Answer
The equation of the line passing through $\left( 4,\ -7 \right)$ and parallel to the line with equation $3x+y-9=0$ in point-slope and slope-intercept form is $y+3x-5=0$ and $y=-3x+5$ respectively.
Work Step by Step
The point on the line is given as $\left( 4,\ -7 \right)$ and the equation of the line parallel to the required line is $3x+y-9=0$.
We know that if two lines are parallel, their slopes are the same.
Thus, the required line and the given line equation have the same slope.
The slope of the given line equation will be:
$\begin{align}
& m=-\frac{a}{b} \\
& m=-\frac{3}{1} \\
& m=-3
\end{align}$
Therefore, the slope of the required line equation will also be $-3$.
Point-Slope form:
It is known that the equation of the line by point-slope form will be:
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Now, substitute the given data in the above equation to get the desired equation of the line:
$\begin{align}
& y-\left( -7 \right)=-3\left( x-4 \right) \\
& y+7=-3\left( x-4 \right) \\
& y+7=-3x+12 \\
& y+3x-5=0
\end{align}$
Slope-Intercept form:
Now, the above equation can be rewritten as:
$y=-3x+5$
Compare the above equation with the general equation of slope-intercept form:
$\begin{align}
& y=mx+c \\
& \text{Therefore,} \\
& m=-3 \\
& c=5.
\end{align}$
