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Apparently, my previous question didn't get no satisfactory answer, when I asked for two equations having a fixed value for each, not necessarily linear. As XenoGraff states, WolframAlpha does the task, but counts permutations of values among variables, and is thus impractical to test any two equations.

Actually, I ask, is it possible that there could be a system of X equations that can be solved for more than X variables, all having whole number values, considering that these X equations have an unique solution?

As Gerry Myerson states in the previous thread, there is the unproven conjecture that $x^{5}+y^{5}=N$ will have only one solution for $x,y$ for a given $N$, which can be modified to satisfy $x^{5}+y^{5}=x+y$ for only one set of values for $x,y$.

So... are there any such equations? What about differential equations (I don't understand them, anyway) and multivariates? And Diophantine equations?

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1 Answer 1

You can have $(x-a)^2+(y-b)^2+(z-c)^2=0$, which can be solved for $x,y,z$, viewing $a,b,c$ as parameters. It works for reals or integers. Is that what you are looking for?

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Seems right, but what exactly are these parameters and what do they do? BTW where are the two equations I'd asked for, containing the same and same number of variables both (eg. $a+b+c+d+e=25, a^{3}+b^{3}+c^{3}+d^{3}+e^{3}$, no solution present for equations as this is just a random example, but assuming here that both equations have only one common unique solution)? Sorry if I hadn't read your answer properly. –  Mach9 May 4 '12 at 9:33
    
This was just one example where you can solve for more variables than the number of equations. It is used to collapse several equations into one so you can feed it to a function minimizer and get an approximate solution. The same strategy can be used to get two equations that you can solve for any number of variables. I can add $x+y+c=N$ if you want. Now there will not be a solution unless $a+b+c=N$. I don't think I understand what you are asking. –  Ross Millikan May 4 '12 at 12:47
    
We can take for an example $a+b+c+d+e+f=X$, $a^{3}+b^{3}+c^{3}+d^{3}+e^{3}+f^{3}=Y$ assuming $X,Y$ are fixed values, and both equations have only one solution (albeit probably false) in common. What other two equations exist such that any number of same variables in both equations share an unique common solution? –  Mach9 May 4 '12 at 14:07

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