# How do you write an equation of the line given (5, 2) and (-7, 3)?

Jan 31, 2017

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- \frac{1}{12}} \left(x - \textcolor{red}{5}\right)$

Or

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{1}{12}} \left(x + \textcolor{red}{7}\right)$

Or

$y = - \frac{1}{12} x + \frac{29}{12}$

#### Explanation:

Given two points we can use the point-slope formula to find an equation for a line.

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:

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

$m = \frac{1}{-} 12 = - \frac{1}{12}$

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 use the first point and the slope we calculate to give:

$\left(y - \textcolor{red}{2}\right) = \textcolor{b l u e}{- \frac{1}{12}} \left(x - \textcolor{red}{5}\right)$

We can also use the second point and the slope we calculate to give:

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{1}{12}} \left(x - \textcolor{red}{- 7}\right)$

$\left(y - \textcolor{red}{3}\right) = \textcolor{b l u e}{- \frac{1}{12}} \left(x + \textcolor{red}{7}\right)$

We can also solve this for $y$ to give an equation in slope-intercept form:

$y - \textcolor{red}{3} = \left(\textcolor{b l u e}{- \frac{1}{12}} \times x\right) + \left(\textcolor{b l u e}{- \frac{1}{12}} \times \textcolor{red}{7}\right)$

$y - \textcolor{red}{3} = \textcolor{b l u e}{- \frac{1}{12}} x - \frac{7}{12}$

$y - \textcolor{red}{3} + 3 = \textcolor{b l u e}{- \frac{1}{12}} x - \frac{7}{12} + 3$

$y - 0 = \textcolor{b l u e}{- \frac{1}{12}} x - \frac{7}{12} + \frac{36}{12}$

$y = - \frac{1}{12} x + \frac{29}{12}$