find the equation of the perpendicular bisector of the line joining A(2, -1) B(8, 3)

To be a perpendicular bisector it must be both perpendicular and a bisector.

To be a bisector, it must pass through the midpoint of the line segment AB.

A formula for finding the midpoint of A(x₁, y₁) and B(x₂, y₂) is  Midpoint  =  ( (x₁ + x₂)/2, (y₁ + y₂)/2 ).

--->   Midpoint  =  ( (2 + 8)/2, (-1 + 3)/2 )  =  (5, 1)

To be perpendicular, the slope of the line must be the negative reciprocal of the original line.

A formula for slope is:  m  =  (y₂ - y₁) / (x₂ - x₁)

--->   Slope  =  (3 - -1) / (8 - 2)  =  4/6  =  2/3

--->  Negative reciprocal of that slope:  m  =  -3/2

Point-slope equation of a line:  y - y₁  =  m(x - x₁)

--->   Point  =  (5, 1)         Slope  =  -3/2

--->   y - 1  =  -3/2(x - 5)

--->   2y - 2  =  -3(x - 5)

--->   2y - 2  =  -3x + 15

--->   2y  =  -3x + 17

--->   3x + 2y  =  17