Answer
$y + 2 = -\frac{1}{3}(x - 3)$
Work Step by Step
With lines that are perpendicular to each other, the product of their slopes is $-1$; one slope is the negative reciprocal of the other. If we want to find the slope of a line that is perpendicular to a given line, we must first find the slope of the given line.
The line given is in the slope-intercept form, which is given by the formula:
$y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
Therefore, the slope of the given line is the coefficient of $x$, so the slope is $3$.
Let us set up an equation to find the slope of the line that is perpendicular to the given line by multiplying the two slopes to yield $-1$. Let $x$ be the slope of the perpendicular line:
$(3)(x) = -1$
Divide both sides by $3$:
$x = -\frac{1}{3}$
Now that we have our slope and a point on the line, we can write the equation for this line using the point-slope form of an equation, which is given by the formula:
$y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on that line.
Let's plug our values into this equation:
$y - (-2) = -\frac{1}{3}(x - 3)$
Simplify the left side of the equation:
$y + 2 = -\frac{1}{3}(x - 3)$
