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

Feb 5, 2017

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{\frac{4}{3}} \left(x - \textcolor{red}{5}\right)$

Or

$\left(y + \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{4}{3}} \left(x + \textcolor{red}{1}\right)$

#### Explanation:

First, we need to determine the slope. 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 equation gives:

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

$m = \frac{- 8}{-} 6 = \frac{- 2 \times 4}{- 2 \times 3} = \frac{\cancel{- 2} \times 4}{\cancel{- 2} \times 3} = \frac{4}{3}$

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 we calculated and the first point to give:

$\left(y - \textcolor{red}{5}\right) = \textcolor{b l u e}{\frac{4}{3}} \left(x - \textcolor{red}{5}\right)$

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

$\left(y - \textcolor{red}{- 3}\right) = \textcolor{b l u e}{\frac{4}{3}} \left(x - \textcolor{red}{- 1}\right)$

$\left(y + \textcolor{red}{3}\right) = \textcolor{b l u e}{\frac{4}{3}} \left(x + \textcolor{red}{1}\right)$