Answer
$y = -2x + 4$
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.
This given line is written in standard form, so we want to rewrite this line in slope-intercept form, which is given by the following formula:
$y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-intercept.
We can rewrite our equation by isolating our $y$ term.
First, we subtract $x$ from each side of the equation:
$-2y = -x - 5$
Dividing both sides of the equation by $-2$ to isolate $y$:
$y = \dfrac{1}{2}x + \dfrac{5}{2}$
Therefore, the slope of the given line is the coefficient of $x$, so the slope is $\dfrac{1}{2}$.
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 $m$ be the slope of the perpendicular line:
$(\frac{1}{2})(m) = -1$
Multiply both sides by $2$:
$m = -2$
Let us plug this slope and the point we are given into the point-slope form of the equation, which is given by the formula:
$y - y_1 = m(x - x_1)$, where $m$ is the slope of the line and $(x_1, y_1)$ is a point on that line.
Let us use the point $(0, 4)$ to plug into the formula:
$y - 4 = -2(x - 0)$
Simplify the equation:
$y - 4= -2x$
We are asked to give the equation either in standard form or slope-intercept form.
Let us rewrite this equation in slope-intercept form. To accomplish this, we isolate $y$ by adding $4$ to each side of the equation:
$y = -2x + 4$
