# How do you write an equation of a line given (-7, 9) and (3, -7)?

Mar 30, 2017

See the entire solution process below:

#### Explanation:

First, we must 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:

$m = \frac{\textcolor{red}{- 7} - \textcolor{b l u e}{9}}{\textcolor{red}{3} - \textcolor{b l u e}{- 7}} = \frac{\textcolor{red}{- 7} - \textcolor{b l u e}{9}}{\textcolor{red}{3} + \textcolor{b l u e}{7}} = - \frac{16}{10} = - \frac{8}{5}$

Now, we can use the point-slope formula to write and 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:

$\left(y - \textcolor{red}{9}\right) = \textcolor{b l u e}{- \frac{8}{5}} \left(x - \textcolor{red}{- 7}\right)$

Solution 1: $\left(y - \textcolor{red}{9}\right) = \textcolor{b l u e}{- \frac{8}{5}} \left(x + \textcolor{red}{7}\right)$

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

$\left(y - \textcolor{red}{- 7}\right) = \textcolor{b l u e}{- \frac{8}{5}} \left(x - \textcolor{red}{3}\right)$

Solution 2: $\left(y + \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{8}{5}} \left(x - \textcolor{red}{3}\right)$

Or, we can solve the first or second equation for $y$ and write 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.

$y + \textcolor{red}{7} = \left(\textcolor{b l u e}{- \frac{8}{5}} \times x\right) - \left(\textcolor{b l u e}{- \frac{8}{5}} \times \textcolor{red}{3}\right)$

$y + \textcolor{red}{7} = - \frac{8}{5} x - \left(- \frac{24}{5}\right)$

$y + \textcolor{red}{7} = - \frac{8}{5} x + \frac{24}{5}$

$y + \textcolor{red}{7} - 7 = - \frac{8}{5} x + \frac{24}{5} - 7$

$y + 0 = - \frac{8}{5} x + \frac{24}{5} - \left(\frac{5}{5} \times 7\right)$

$y + 0 = - \frac{8}{5} x + \frac{24}{5} - \frac{35}{5}$

Solution 3: $y = \textcolor{red}{- \frac{8}{5}} x - \textcolor{b l u e}{\frac{11}{5}}$