Answer
Point-slope form is $y+2=-\frac{2}{3}\left( x-6 \right)$.
Slope-intercept form is $y=-\frac{2}{3}x+2$.
Work Step by Step
The point slope form of a line is obtained with the help of the slope of the line and any one point that lies in the line.
For a line with slope m and passing through the point $\left( {{x}_{1}},{{y}_{1}} \right)$, point-slope form is given as follows:
$y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Consider $\left( {{x}_{1}},{{y}_{1}} \right)=\left( 6,-2 \right)$
Since the point-slope form is given by $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$, replace the values of m and $\left( {{x}_{1}},{{y}_{1}} \right)$ in it:
$\begin{align}
& y-\left( -2 \right)=-\frac{2}{3}\left( x-6 \right) \\
& y+2=-\frac{2}{3}\left( x-6 \right) \\
& y+2=-\frac{2}{3}x+4
\end{align}$
Subtract $2$ from both sides of the equation:
$\begin{align}
& y=-\frac{2}{3}x+4-2 \\
& =-\frac{2}{3}x+2
\end{align}$
The point-slope form of line is $y=-\frac{2}{3}x+2$.
The slope-intercept form of a line is provided by $y=mx+b$ ; here, m is the slope and b is the y-intercept, and the y-intercept is the y-coordinate of a point where the line intersects the y-axis.
Change the values of m and $\left( x,y \right)=\left( 6,-2 \right)$ in $y=mx+b$ to find the value of b.
$\begin{align}
& -2=-\frac{2}{3}\left( 6 \right)+b \\
& b=2
\end{align}$
Now, change the values of b and m in $y=mx+b$.
So
$y=-\frac{2}{3}x+2$
The slope-intercept form of the equation is $y=-\frac{2}{3}x+2$.
Therefore, the point-slope form of the line with $m=-\frac{2}{3}$ and passing through the point $\left( 6,-2 \right)$ is $y+2=-\frac{2}{3}\left( x-6 \right)$, and the slope-intercept form is $y=-\frac{2}{3}x+2$.
