Algebraic solution for a system of algebraic equations?

Question

I was asked about solving algebraically the following system of algebraic equations. $$f(a,b):=a(1-b)+ab\frac a{a+b}.$$ $$u = f(a,b),\quad v = f(b,a).$$ Solve algebraically $(a,b)$ in terms of $(u,v)$

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Multiplying both sides of the equations by $a+b$ would give us a system of cubic equations.

Answer 1

After multiplying by $$a+b\neq 0$$ we get $$a^2+ab-ab^2=u(a+b)$$ $$b^2-a^2b+ab=v(a+b)$$

Answer 2

This is answered with the aide of a Computer Algebraic System (Mathematica or Maple) on Mathoverflow.net after I cross-posted the question there.