Some equations have no algorithm that always works
Solving an equation means finding values that make both sides equal when substituted for unknowns. Solutions may be numbers, functions, or infinite sets. Hilbert's tenth problem was proved unsolvable in 1970, yet elementary algebra still handles linear equations and quadratics with pencil and paper.
To solve an equation is to find its solutions—the assignments to unknown variables that make the equality true. Substituting those values turns the equation into a verified statement. Solutions are often called roots, especially for polynomials. The collection of all solutions is the solution set, which may be empty, a singleton, finite, or infinite.
Equations may be solved numerically, admitting only numbers, or symbolically, allowing expressions. For x + y = 2x − 1, rearranging gives x = y + 1 or y = x − 1 depending on which variable is unknown. Treating both as unknowns yields infinitely many pairs (a + 1, a) for any parameter a.
Methods depend on equation type. Linear and simple rational equations in one variable yield to elementary algebra. Larger linear systems use Gaussian elimination. Polynomials up to degree four have exact algebraic formulas; degree five and higher generally need numerical methods or special functions. Diophantine equations demand integer solutions and rank among the hardest problems.
No universal algorithm exists for all equation classes. Hilbert's tenth problem was shown unsolvable in 1970. When solution sets are finite, brute force can test every candidate—impractical for encryption-sized spaces. Infinite solution sets often become geometric objects: a linear equation in three variables describes a plane through points like (0, 0, 0), (3, 6, 1), and (8, 9, 2). Algebraic geometry studies such shapes as solution sets of polynomial equations. Trial and error may yield solutions when inspired guesses follow the form of a similar solved equation.
Source: Equation solving
