
We know A.x1 + B.y1 = C and A.x2 + B.y2 = C
so combining these gives A.x1 + B.y1 = A.x2 + B.y2
=> A(x1  x2) = B(y2  y1)
The thing to remember is that A, B and C can all be scaled by a constant as much as we like  i.e. if {A, B, C} represent the line, so does {2A, 2B, 2C}  so we just have to find a solution to the above equation. One solution happens to be A = y2  y1 and B = x1  x2
You can then work out C by putting those values into one of the original equations (the example you give uses the first one). 