Question

Which of the following is an equation for the line that passes through$\left(\mathbf{2}\mathbf{,}-\mathbf{5}\right)$ and is perpendicular to $\mathbf{2}x+\mathbf{4}y=\mathbf{8}$?

$$\begin{gathered} \mathbf{A}\mathit{y}\mathbf{=}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{10} \\ \mathbf{B}\mathbf{y}\mathbf{=}\mathbf{-}\frac{1}{2}\mathbf{x}\mathbf{-}\mathbf{4} \\ \mathbf{C}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{9} \\ \mathbf{D}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{x}\mathbf{-}\mathbf{1} \end{gathered}$$

Short answer

The equation for line passing through$\left(2,-5\right)$ and perpendicular to$2x+4y=8$ is $y=\mathbf{2}x-\mathbf{9}$.

Step-by-step solution

Step 1. Write equation in slope-intercept form.

Write an equation of line in slope-intercept form.

$$\begin{gathered} 2x+4y-2x=8-2x \\ \frac{4y}{4}=\frac{8-2x}{4} \\ y=-\frac{1}{2}x+2 \end{gathered}$$

Step 2. Find slope of a line.

The slope of line with equation$y=-\frac{1}{2}x+2$ is $-\frac{1}{2}$. The slope of perpendicular line is the opposite reciprocal of$-\frac{1}{2}$ or 2.

Step 3. Find the y-intercept.

Substitute 2 for m and 2 for x and$-5$ for y in $y=mx+c$.

$$\begin{gathered} -5=\left(2\right)\left(2\right)+c \\ c=-5-4 \\ =-9 \end{gathered}$$

The equation in the form of$y=mx+c$ is $y=2x-9$.

Therefore, option C is the required answer.