When solving the equation such as 2/3x = 4/5 , why would you multiply both sides of the equation by the reciprocal of 2/3 ?

Well you’re solving for “x,” so you’ll need to isolate this variable. This means we need to “get rid of” the fraction 2/3 that is attached to it. The faction is attached by the operation multiplication, so to remove it we must divide it on both sides. And division with fractions requires the reciprocal. (Try watching an explanation if you’re confused on why division requires reciprocals).