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Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example , which simplifies this to a quadratic equation in z).
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.
Equations involving linear or simple rational functions of a single real-valued unknown, say x, such as can be solved using the methods of elementary algebra.
Smaller systems of linear equations can be solved likewise by methods of elementary algebra.
However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation.
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So, if we were instead to define that the domain of ƒ consists of the real numbers, the equation above has two solutions, and its solution set is (with a, b, c, d, and k real-valued constants) is a hyperplane.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.Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2.In fact, this particular set of solutions describes a plane in three-dimensional space, which passes through the three points with these coordinates.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.However, in some areas of mathematics the convention is to reserve some variables as known and others as unknown.When writing polynomials, the coefficients are usually taken to be known and the indeterminates to be unknown, but depending on the problem, all variables may assume either role.