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Is their a universal method to solve systems of equation, eg do methods such as 'elimination' work for ALL types of simultaneous equations (I am specifically referring to 2 and 3 equation simultaneous equations).

If their is not, why?

I am referring to highschool level equations, to likes of this:

enter image description here

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Yes there is. It is called Gaussian elimination and it involves matrices. See: en.wikipedia.org/wiki/Gaussian_elimination. Also linear algebra is the part of math, which deals with these kind of questions. –  05storm26 May 31 at 8:30
    
@05storm26 that is beautiful, i wonder if it would be easy to learn. thanks –  mahat May 31 at 8:31
    
@05storm26 I don't think gaussian elimination would deal with the $xy=4$ equation, but it will work with the linear equations –  ellya May 31 at 8:39
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For, linear equations, there are systematic methods like Gaussian elimination that will either give you the answers, or prove that there are none, as the comment above points out.

But one of the equations you gave, the one that says $xy=4$, is not linear. As soon as you start allowing non-linear equations, big troubles arise. If you have even a single equation that contains terms like $x^5$ or higher powers, then it is known that there can not be any simple formula for the solutions. These sorts of equations are called "quintic"; see here for more information. Similarly, equations like $\sin x = x$ can't be solved by any sort of formula. And, naturally, things get even worse if you have several equations in several variables.

In practice (rather than theory), things work out OK because there are well-developed numerical methods that can give us solutions even though we don't have any closed-form formulae.

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I can assure You that to obtain a formula for solving systems of nonlinear equations. And not just algebraic, but Diophantine. math.stackexchange.com/questions/757664/… –  individ Jun 1 at 16:20
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