# How to show some systems of equations do not have a closed form solution?

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.

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@QED : yes,now solve that for y :) the point is NOT solution to this problem either, but to show that some problems can not be explicitly solved, they might look as simple as this one. –  Arjang Dec 30 '11 at 12:08
Theoretically, as long as your system's entirely algebraic, you can reduce to the solution of some polynomial. When transcendental functions are involved, all bets are off. –  Guess who it is. Dec 30 '11 at 12:20
At the heart of the matter: What is a closed form solution? In Abel-Ruffini only radicals and basic arithmetic operations are allowed, but if hypergeometric (or theta) series are included, then we have "closed form solutions". Will you allow Bessel-functions? Lambert? Where do you draw the line? Many an interesting equation (or a DE) defines a function as a solution (by implicit function theorem or another relatively high-powered result). How familiar do we need to be with a function to call it "closed form"? In this sense the original question is unfortunately ill-defined. –  Jyrki Lahtonen Dec 30 '11 at 12:52
That's beautifully written, @Jyrki. I think that's admissible as an answer here. –  Guess who it is. Dec 30 '11 at 12:57
Well, I had Gröbner in mind when I said "you can reduce to the solution of some polynomial". As long as you have an algebraic system, I don't see why the triangularization isn't possible. –  Guess who it is. Dec 30 '11 at 13:11