# How do you write the equation in point slope form given (-1/2, 1/2) and (1/4, 3/4)?

May 9, 2017

See a solution process below:

#### Explanation:

First, we need to determine the slope of the line which passes through the two points in the problem. 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:

$m = \frac{\textcolor{red}{\frac{3}{4}} - \textcolor{b l u e}{\frac{1}{2}}}{\textcolor{red}{\frac{1}{4}} - \textcolor{b l u e}{- \frac{1}{2}}} = \frac{\textcolor{red}{\frac{3}{4}} - \textcolor{b l u e}{\frac{1}{2}}}{\textcolor{red}{\frac{1}{4}} + \textcolor{b l u e}{\frac{1}{2}}} = \frac{\textcolor{red}{\frac{3}{4}} - \left(\frac{2}{2} \times \textcolor{b l u e}{\frac{1}{2}}\right)}{\textcolor{red}{\frac{1}{4}} + \left(\frac{2}{2} \times \textcolor{b l u e}{\frac{1}{2}}\right)} =$

$\frac{\textcolor{red}{\frac{3}{4}} - \textcolor{b l u e}{\frac{2}{4}}}{\textcolor{red}{\frac{1}{4}} + \textcolor{b l u e}{\frac{2}{4}}} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1 \times 4}{4 \times 3} = \frac{4}{12} = \frac{1}{3}$

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:

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

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

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

$\left(y - \textcolor{red}{\frac{3}{4}}\right) = \textcolor{b l u e}{\frac{1}{3}} \left(x - \textcolor{red}{\frac{1}{4}}\right)$