Question

How do you find the slope of a line that is a) parallel to the line given and b)perpendicular to the line with the given equation `2x-5y = -7`?

Answer 1

If a line is expressed by the equation `y=mx+c` then `m` is the slope and `c` is the intercept.

Starting with the equation given, first add `5y` to both sides to get:

`2x=5y-7`

Add `7` to both sides to get:

`2x+7 =5y`

Divide both sides by `5` to get:

`2/5x+7/5=y`

In other words `y = 2/5x + 7/5`

Comparing with `y=mx+c`, we see that the slope is `2/5` and the intercept is `7/5`.

Any line parallel to this line will have the same slope: `2/5`.

Any line perpendicular to a line of slope `2/5` will have a slope of `-5/2`.

There are several ways to see this.

For example, consider the following steps:
(1) Reflect the original line in the line `y = x`. This will have the effect of swapping `x` and `y` in the original equation.
(2) Reflect the line in the `x` axis, given by the equation `y = 0`. This will have the effect of inverting the sign of the `y` coordinate.

The total effect of (1) followed by (2) is a rotation of `pi/2`. That is, the resulting line is perpendicular to the line you started with.

Starting with your original line this will result in a line with equation `-2y-5x=-7`, which can be reformulated as:
`y = -(5/2)x+7/2`. This has slope `-5/2` as predicted.