After formulating a problem I found that the solution should be the one that is given by the pair of equations \begin{align*} 990 x^7 - 112 x^8 + 165 y^7 - 35 x y^7 + 21 x^3 y^4 (-25 + 7 y) = 0, \\ 75 x^7 - 35 x^7 y + 21 x^4 y^3(-55 + 7 x) + 450 y^7 - 112 y^8 = 0, \end{align*} However, I'm having trouble to see that this pair really has an unique solution in $\mathbb{R}_+^2 \setminus (0,0)$ (as it is most probably impossible to solve it explicitly). Here $\mathbb{R}_+^2 \setminus (0,0)$ means that both $x$ and $y$ should be strictly positive. According to numerical calculations it does, but how can I prove it?
I'm very interested if someone can come up with a method that also includes cases where the two equations do not only include polynomials.
To be more precise, I bet doing this will be very cumbersome and I'm more interested in ideas how to proceed and tackle problems like this than solving this one explicit case! Any references and ideas are welcome.
Alright. There is an evident solution at about $(9,2).$ Then have curves have asymptotes near $y=x$ in the first quadrant. It is fairly likely that these asymptotes do not intersect, and this can be investigated more carefully. We usually think in terms of slope and $y-x,$ so let $x = -u+v$ and $y = u+v,$ so that $y-x = 2u$ and $y+x = 2v.$ Both curves have arcs with large positive $v$ and small $u,$ I think slightly negative. So: more work is needed, but it can be done.