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

##### 1 Answer

Take the negative reciprocal of the given line's slope. The new slope will be

#### Explanation:

Lines that are perpendicular will have *negative reciprocal* slopes. Meaning, if one line's slope is

Why? A line's slope is equal to its rise over its run—also written as

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

#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_"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".