How to show some systems of equations do not have a closed form solution?
for example I was once given something similar to ( this might not be the exact problem but i am just using it as an example to convey what type of problems I am talking about):
$xy^2+y^2x=10 \tag{I}$ and $\frac{1}{x}+\frac{1}{y}=15 \tag{II}$
What are the numerical values that would satisfy both I and II.
(again I reiterate that solving the above is not the main issue, if somebody can give a simpler example where it can be shown that it can NOT be explicitly solved for x and y they are welcome to edit this post), maybe the fact is that a solution does not exist (easy way to look at their graphs).
But is there a way to not end up wasting time trying to explicitly come up with solutions?
Is there a simple example similar to above that can be shown that it can NOT be solved explicitly? I am aware of a computational algebraic result to show that some function do NOT have closed form integrals but nothing to suggest the similar for simple systems of non-linear equations being explicitly solvable.