equationsolving

Equation solving is the process of determining the values of unknown quantities that satisfy one or more equations. It is a foundational activity in mathematics and appears in applied disciplines such as physics, engineering, economics, and computer science. An equation expresses equality between expressions containing variables; solving means finding the variable assignments that make the equality true. When a problem has multiple unknowns, a system of equations is solved, often representing constraints or relationships among quantities.

Equations are classified by form: linear equations (polynomial degree one), polynomial equations (degree higher than one), rational, exponential, logarithmic, and differential equations, among others. Systems can be linear or nonlinear and may be subject to additional conditions such as bounds or initial values. Differential equations involve derivatives and describe change; their solutions are functions rather than simple numbers.

Methods of solving range from exact symbolic techniques to numerical approximations. For linear systems Ax=b, methods include Gaussian elimination, LU decomposition, and, when applicable, Cramer's rule. For nonlinear equations, iterative methods such as the Newton-Raphson method, fixed-point iteration, or bisection are common, often together with convergence criteria. Symbolic computation uses algebraic manipulation, factorizations, and, for systems, Groebner bases or resultants to obtain closed-form solutions when possible. In cases where exact solutions do not exist or are intractable, numerical solvers approximate the solution to a specified precision.

Applications include designing circuits, modeling physical phenomena, optimizing resources, and solving puzzles. The study of equation solving also intersects with numerical analysis, linear algebra, and computational algebra.