y(b-x) = a
   by-xy = a
     -xy = -by+a 
      xy = by-a

Let xy = Y and y = X

       Y = bX-a

Let b = m and -a = c

       Y = mX+c

  Let y(y-a) = Y, X = x

             Y = X-b

Let m = 1 and -b=c
             
             Y = mX+c

You already have m in there so it would be confusing to have m represent two
different constants, so let's change the problem with another letter besides
m. I'll use p:

      px2 + ny2 = x
        px2 - x = -ny2

Let px2 - x = Y and let y2 = X

              Y = -nX 
              Y = -nX + 0

Let -n=m and let c=0

              Y = mX + c

Edwin