What is the equation of the line that passes through (-2,1)  and is perpendicular to the line that passes through the following points: (1,4),(-2,3) ?

Mar 30, 2016

First step is to find the slope of the line through $\left(1 , 4\right)$ and $\left(- 2 , 3\right)$, which is $\frac{1}{3}$. Then all lines perpendicular to this line have slope $- 3$. Finding the y-intercept tells us the equation of the line we are looking for is $y = - 3 x - 5$.

Explanation:

Slope of the line through $\left(1 , 4\right)$ and $\left(- 2 , 3\right)$ is given by:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{3 - 4}{\left(- 2\right) - 1} = \frac{- 1}{- 3} = \frac{1}{3}$

If the slope of a line is $m$, lines perpendicular to it have slope $- \frac{1}{m}$. In this case, the slope of the perpendicular lines will be $- 3$.

The form of a line is $y = m x + c$ where $c$ is the y-intercept, so if we substitute in $- 3$ as the slope and the given points $\left(- 2 , 1\right)$ for $x$ and $y$, we can solve to find the value of $c$:

$1 = - 3 \left(- 2\right) + c$

$c = - 5$

So the equation of the line we want is $y = - 3 x - 5$