Question

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

Answer 1

Take the negative reciprocal of the given line's slope. The new slope will be `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 `-1/m`.

Why? A line's slope is equal to its rise over its run—also written as `m=[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_"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 `(-1)/(-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 `2x+3y=5`? If we rearrange this into slope-intercept form, we get

`y=-2/3 x+5/3`,

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

The negative reciprocal of the slope `m=-2/3` is

`m_"new"=-1/m=(-1)/(- 2/3)=3/2`,

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