Question

The equation of a line is `3y+2x=12`. What is the slope of the line perpendicular to the given line?

Answer 1

The perpendicular slope would be `m=3/2`

Explanation:

If we convert the equation to slope-intercept form, `y=mx+b` we can determine the slope this line.

`3y+2x=12`

Begin by using the additive inverse to isolate the `y-term`.

`3y cancel(+2x) cancel(-2x)=12-2x`

`3y=-2x +12`

Now use the multiplicative inverse to isolate the `y`

`(cancel3y)/cancel3=(-2x)/3 +12/3`

`y=-2/3x +4`

For this equation of the line the slope is `m=-2/3`

The perpendicular slope to this would be the inverse reciprocal.

The perpendicular slope would be `m=3/2`

Answer 2

`+3/2`

Explanation:

Convert to the standard form `y=mx+c` where `m` is the gradient.

The gradient of a line perpendicular to this one is:

`(-1)xx1/m`

Divide both sides by `color(blue)(3)` so that `3y" becomes "y`

`color(brown)(3y+2x=12" "->" "3/(color(blue)(3))y+2/(color(blue)(3))x=12/(color(blue)(3))`

`y+2/3x=4`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~
Subtract `2/3x` from both sides

`y=-2/3x+4`

Thus the gradient of this line is `-2/3`

So the gradient of the line perpendicular to it is:

`(-1)xx (color(white)(..)1color(white)(..))/(-2/3)`

`+3/2`