Question

How do you find the slope that is perpendicular to the line `4x+5y= -5`?

Answer 1

The slope of the line perpendicular to the line `4x+5y=-5` is `5/4`.

Explanation:

Let's start with the original equation:

`4x+5y=-5`

From here, we can manipulate the equation into the slope-intercept form. We first move the `4x` over to the right side by subtracting `4x` from both sides:

`5y=-5-4x`

Next, we divide both sides by `5` to isolate the `y` variable:

`y=(-5-4x)/5`

We then simplify the right portion of the equation:

`y=-5/5-(4x)/5`

And further simplification follows:

`y=-1-(4x)/5`

We then rearrange the entire equation to clearly show the equation in slope-intercept form:

`y=-(4x)/5-1`

Now that we have the equation in slope-intercept form, we can clearly see that `-4/5` is the slope here.

From here, it is easy. The product of a slope and its perpendicular slope is always `-1`. (Proof: perpendicular lines have negative reciprocal slope)

If we set `a` to be the perpendicular slope of `-4/5`, then

`-4/5a=-1`

Then, we can isolate `a` by dividing by `-4/5` on both sides and then simplifying the result:

`a=(-1)/(-4/5)=-1*-5/4=5/4`

Thus, the slope of the line perpendicular to the line `4x+5y=-5` is `5/4`.

Answer 2

The slope that is perpendicular to the line `4x + 5y = -5` is `5/4`.

Explanation:

The slope perpendicular to a line is the opposite reciprocal of the slope of the given line.

For example, if the slope of a line is 2, the slope of a line perpendicular to the line with slope 2 must be `-1/2`. "Opposite" refers to the opposite sign (e.x. the opposite of `-5` is `5`), and "reciprocal" just means 1 over whatever you're given (e.x. the reciprocal of 20 is just `1/20`).

Because in your case the line is in standard form (i.e. `ax + by = c`, such that `a` is a positive integer, and `b` and `c` are integers), the slope of the line is given by the equation: `m=-a/b`. Thus, the slope of the line is `-4/5`.

You can also find the slope, `m`, by putting the equation into slope-intercept form (`y=mx+b`):
1. `4x + 5y = -5`
2. `5y = -4x - 5`
3. `y = -4/5x -1`
4. `m=-4/5`

The slope of a line perpendicular to this line must be the opposite reciprocal, so `-(1/(-4/5)) = 5/4`. The slope that is perpendicular to the line `4x + 5y = -5` is `5/4`.