The midpoint of the segment is

((-1+7)/2, (1+5)/2) = (3, 3)

The slope of the given segment is

(5-1)/(7-(-1)) = 4/8 = 1/2

The slope of the bisector will be

-1/(1/2) = -2

One way to write the equation you want is

y = -2(x-3) + 3

Another way to write it is

2x + y = 9

((-1+7)/2, (1+5)/2) = (3, 3)

The slope of the given segment is

(5-1)/(7-(-1)) = 4/8 = 1/2

The slope of the bisector will be

-1/(1/2) = -2

One way to write the equation you want is

y = -2(x-3) + 3

Another way to write it is

2x + y = 9