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,


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