Answer
$3y=-6x+1$
Work Step by Step
$\underline {\textbf{Background :}}$
$\textbf{(1)}$ The slope-intercept form of a line's equation is
$y=mx+b$ where $m$ is the slope of the line and $b$ is the $y-$intercept.
$\textbf{(2)}$ When one line has a slope of $m$, a perpendicular line has a slope of $\frac{1}{m}$
$\textbf{(3)}$ The equation of the line that passes through the point $(x_1,y_1)$ and has slope $m$ is $y-y_1=m(x-x_1)$
$\textbf{(I)}\underline {\textbf{ Transform the line $4x-8y=1$ to slope-intercept form:}}$
$-8y=-4x+1\Rightarrow y=\frac{1}{2}x-\frac{1}{8}$
The slope of this line is $m=\frac{1}{2}$
Hence we are looking for a line which is perpendicular to the line above.
This means the slope of the required line is $m=-2$
$\textbf{(II)}\underline {\textbf{ The required line has slope $m=-2$}}$
$\underline {\textbf{ and passes through}(\frac{1}{2},-\frac{2}{3})}$
The equation of the required line is
$y-(-\frac{2}{3})=-2(x-\frac{1}{2})$
$y+\frac{2}{3}=-2x+1$
$3y+2=-6x+3$
$3y=-6x+1$
Therefore, the equation of the line we are looking for is:
$3y=-6x+1$
