Question

Consider the line 3x+5y=-1 What is the slope of a line parallel to this line? What is the slope of a line perpendicular to this line?

Answer 1

Parallel: m = -3/5
Perpendicular: m = 5/3

Explanation:

First let's find the slope of the line.
To do so, we must make
`y` the subject of the line's equation. The slope would then be the coefficient of
`x`.

`3x + 5y = -1 \text{ // -3x} \\3x + 5y - 3x = -3x - 1\\\to 5y = -3x - 1 \text{ // :5}\\\\\frac{5y}5 = \frac{-3x - 1}5\\\\y = -\frac35x - \frac15`

Therefore, the slope of the line is
`m = -\frac35`.

Parallel lines have similar slopes, therefore a line parallel to the line 3x + 5y = -1 would have a slope of -3/5.
If a line is perpendicular to our line, then its slope would be the negative invert of our line's slope. Therefore, its slope would be
`m = -\left(-\frac35\right)^(-1) \to -\left(-\frac53\right) \to \frac53`