Question

How do you find the slope of a line that is perpendicular to the line `y= x/-5 - 7`?

Answer 1

`y = x/(-5)-7 = -1/5x-7`

is the equation of a line with slope `-1/5` and intercept `-7`

If a line has slope `m`, then any line perpendicular to it will have slope `-1/m`.

So a line perpendicular to your line of slope `-1/5` will have slope `5`.

One way I like to picture this is as follows:

Suppose a line is given in slope-intercept form as

`y = mx+c`

where `m` is the slope and `c` the intercept.

If we reflect that line in the line `y=x` then the effect will be to swap the `x` and `y` coordinates, giving a line with equation:

`x = my+c`

If we reflect this new line in the `x`-axis then the result is to reverse the sign of the `y` coordinate, resulting in a line with equation:

`x = -my + c`

The total geometric effect of these two reflections is a rotation around the origin by a right angle, that is our new line is perpendicular to the old line.

Next let's rearrange into slope, intercept format. First subtract `c` from both sides to get:

`-my = x - c`

Then divide both sides by `-m` to get:

`y = -1/mx + c/m`