Expression vs. Equation
An expression is a mathematical phrase with no equals sign that can be evaluated; an equation is a statement that two expressions are equal and can be solved.
Definition
An expression is a math phrase with no equals sign, while an equation is a statement with an equals sign saying two things are the same; you simplify expressions, but you solve equations, since the equals sign is the dividing line. An expression is a combination of numbers, variables, and operations with no relational symbol, while an equation asserts that two expressions are equal, and the critical difference remains: expressions are evaluated or simplified, equations are solved for a variable value. In formal logic, an expression (or term) is a syntactic object built from variables, constants, and function symbols, while an equation is a formula $P = Q$ asserting equality between two terms and has a truth value relative to an interpretation; expressions are interpreted (mapped to values) while equations are verified (proven true or false), a distinction foundational to universal algebra and model theory.
Example
$3x + 7$ is an expression; $3x + 7 = 16$ is an equation. You cannot "solve" the expression since there is nothing to solve for, but you can evaluate or simplify it. Expression: $2(x + 3) - x$ simplifies to $x + 6$; equation: $2(x + 3) - x = 10$ is solved by getting $x + 6 = 10$, then $x = 4$. In universal algebra, a term $t(x_1, \ldots, x_n)$ is an expression, and an identity is an equation $t_1 = t_2$ satisfied by all elements of an algebraic structure; commutativity, $x + y = y + x$, is an identity true in all abelian groups.
Key Insight
The equals sign is the dividing line: no equals sign means expression, equals sign means equation, one gets simplified and the other gets solved. A common error is adding an equals sign to "simplify" an expression, accidentally creating a false equation, so never chain equals signs across separate simplification steps. Birkhoff's theorem states that a class of algebras is definable by a set of identities (equations) if and only if it is closed under homomorphic images, subalgebras, and direct products, a cornerstone of equational logic.