# How do you write an equation of a line given (5, -2); (-16, 4)?

Aug 24, 2017

See a solution process below:

#### Explanation:

First, we need to 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:

$m = \frac{\textcolor{red}{4} - \textcolor{b l u e}{- 2}}{\textcolor{red}{- 16} - \textcolor{b l u e}{5}} = \frac{\textcolor{red}{4} + \textcolor{b l u e}{2}}{\textcolor{red}{- 16} - \textcolor{b l u e}{5}} = \frac{6}{-} 21 = - \frac{2}{7}$

We can now use the point-slope formula to write an equation for the line. The point-slope form of a linear equation is: $\left(y - \textcolor{b l u e}{{y}_{1}}\right) = \textcolor{red}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

Where $\left(\textcolor{b l u e}{{x}_{1}} , \textcolor{b l u e}{{y}_{1}}\right)$ is a point on the line and $\textcolor{red}{m}$ is the slope.

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

$\left(y - \textcolor{b l u e}{- 2}\right) = \textcolor{red}{- \frac{2}{7}} \left(x - \textcolor{b l u e}{5}\right)$

$\left(y + \textcolor{b l u e}{2}\right) = \textcolor{red}{- \frac{2}{7}} \left(x - \textcolor{b l u e}{5}\right)$

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

$\left(y - \textcolor{b l u e}{4}\right) = \textcolor{red}{- \frac{2}{7}} \left(x - \textcolor{b l u e}{- 16}\right)$

$\left(y - \textcolor{b l u e}{4}\right) = \textcolor{red}{- \frac{2}{7}} \left(x + \textcolor{b l u e}{16}\right)$