How To Solve Algebraic Problems
A solution of an equation is often also called a root of the equation, particularly but not only for algebraic or numerical equations.A problem of solving an equation may be numeric or symbolic.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.For solving larger systems, algorithms are used that are based on linear algebra.Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example.The methods for solving equations generally depend on the type of equation, both the kind of expressions in the equation and the kind of values that may be assumed by the unknowns.The variety in types of equations is large, and so are the corresponding methods. In general, given a class of equations, there may be no known systematic method (algorithm) that is guaranteed to work.
Solving an equation symbolically means that expressions that may contain known variables or possibly also variables not in the original equation are admitted as solutions. Note that the distinction between known variables and unknown variables is made in the statement of the problem, rather than the equation.
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.
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.