Answer
$y=\frac{3}{2}x-\frac{13}{2}$
Work Step by Step
Write $3x-2y=6$ in slope-intercept form by isolating $y$ to obtain:
$-2y=-3x+6
\\\frac{-2y}{-2} = \frac{-3x+6}{-2}
\\y=\frac{3}{2}x-3$
This means that the equation $3x-2y=6$ is equivalent to $y=\frac{3}{2}x-3$.
The graph of this equation is a line whose slope is $\frac{3}{2}$.
RECALL:
Two lines are parallel if they have the same slope but different y-intercepts.
Thus, the slope of the line we are looking for is also $\frac{3}{2}$.
This means that the tentative equation of the line we are looking for is $y=\frac{3}{2}x+b$.
The line contains the point $(3, -2)$.
This means that the coordinates of this point satisfy the equation of the line we are looking for.
Substitute the x and y values of this point into the tentative equation to obtain:
$y=\frac{3}{2}x+b
\\-2=\frac{3}{2}(3) + b
\\-2=\frac{9}{2}+b
\\-2-\frac{9}{2}=b
\\-\frac{4}{2} - \frac{9}{2}=b
\\-\frac{13}{2} = b$
Therefore, the equation of the line that passes through the point $(3, -2)$ and is parallel to $3x-2y=6$ is $\color{blue}{y=\frac{3}{2}x-\frac{13}{2}}$.
