Write the equation for the perpendicular bisector of the line segment connecting the points $(3,2)$ and $(-1,7)$ in the form $y = mx + b$.
Note: The perpendicular bisector of the line segment $\overline{AB}$ is the line that passes through the midpoint of $\overline{AB}$ and is perpendicular to $\overline{AB}$.
To find the midpoint of the line segment connecting the points $(3,2)$ and $(-1,7)$, we average the $x$-coordinates and average the $y$-coordinates: $$\left(\frac{3+(-1)}{2},\frac{2+7}{2}\right)=(1,4.5).$$The slope of the line segment connecting these two points is $$\frac{7-2}{-1-3}=\frac{5}{-4}=-\frac{5}{4}.$$The negative reciprocal of $-\frac{5}{4}$ is $\frac{4}{5}$, so the slope of the perpendicular bisector is $\frac{4}{5}$. Since the line passes through the point $(1,4.5)$, an equation for the perpendicular bisector is $$y-4.5=\frac{4}{5}(x-1).$$We can write this in slope-intercept form: \begin{align*}
y-4.5&=\frac45(x-1)\\
y&=\frac45x-\frac45+\frac95\\
y&=\frac45x+\frac55\\
y&=\boxed{\frac45x+1}.
\end{align*}