# How do you write an equation of a line that contains the given point (-5,5) and is perpendicular to the given line y=-5x+9?

Jan 14, 2017

$\left(y - 5\right) = \frac{1}{5} \left(x + 5\right)$

or

$y = \frac{1}{5} x + 6$

#### Explanation:

To find the equation of the line perpendicular to the given line and going through the given point we will use the point-slope formula. We have been given a point so what we are missing is the slope.

The given line is in slope-intercept form.

The slope-intercept form of a linear equation is:

$y = \textcolor{red}{m} x + \textcolor{b l u e}{b}$

Where $\textcolor{red}{m}$ is the slope and color(blue)(b is the y-intercept value.

Therefore we know the slope of the given line, $\textcolor{red}{m}$ is $\textcolor{red}{- 5}$ - it is the coefficient of the $x$ term.

A perpendicular line will have a slope which is the negative inverse of this line, or color(red)(m = - 1/-5 = 1/5.

We can now use the point-slope formula to write the equation for the perpendicular line.

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 point we were given and the slope we calculated gives:

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

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

or, we can solve for $y$ to convert to the more familiar slope-intercept form:

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

$y - \textcolor{red}{5} = \textcolor{b l u e}{\frac{1}{5}} x + 1$

$y - \textcolor{red}{5} + 5 = \textcolor{b l u e}{\frac{1}{5}} x + 1 + 5$

$y - 0 = \textcolor{b l u e}{\frac{1}{5}} x + 6$

$y = \textcolor{b l u e}{\frac{1}{5}} x + 6$