# How do you write an equation of a line with points (-5,3), (0,-7)?

Feb 9, 2017

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

or

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

Or

$y = \textcolor{b l u e}{- 2} x - 7$

#### Explanation:

We can use the point-slope formula to write an equation. However, we must first 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 problem gives:

$m = \frac{\textcolor{red}{- 7} - \textcolor{b l u e}{3}}{\textcolor{red}{0} - \textcolor{b l u e}{- 5}}$

$m = \frac{\textcolor{red}{- 7} - \textcolor{b l u e}{3}}{\textcolor{red}{0} + \textcolor{b l u e}{5}} = - \frac{10}{5} = - 2$

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 giving:

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

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

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

$\left(y - \textcolor{red}{- 7}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{0}\right)$

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

Or, we can solve this equation for $y$ to give an equation in slope-intercept form:

$y + \textcolor{red}{7} = \textcolor{b l u e}{- 2} x$

$y + \textcolor{red}{7} - 7 = \textcolor{b l u e}{- 2} x - 7$

$y + 0 = \textcolor{b l u e}{- 2} x - 7$

$y = \textcolor{b l u e}{- 2} x - 7$