# How do you write an equation in point slope form given (8,3), (-1/3,-2)?

Mar 21, 2017

See the entire solution process below:

#### Explanation:

First, we need to determine the slope f 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:

$m = \frac{\textcolor{red}{- 2} - \textcolor{b l u e}{3}}{\textcolor{red}{- \frac{1}{3}} - \textcolor{b l u e}{8}} = \frac{- 5}{\textcolor{red}{- \frac{1}{3}} - \textcolor{b l u e}{\frac{24}{3}}} = \frac{\frac{- 5}{1}}{\frac{- 25}{3}} = \frac{- 5 \times 3}{1 \times - 25} =$

$\frac{- 1 \times 3}{1 \times - 5} = \frac{- 3}{-} 5 = \frac{3}{5}$

Now, we can use the point-slope formula to write the 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 first point from the problem gives:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x - \textcolor{red}{8}\right)$

Or. we can substitute the slope we calculated and the second point from the problem giving:

$\left(y - \textcolor{red}{- 2}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x - \textcolor{red}{- \frac{1}{3}}\right)$

$\left(y + \textcolor{red}{2}\right) = \textcolor{b l u e}{\frac{3}{5}} \left(x + \textcolor{red}{\frac{1}{3}}\right)$