Answer
The linear equation in slope-intercept form is $ y=3x+3$.
Work Step by Step
We know that if two lines are perpendicular, then the product of their slopes must be $-1$. Considering that ${{m}_{1}}$ and ${{m}_{2}}$ are slopes, the condition of perpendicularity is:
${{m}_{1}}\times {{m}_{2}}=-1$
Therefore, the slope of the line $ x+3y-6=0$ can be computed as:
$\begin{align}
& x+3y-6=0 \\
& 3y=-x+6 \\
& y=-\frac{x}{3}+2
\end{align}$
So, the slope of the line is ${{m}_{1}}=-\frac{1}{3}$.
And,
$\begin{align}
& -\frac{1}{3}\times {{m}_{2}}=-1 \\
& {{m}_{2}}=3
\end{align}$
Therefore, the equation of the line is written as given below:
$\begin{align}
& y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\
& y-0=3\left( x+1 \right) \\
& y=3x+3
\end{align}$
Thus, the linear equation in slope-intercept form is $ y=3x+3$.
