# How do you find the slope of a line that is perpendicular to the line y= x/-5 - 7?

Jun 7, 2015

$y = \frac{x}{- 5} - 7 = - \frac{1}{5} x - 7$

is the equation of a line with slope $- \frac{1}{5}$ and intercept $- 7$

If a line has slope $m$, then any line perpendicular to it will have slope $- \frac{1}{m}$.

So a line perpendicular to your line of slope $- \frac{1}{5}$ will have slope $5$.

One way I like to picture this is as follows:

Suppose a line is given in slope-intercept form as

$y = m x + c$

where $m$ is the slope and $c$ the intercept.

If we reflect that line in the line $y = x$ then the effect will be to swap the $x$ and $y$ coordinates, giving a line with equation:

$x = m y + c$

If we reflect this new line in the $x$-axis then the result is to reverse the sign of the $y$ coordinate, resulting in a line with equation:

$x = - m y + c$

The total geometric effect of these two reflections is a rotation around the origin by a right angle, that is our new line is perpendicular to the old line.

Next let's rearrange into slope, intercept format. First subtract $c$ from both sides to get:

$- m y = x - c$

Then divide both sides by $- m$ to get:

$y = - \frac{1}{m} x + \frac{c}{m}$