# How do you write the point-slope form of an equation of the line that passes through the given point (-5,7), (0, 1/2)?

May 9, 2017

See a 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:

$m = \frac{\textcolor{red}{\frac{1}{2}} - \textcolor{b l u e}{7}}{\textcolor{red}{0} - \textcolor{b l u e}{- 5}} = \frac{\textcolor{red}{\frac{1}{2}} - \textcolor{b l u e}{7}}{\textcolor{red}{0} + \textcolor{b l u e}{5}} = \frac{\textcolor{red}{\frac{1}{2}} - \left(\frac{2}{2} \times \textcolor{b l u e}{7}\right)}{5} =$

$\frac{\textcolor{red}{\frac{1}{2}} - \frac{14}{2}}{5} = \frac{- \frac{13}{2}}{5} = - \frac{13}{10}$

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 values from the first point in the problem gives:

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{13}{10}} \left(x - \textcolor{red}{- 5}\right)$

$\left(y - \textcolor{red}{7}\right) = \textcolor{b l u e}{- \frac{13}{10}} \left(x + \textcolor{red}{5}\right)$

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

$\left(y - \textcolor{red}{\frac{1}{2}}\right) = \textcolor{b l u e}{- \frac{13}{10}} \left(x - \textcolor{red}{0}\right)$

Or

$\left(y - \textcolor{red}{\frac{1}{2}}\right) = \textcolor{b l u e}{- \frac{13}{10}} x$