Precalculus, Chapter 1, Review Exercises, Section Review Exercises, Problem 36

Determine the equation of the line perpendicular to the line $3x - y = -4$ and containing the point $(-2,4)$. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer.

We know that if the two lines are perpendicular, the product of their slopes is $-1$. We write the equation $3x - y = -4$ in slope intercept form to find the slope.


$
\begin{equation}
\begin{aligned}

3x - y =& -4
\\
-y =& -3x - 4
\\
y =& 3x + 4

\end{aligned}
\end{equation}
$


The slope is $3$. The slope of the other line is $\displaystyle \frac{-1}{3}$.

Using Point Slope Form,


$
\begin{equation}
\begin{aligned}

y - y_1 =& m (x - x_1)
&&
\\
\\
y - 4 =& \frac{-1}{3} [x - (-2)]
&& \text{Substitute } m = \frac{-1}{3}, x = -2 \text{ and } y = 4
\\
\\
y =& \frac{-1}{3}x - \frac{2}{3} + 4
&& \text{Simplify}
\\
y =& \frac{-1}{3}x + \frac{10}{3}
&& \text{Slope Intercept Form}
\\
x + 3y =& 10
&& \text{General Form}

\end{aligned}
\end{equation}
$