Answer
$y = -\dfrac{1}{2}x - \dfrac{3}{2}$
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 $\dfrac{1}{2}$.
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:
$(2)(m) = -1$
Divide both sides by $2$:
$m = -\dfrac{1}{2}$
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 $(1, -2)$ to plug into the formula:
$y - (-2) = -\dfrac{1}{2}(x - 1)$
Simplify the equation:
$y + 2 = -\dfrac{1}{2}(x - 1)$
We are asked to give the equation either in standard form or slope-intercept form.
Let's rewrite this equation in slope-intercept form. First, we distribute the terms on the right side of the equation:
$y + 2 = -\dfrac{1}{2}x + \dfrac{1}{2}$
Isolate $y$ by subtracting $2$ from each side of the equation:
$y = -\dfrac{1}{2}x - \dfrac{3}{2}$
