Answer
$$y=x-3$$
Work Step by Step
First, we have to find slope of the line connecting $A(1,4)$ and $B(7,-2)$
$m_{AB}=\frac{-2-4}{7-1}=\frac{-6}{6}=-1$
To find the equation, we have to first find slope of the perpendicular bisector of this line (Let the bisector be $b$). According to one of the slope rules, product of perpendicular line slopes is $-1$:
$m_b \times m_{AB}=-1$
$m_b\times -1 = -1$
$m_b=1$
To write the equation, we also need any point, located on this line. Since we know, that it is a bisector, then one of its point will be in the middle of $A$ and $B$ (Let the point be $C$):
$C(x, y)=(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})=(\frac{1+7}{2}, \frac{4-2}{2})=(4, 1)$
We have $C(4,1)$
$m=\frac{y-y_0}{x-x_0}$
$y-y_0=m(x-x_0)$
$y-1=1(x-4)$
$$y=x-3$$
