Equation

An equation is a statement that the two expressions on its sides are equal. It typically involves variables, constants, and operators, and is written with an equality sign (=). Equations express a relation to be satisfied by the values of the variables, as opposed to an expression, which evaluates to a value without asserting equality.

Equations have structure: a left-hand side and a right-hand side. They can be purely algebraic, numerical, functional, or accompanied by derivatives or integrals. Common classes include linear equations, polynomial equations, and differential equations. An equation may be conditional, having solutions only for certain values, or an identity, true for all values of the variables.

Examples include x + 2 = 5, a linear equation with one unknown; x^2 − 3x + 2 = 0, a quadratic; and dy/dx = y, a first-order differential equation. Systems of equations involve multiple unknowns and are often solved together, for example 2x + y = 3 and x − y = 1.

Solving an equation means finding the values of the variables that make the equality hold. Methods include algebraic manipulation, factoring, completing the square, substitution, elimination, and, for more complex cases, numerical methods or graphical solutions. Some equations have no solutions, some have a finite set, and some have infinitely many.

Equations are foundational in mathematics and its applications. They model relationships in physics, engineering, economics, and computer science, among others, providing a formal framework for describing constraints, dynamics, and optimization problems.