How do you derive the equation of the line that has slope 1/4 and passes through the point (1,-1/2)?

Dec 28, 2016

$\left(y + \frac{1}{2}\right) = \frac{1}{4} \left(x - 1\right)$

or

$y = \frac{1}{4} x - \frac{3}{4}$

Explanation:

To derive this equation we can use the point slope formula.

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 data from the problem gives:

$\left(y - \textcolor{red}{- \frac{1}{2}}\right) = \textcolor{b l u e}{\frac{1}{4}} \left(x - \textcolor{red}{1}\right)$

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

If we want to convert this to slope-intercept form we can solve for $y$ as follows:

$y + \textcolor{red}{\frac{1}{2}} = \textcolor{b l u e}{\frac{1}{4}} x - \left(\textcolor{b l u e}{\frac{1}{4}} \cdot \textcolor{red}{1}\right)$

$y + \textcolor{red}{\frac{1}{2}} = \textcolor{b l u e}{\frac{1}{4}} x - \frac{1}{4}$

$y + \textcolor{red}{\frac{1}{2}} - \textcolor{g r e e n}{\frac{1}{2}} = \textcolor{b l u e}{\frac{1}{4}} x - \frac{1}{4} - \textcolor{g r e e n}{\frac{1}{2}}$

$y + 0 = \textcolor{b l u e}{\frac{1}{4}} x - \frac{1}{4} - \left(\frac{2}{2} \cdot \textcolor{g r e e n}{\frac{1}{2}}\right)$

$y = \textcolor{b l u e}{\frac{1}{4}} x - \frac{1}{4} - \frac{2}{4}$

$y = \textcolor{b l u e}{\frac{1}{4}} x - \frac{3}{4}$