# How do you find the slope that is perpendicular to the line 2x +3y = 5?

Jan 17, 2017

Take the negative reciprocal of the given line's slope. The new slope will be $\frac{3}{2}$.

#### Explanation:

Lines that are perpendicular will have negative reciprocal slopes. Meaning, if one line's slope is $m$, then a perpendicular line will have a slope of $- \frac{1}{m}$.

Why? A line's slope is equal to its rise over its run—also written as $m = \frac{\Delta y}{\Delta x}$. If we rotate that line 90° counterclockwise (making it perpendicular to its old self), the run (to the right) becomes a rise (up), and the rise (up) becomes a backwards run (to the left):

In math terms:

$\Delta {y}_{\text{new"=Delta x" "and" "Delta x_"new}} = - \Delta y$

thus

m_"new"=(Delta y_"new")/(Delta x_"new")=(Delta x)/(-Delta y)=-(Delta x)/(Delta y)=-1/m

(Note: if we rotate this new line another 90° (180° total from the beginning), this 3rd line will have a slope of $\frac{- 1}{- \frac{1}{m}}$, which simplifies to $m$—the same slope of the first line, which is what we would expect.)

Okay, great—so what's the slope of $2 x + 3 y = 5$? If we rearrange this into slope-intercept form, we get

$y = - \frac{2}{3} x + \frac{5}{3}$,

meaning that for every step of "2 down", we have a step of "3 right".

The negative reciprocal of the slope $m = - \frac{2}{3}$ is

${m}_{\text{new}} = - \frac{1}{m} = \frac{- 1}{- \frac{2}{3}} = \frac{3}{2}$,

meaning that, for a perpendicular line, a step of "3 up" comes with a step of "2 right".