Question

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

Answer 1

See a solution process below:

Explanation:

First, this line is in Standard Form for a Linear Equation. The standard form of a linear equation is: `color(red)(A)x + color(blue)(B)y = color(green)(C)`

Where, if at all possible, `color(red)(A)`, `color(blue)(B)`, and `color(green)(C)`are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

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

The slope of an equation in standard form is: `m = -color(red)(A)/color(blue)(B)`

Substituting the values from the equation gives the slope of this line as:

`m = (-color(red)(2))/color(blue)(-5) = 2/5`

The slope of a perpendicular line is the negative inverse of the slope.

So, if we can the slope of the perpendicular line `m_p` it's slope would be:

`m_p = -1/m`

Substituting the slope we calculated and calculating the perpendicular slope gives:

`m_p = -1/(2/5) = -5/2`

The slope of any line perpendicular to the line in the problem will have a slope of:

`-5/2`

Answer 2

Slope of perpendicular: `color(blue)(-5/2)`

Explanation:

Any linear equation in standard form: `Ax+By=C`
has a slope of `m=-A/B`
`rArr 2x-5y=3`
has a slope of `m=2/5`

The perpendicular to a line with a slope of `m`
has a slope of `-1/m`
`rArr` any perpendicular to `2x-5y=3`
has a slope of `-5/2`