Answer
$\text{slope-intercept form: }
y=-\dfrac{5}{6}x+\dfrac{13}{3}
\\\\
\text{standard form: }
5x+6y=26$
Work Step by Step
Using $y-y_1=\dfrac{y_1-y_2}{x_1-x_2}(x-x_1)$ or the two-point form of linear equations, the equation of the line passing through $\left(
-2,6
\right)$ and $\left(
4,1
\right)$ is
\begin{array}{l}\require{cancel}
y-6=\dfrac{6-1}{-2-4}\left( x-(-2) \right)
\\\\
y-6=\dfrac{5}{-6}\left( x+2 \right)
\\\\
y-6=-\dfrac{5}{6}\left( x+2 \right)
.\end{array}
In the form $y=mx+b$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
y-6=-\dfrac{5}{6}x-\dfrac{5}{3}
\\\\
y=-\dfrac{5}{6}x-\dfrac{5}{3}+6
\\\\
y=-\dfrac{5}{6}x-\dfrac{5}{3}+\dfrac{18}{3}
\\\\
y=-\dfrac{5}{6}x+\dfrac{13}{3}
.\end{array}
In the form $Ax+By=C$, the equation above is equivalent to
\begin{array}{l}\require{cancel}
y=-\dfrac{5}{6}x+\dfrac{13}{3}
\\\\
6(y)=6\left( -\dfrac{5}{6}x+\dfrac{13}{3} \right)
\\\\
6y=-5x+26
\\\\
5x+6y=26
.\end{array}
Hence, the different forms of the equation of the line with the given conditions are
\begin{array}{l}\require{cancel}
\text{slope-intercept form: }
y=-\dfrac{5}{6}x+\dfrac{13}{3}
\\\\
\text{standard form: }
5x+6y=26
.\end{array}
