# How do you determine the equation of a line passing through (2, -3) that is perpendicular to the line 4x-y=22?

Dec 15, 2016

Equation of a line passing through $\left(2 , - 3\right)$ that is perpendicular to the line $4 x - y = 22$ is $x + 4 y + 10 = 0$

#### Explanation:

Equation of a line that is perpendicular to a line $a x + b y + c = 0$, will be of type

$b x - a y + k = 0$

Note that coefficients of $x$ and $y$ have changed and sign before $y$ has been changed, keeping sign before $x$ to be same.

Hence, equation of a line that is perpendicular to a line $4 x - y = 22$, will be of type

$x + 4 y + k = 0$

As it passes through $\left(2 , - 3\right)$, we have $2 + 4 \times \left(- 3\right) + k = 0$ or $2 - 12 + k = 0$ i.e. $k = 12 - 2 = 10$.

Hence, equation of a line passing through $\left(2 , - 3\right)$ that is perpendicular to the line $4 x - y = 22$ is $x + 4 y + 10 = 0$.
graph{(4x-y-22)(x+4y+10)=0 [-7.89, 9.89, -5.564, 3.325]}
Note: Equation of a line that is parallel to a line $a x + b y + c = 0$, will be of type $a x + b y + k = 0$, i.e. no change in coefficients or sign. Only constant term changes. Also observe that slopes of parallel llines are equal but product of slopes of perpendicular lines is $- 1$.