# How do you determine the equation of a line passing through (2, -3) that is perpendicular to the 4x-y=22?

Dec 18, 2016

$y + 3 = - \frac{1}{4} x + \frac{1}{2}$

or

$y = - \frac{1}{4} x - \frac{5}{2}$

#### Explanation:

To determine this perpendicular line we can use the point-slope formula. We already have a point (2, -3), now we need to determine the slope. The slope of a perpendicular line is the negative inverse of the line it is perpendicular to. If we convert the equation we are given to the slope-intercept form we will have a slope we can take the negative inverse of.

$4 x - y = 22$

$4 x - y \textcolor{red}{+ y - 22} = 22 \textcolor{red}{+ y - 22}$

$4 x - 22 = y$

$y = 4 x - 22$

The slope-intercept form is $\textcolor{red}{y = m x + b}$ where $\textcolor{red}{m}$ is the slope. Therefore the slope of the line we were give is $m = 4$

Given a slope $\textcolor{red}{m}$ the negative inverse is $\textcolor{red}{- \frac{1}{m}}$

For our slope of $m = 4$ the negative inverse is $- \frac{1}{4}$

Now we can use this slope and the point we were given and use the point-slope formula to find the equation of the perpendicular line.

The point-slope formula states: $\textcolor{red}{\left(y - {y}_{1}\right) = m \left(x - {x}_{1}\right)}$
Where $\textcolor{red}{m}$ is the slope and $\textcolor{red}{\left({x}_{1} , {y}_{1}\right)}$ is a point the line passes through.

Substituting the information we have gives:

$y - - 3 = - \frac{1}{4} \left(x - 2\right)$

$y + 3 = - \frac{1}{4} x + \frac{2}{4}$

$y + 3 = - \frac{1}{4} x + \frac{1}{2}$

If we want to convert to the more standard slope-intercept form we would get:

$y + 3 \textcolor{red}{- 3} = - \frac{1}{4} x + \frac{1}{2} \textcolor{red}{- 3}$

$y = - \frac{1}{4} x + \frac{1}{2} - \frac{6}{2}$

$y = - \frac{1}{4} x - \frac{5}{2}$