Answer
Point slope form $y+7=-2(x-4)$.
Slope intercept form $y=-2x+1$.
or $f(x)=-2x+1$
Work Step by Step
The given equation of line is
$\Rightarrow x-2y=3$
Subtract $x$ from both sides.
$\Rightarrow x-2y-x=3-x$
Simplify.
$\Rightarrow -2y=3-x$
Divide both sides by $-2$.
$\Rightarrow \frac{-2y}{-2}=\frac{3-x}{-2}$
Simplify.
$\Rightarrow y=-\frac{3}{2}+\frac{x}{2}$
Rearrange.
$\Rightarrow y=\frac{1}{2}x-\frac{3}{2}$
This is in the standard form of slope-intercept form
$y=mx+c$
Where $m$ is a slope of the line.
We have $m_1=\frac{1}{2}$.
Two lines are perpendicular if their slopes are negative reciprocals.
$m_2=-\frac{1}{\frac{1}{2}}$
Simplify.
$m_2=-2$
If the line passes through a point $(x_1,y_1)$, then point-slope form of the perpendicular line's equation is:
$\Rightarrow y-y_1=m_2(x-x_1)$
We have the point $(x_1,y_1)=(4,-7)$.
Plug all values into point-slope form.
$\Rightarrow y-(-7)=(-2)(x-4)$
Simplify.
$\Rightarrow y+7=-2(x-4)$
Now subtract $7$ from both sides.
$\Rightarrow y+7-7=-2(x-4)-7$
Simplify.
$\Rightarrow y=-2x+8-7$
$\Rightarrow y=-2x+1$
The above equation is in slope intercept form.
$f(x)=-2x+1$.
