given
u=y-x
v=xy
Find x(u,v) and y(u,v) .
Thank you.
Use substitution:
$\displaystyle y = u + x$
Thus
$\displaystyle v = x(u + x)$
$\displaystyle x^2 + ux - v = 0$
So
$\displaystyle x = \frac{-u \pm \sqrt{u^2 + 4v}}{2}$
Thus
$\displaystyle y = u + \frac{-u \pm \sqrt{u^2 + 4v}}{2} = \frac{u \pm \sqrt{u^2 + 4v}}{2}$
Which of the "+" or "-" signs to use is going to depend on the specific nature of the problem where the u and v come from. At this stage all we can say is that there are two acceptable solutions.
-Dan