# How do you write an equation of a line that passes through points (-1,3), (2,-3)?

Feb 14, 2017

See the entire 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:

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

We can now use the point-slope formula to write the equation for a 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 gives:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x - \textcolor{red}{- 1}\right)$

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{1}\right)$

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

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

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

We can also solve this equation for $y$ 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.

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

$y + \textcolor{red}{3} = - 2 x + 4$

$y + \textcolor{red}{3} - 3 = - 2 x + 4 - 3$

$y + 0 = - 2 x + 1$

$y = \textcolor{red}{- 2} x + \textcolor{b l u e}{1}$

Three equations which solve this problem are:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- 2} \left(x + \textcolor{red}{1}\right)$

Or

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

Or

$y = \textcolor{red}{- 2} x + \textcolor{b l u e}{1}$