Answer
The linear function in slope-intercept form is $ y=-5x-2$.
Work Step by Step
Let us consider the coordinates $\left( 0,-2 \right)$ as $\left( {{x}_{1}},{{y}_{1}} \right)$.
We have to find the value of the slope of the equation $ x-5y-20=0$:
$\begin{align}
& x-5y-20=0 \\
& x-20=5y \\
& y=\frac{x}{5}-\frac{20}{5}
\end{align}$
And compare this equation with the form $ y=mx+c $ to get $ m=\frac{1}{5}$.
Here, ${{m}_{1}}$ is the slope of the required line that is perpendicular to $ x-5y-20=0$.
And for the perpendicular lines, the condition of the slope is $ m\cdot {{m}_{1}}=-1$.
So,
$\begin{align}
& {{m}_{1}}=\frac{-1}{m} \\
& =\frac{-1}{\frac{1}{5}} \\
& =-5
\end{align}$
Then, for the equation of the function to be determined:
$\begin{align}
& \frac{y-{{y}_{1}}}{x-{{x}_{1}}}={{m}_{1}} \\
& \frac{y-\left( -2 \right)}{x-0}=-5 \\
& \frac{y+2}{x}=-5
\end{align}$
And simplify it further, to get
$\begin{align}
& y+2=-5x \\
& y=-5x-2 \\
\end{align}$
Thus, the slope-intercept form is $ y=-5x-2$.
