Question

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

Answer 1

Convert to the slope-intercept form and then, using the slope from this form of the equation, take the negative inverse of the slope.

Explanation:

First, we need to find the slope of the line given in the problem. To do this we need to put this equation in the slope-intercept form.

The slope-intercept form of a linear equation is:

`y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b` is the y-intercept value.

We can convert the equation from our problem to this form by solving for `y`:

`5x - 3y = 2`

`5x - color(red)(5x) - 3y = - color(red)(5x) + 2`

`0 - 3y = - color(red)(5x) + 2`

`-3y = -5x + 2`

`(-3y)/color(red)(-3) = (-5x + 2)/color(red)(-3)`

`(color(red)(cancel(color(black)(-3)))y)/cancel(color(red)(-3)) = (-5x)/color(red)(-3) + 2/color(red)(-3)`

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

The slope of this line is `m = 5/3`

For any line with slope `color(red)(m)` any line perpendicular to this line will have the negative inverse of the original line or `-1/color(red)(m)`

Therefore the slope of a line perpendicular to the line in the problem will be:

`-3/5`