# How do you write the point slope form of the equation given (5,-6) and (2,3)?

Apr 30, 2017

See the 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:

$m = \frac{\textcolor{red}{3} - \textcolor{b l u e}{- 6}}{\textcolor{red}{2} - \textcolor{b l u e}{5}} = \frac{\textcolor{red}{3} + \textcolor{b l u e}{6}}{\textcolor{red}{2} - \textcolor{b l u e}{5}} = \frac{9}{-} 3 = - 3$

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.

Substituting the slope we calculated and the values from the first point in the problem gives:

$\left(y - \textcolor{red}{- 6}\right) = \textcolor{b l u e}{- 3} \left(x - \textcolor{red}{5}\right)$

Solution 1) $\left(y + \textcolor{red}{6}\right) = \textcolor{b l u e}{- 3} \left(x - \textcolor{red}{5}\right)$

We can also substitute the slope we calculated and the values from the second point in the problem giving:

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

Apr 30, 2017

$y = - 3 x + 9$

#### Explanation:

First, find the slope between the points. You can find slope using:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Plug into the formula:

$m = \frac{3 - \left(- 6\right)}{2 - 5} = \frac{9}{-} 3 = - 3$

Now we have the slope. We can now plug into the point-slope form which is:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

It doesn't really matter which point we plug in so I'll plug in $\left(2 , 3\right)$ since I don't like dealing with negative numbers:

$y - 3 = - 3 \left(x - 2\right)$

Distribute the $- 3$:

$y - 3 = - 3 x + 6$

Add $3$ to both sides:

$y = - 3 x + 9$