# How do you write an equation of a line given (-3, 1) and (-2, -5)?

Apr 13, 2017

See the entire solution process below:

#### Explanation:

First, we must 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:

$m = \frac{\textcolor{red}{- 5} - \textcolor{b l u e}{1}}{\textcolor{red}{- 2} - \textcolor{b l u e}{- 3}} = \frac{\textcolor{red}{- 5} - \textcolor{b l u e}{1}}{\textcolor{red}{- 2} + \textcolor{b l u e}{3}} = - \frac{6}{1} = - 6$

We can now use the point slope formula to find an equation for the line. 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:

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

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

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

$\left(y - \textcolor{red}{- 5}\right) = \textcolor{b l u e}{- 6} \left(x - \textcolor{red}{- 2}\right)$

Solution 2) $\left(y + \textcolor{red}{5}\right) = \textcolor{b l u e}{- 6} \left(x + \textcolor{red}{2}\right)$

We can also solve this for $y$ to put the equation in slope intercept form. The slope-intercept form of a linear equation is: $y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and $\textcolor{b l u e}{b}$ is the y-intercept value.

$y + \textcolor{red}{5} = \left(\textcolor{b l u e}{- 6} \times x\right) + \left(\textcolor{b l u e}{- 6} \times \textcolor{red}{2}\right)$

$y + \textcolor{red}{5} = - 6 x - 12$

$y + \textcolor{red}{5} - 5 = - 6 x - 12 - 5$

$y + 0 = - 6 x - 17$

Solution 3) $y = \textcolor{red}{- 6} x - \textcolor{b l u e}{17}$