac{4}{3}} \left(x - \textcolor{red}{3}\right)$

$\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{- \frac{4}{3}} \left(x - \textcolor{red}{3}\right)$

We can write another equation in point-slope form by substituting the slope we calculated and the values from the second point in the problem giving:

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- \frac{4}{3}} \left(x - \textcolor{red}{- 3}\right)$

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- \frac{4}{3}} \left(x + \textcolor{red}{3}\right)$

We can solve this equation for $y$ to transform the equation to slope-intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y - \textcolor{red}{2} = \left(\textcolor{b l u e}{- \frac{4}{3}} \cdot x\right) + \left(\textcolor{b l u e}{- \frac{4}{3}} \cdot \textcolor{red}{3}\right)$

$y - \textcolor{red}{2} = - \frac{4}{3} x + \left(- 4\right)$

$y - \textcolor{red}{2} = - \frac{4}{3} x - 4$

$y - \textcolor{red}{2} + 2 = - \frac{4}{3} x - 4 + 2$

$y - 0 = - \frac{4}{3} x - 2$

$y = \textcolor{red}{- \frac{4}{3}} x - \textcolor{b l u e}{2}$