Expression vs. Equation
Pre-AlgebraAn 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. An equation is a statement with an equals sign saying two things are the same. You simplify expressions. You solve equations.
Example
$3x + 7$ is an expression. $3x + 7 = 16$ is an equation. You cannot "solve" the expression (nothing to solve for), but you can evaluate it or simplify it.
Key Insight
The equals sign is the dividing line. No equals sign: expression. Equals sign: equation. One gets simplified, the other gets solved.
Definition
An expression is a combination of numbers, variables, and operations with no relational symbol. An equation asserts that two expressions are equal. The critical difference: expressions are evaluated or simplified; equations are solved for a variable value.
Example
Expression: $2(x + 3) - x$ simplifies to $x + 6$. Equation: $2(x + 3) - x = 10$ is solved to get $x + 6 = 10$, then $x = 4$.
Key Insight
A common error is adding an equals sign to simplify an expression, accidentally creating a false equation. Never write "$3x + 2x = 5x = 10x - 5x$." Each line of simplification should use a new line, not a chain of equals signs connecting unequal quantities.
Definition
In formal logic, an expression (or term) is a syntactic object built from variables, constants, and function symbols. 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). This distinction is foundational to universal algebra and model theory.
Example
In universal algebra, a term $t(x_1, \ldots, x_n)$ is an expression. An identity is an equation $t_1 = t_2$ satisfied by all elements of an algebraic structure. For example, commutativity $x + y = y + x$ is an identity (equation) true in all abelian groups.
Key Insight
The study of which equations (identities) hold in an algebraic structure is the subject of equational logic and Birkhoff's theorem: 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.