Simplify [2/xy + x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2]

To simplify [2/xy + x/(x^2*y)] / [ (x-y)/xy^2 +
(x-xy)/x^2*y^2]

, we multiply both numerator and
denominator by x^2y^2 .

Numerator *x^2y^2 = 2x^2y^2/xy
+x*x^2y^2/x^2y

Numerator *x^2y^2 =
2xy+xy

Numerator*x^2y^2 =
3xy...................(1)

Denominator*x^2y^2 =
(x-y)x^2y^2/xy^2 +(x-xy)x^2y^2/x^2y^2.

Denominator*x^2y^2 =
(x-y)x +(x-xy) = x^2-xy +x-xy.

Denominator*x^2y^2=
x^2+x-2xy .

Denominator*x^2y^2=
x(x-2y+1)......(2).

We use the simplified results at (1)
and (2) to rewite the given rational fraction:

[2/xy +
x/(x^2*y)] / [ (x-y)/xy^2 + (x-xy)/x^2*y^2] =
3xy/x(x-2y+1.)

[2/xy + x/(x^2*y)] / [ (x-y)/xy^2 +
(x-xy)/x^2*y^2] = 3y/(x-2y+1).