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.