Answer
$4x-y=-4$
Work Step by Step
Using $\left( \dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2} \right)$ or the Midpoint Formula, then the midpoint of the line segment joining the points $
(3,-1) \text{ and } (-5,1)
$ is
\begin{array}{l}\require{cancel}
\left( \dfrac{3+(-5)}{2},\dfrac{-1+1}{2} \right)
\\\\
\left( \dfrac{-2}{2},\dfrac{0}{2} \right)
\\\\
\left( -1,0 \right)
.\end{array}
Using $m=\dfrac{y_1-y_2}{x_1-x_2}$ or the Slope Formula, then \begin{array}{l}\require{cancel}
m=\dfrac{-1-1}{3-(-5)}
\\\\
m=\dfrac{-1-1}{3+5}
\\\\
m=\dfrac{-2}{8}
\\\\
m=-\dfrac{1}{4}
.\end{array}
Taking the negative reciprocal of $m$, then the slope of the perpendicular bisector is $m_p=
4
.$
Using $
(-1,0)
$ and $m_p=
4
,$ the equation of the perpendicular bisector is
\begin{array}{l}\require{cancel}
y-0=4(x-(-1))
\\\\
y=4(x+1)
\\\\
y=4x+4
\\\\
-4x+y=4
\\\\
4x-y=-4
.\end{array}
