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As with all kinds of problem solving, trial and error may sometimes yield a solution, in particular where the form of the equation, or its similarity to another equation with a known solution, may lead to an "inspired guess" at the solution.
The solution set of a given set of equations or inequalities is the set of all its solutions, a solution being a tuple of values, one for each unknown, that satisfies all equations or inequalities.
If the solution set is empty, then there are no values x such that the equations or inequalities becomes true simultaneously.
For solutions of constraint satisfaction problems, see Constraint satisfaction problem § Resolution.
For solutions of mathematical optimization problems, see Feasible solution.
A wording such as "an equation in x and y", or "solve for x and y", implies that the unknowns are as indicated: in these cases x and y. Note that the set of solutions can be the empty set (there are no solutions), a singleton (there is exactly one solution), finite, or infinite (there are infinitely many solutions).
For example, an equation such as One particular solution is x = 0, y = 0, z = 0.
This may be due to a lack of mathematical knowledge; some problems were only solved after centuries of effort.
But this also reflects that, in general, no such method can exist: some problems are known to be unsolvable by an algorithm, such as Hilbert's tenth problem, which was proved unsolvable in 1970.
For several classes of equations, algorithms have been found for solving them, some of which have been implemented and incorporated in computer algebra systems, but often require no more sophisticated technology than pencil and paper.
In some other cases, heuristic methods are known that are often successful but that are not guaranteed to lead to success.