# How do you write an equation for the line that passes through (3, -5) and (-2, 1) in point-slope form and point-intercept form?

##### 1 Answer
Mar 14, 2017

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

Or

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{- \frac{6}{5}} \left(x + \textcolor{red}{2}\right)$

And

$y = \textcolor{red}{- \frac{6}{5}} x - \textcolor{b l u e}{\frac{7}{5}}$

#### 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:

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

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:

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

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

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

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{- \frac{6}{5}} \left(x - \textcolor{red}{- 2}\right)$

$\left(y - \textcolor{red}{1}\right) = \textcolor{b l u e}{- \frac{6}{5}} \left(x + \textcolor{red}{2}\right)$

We can solve this equation to put the formula 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.

$y - \textcolor{red}{1} = \left(\textcolor{b l u e}{- \frac{6}{5}} \times x\right) + \left(\textcolor{b l u e}{- \frac{6}{5}} \times \textcolor{red}{2}\right)$

$y - \textcolor{red}{1} = - \frac{6}{5} x + \left(- \frac{12}{5}\right)$

$y - \textcolor{red}{1} = - \frac{6}{5} x - \frac{12}{5}$

$y - \textcolor{red}{1} + 1 = - \frac{6}{5} x - \frac{12}{5} + 1$

$y - 0 = - \frac{6}{5} x - \frac{12}{5} + \left(1 \times \frac{5}{5}\right)$

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

$y = \textcolor{red}{- \frac{6}{5}} x - \textcolor{b l u e}{\frac{7}{5}}$