Expression definition
An expression is a mathematical phrase made from numbers, letters and operations. It can include addition, subtraction, multiplication, division, powers and brackets. An expression does not contain an equals sign. For example, 3x + 5 is an expression. It tells us to multiply x by 3 and then add 5, but it does not say that the result is equal to another value.
Expressions are different from equations. The expression 2x + 7 has no equals sign, so it cannot be solved on its own. The equation 2x + 7 = 15 can be solved because it says the expression is equal to 15. This difference is important. Students often confuse expressions and equations, but an expression is something you can simplify or evaluate, while an equation is something you can solve.
Expressions are used in algebra, formulas, sequences, functions, graphs, geometry, probability, statistics and calculus. At GCSE level, students simplify expressions, substitute values, expand brackets, factorise expressions and form expressions from worded problems. At A-Level, expressions appear in polynomial algebra, trigonometry, logarithms, exponentials, differentiation, integration and modelling.
An expression is made from terms. In the expression 4x + 3y - 8, the terms are 4x, 3y and -8. The numbers multiplying the variables are coefficients, so 4 is the coefficient of x and 3 is the coefficient of y. The term -8 is a constant because it is a fixed number. Understanding terms, coefficients and constants helps students simplify algebra correctly.
Expressions can be evaluated by substituting values for the variables. For example, if the expression is 3x + 5 and x = 4, then the value is 3 times 4 plus 5, which equals 17. The expression itself has many possible values because x can change. This is one reason expressions are useful: they can describe a general rule instead of just one calculation.
Expressions can also be simplified. For example, 2a + 5a simplifies to 7a because the terms are like terms. The expression 3x + 4 - x + 2 simplifies to 2x + 6. Simplifying makes an expression shorter and clearer, but it does not change its value. A simplified expression is equivalent to the original expression.
Expressions are often formed from words. If a question says “five more than a number”, this can be written as n + 5. If it says “three times a number minus two”, this can be written as 3n - 2. Turning words into expressions is an important problem-solving skill because many exam questions describe relationships in sentences before asking for algebra.
Related ideas include terms, variables, coefficients, constants, equations, identities, formulas, substitution, simplifying, expanding brackets and factorising. A strong definition of expression helps students know whether they need to simplify, substitute or solve. In exams, first check whether there is an equals sign. If there is no equals sign, you are usually working with an expression, not an equation.