Equations

An equation is a statement that asserts the equality of two expressions, typically involving variables, constants, and mathematical operations. Equations express relationships between quantities and can be solved to determine the values of the unknowns that satisfy the equality. They differ from expressions, which are terms without an assertion of truth, and from identities, which are true for all values of the variables involved.

Common classes include linear equations, polynomial equations, and equations that involve derivatives or integrals. Linear equations have constant coefficients and degree one, such as ax + b = 0, or systems of linear equations written in matrix form. Polynomial equations take the form p(x) = 0, where p is a polynomial. Differential equations relate a function to its derivatives and describe rates of change. Functional equations define unknown functions by relations between their values at different inputs. Algebraic equations involve only algebraic operations, while transcendental equations include non-algebraic functions such as exponentials or trigonometric functions.

A solution to an equation is a value or set of values that satisfy it. Solutions may be unique, multiple, or infinite, and may be real or complex depending on the context and domain. Methods for solving include algebraic manipulation, factoring, substitution, elimination, and graphical or numerical techniques such as Newton-Raphson.

Equations are fundamental across disciplines, used to model physical laws, economic constraints, engineering systems, and computational problems. They have a long history in mathematics, with developments in symbolic notation and solution techniques contributing to advances in science and technology.