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

Mar 4, 2018

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: $\textcolor{red}{A} x + \textcolor{b l u e}{B} y = \textcolor{g r e e n}{C}$

Where, if at all possible, $\textcolor{red}{A}$, $\textcolor{b l u e}{B}$, and $\textcolor{g r e e n}{C}$are integers, and A is non-negative, and, A, B, and C have no common factors other than 1

$\textcolor{red}{2} x - \textcolor{b l u e}{5} y = \textcolor{g r e e n}{3}$

The slope of an equation in standard form is: $m = - \frac{\textcolor{red}{A}}{\textcolor{b l u e}{B}}$

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

$m = \frac{- \textcolor{red}{2}}{\textcolor{b l u e}{- 5}} = \frac{2}{5}$

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:

${m}_{p} = - \frac{1}{m}$

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

${m}_{p} = - \frac{1}{\frac{2}{5}} = - \frac{5}{2}$

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

$- \frac{5}{2}$

Mar 4, 2018

Slope of perpendicular: $\textcolor{b l u e}{- \frac{5}{2}}$

#### Explanation:

Any linear equation in standard form: $A x + B y = C$
has a slope of $m = - \frac{A}{B}$
$\Rightarrow 2 x - 5 y = 3$
has a slope of $m = \frac{2}{5}$

The perpendicular to a line with a slope of $m$
has a slope of $- \frac{1}{m}$
$\Rightarrow$ any perpendicular to $2 x - 5 y = 3$
has a slope of $- \frac{5}{2}$