An expression is a number, variable, or combination of numbers and variables joined by operations, without an equal sign. Examples are 7, 4x + 9 and 3(a − 2). Expressions can be simplified, but they are not solved unless they are part of an equation. You evaluate one by replacing each variable with a number.
An expression is a math phrase: terms joined by plus and minus signs, with no equal sign. Add = 19 and it becomes an equation, a sentence you can solve.
Formal definition and notation
An expression is a meaningful combination of numbers, variables and operation symbols (+, −, ×, ÷, exponents, roots) with grouping symbols, but no relation symbol such as =, < or >. A numerical expression, like 8 − 2 × 3, has only numbers and names a single value. An algebraic expression, like 2x2 − 5, contains at least one variable, and its value depends on the value of the variable. Two expressions are equivalent if they give the same value for every value of the variable.
Symbol
Read it as
Meaning
3x
three x
3 times x; a number written next to a variable means multiply
3(a − 2)
three times the quantity a minus two
parentheses group a − 2 into one quantity
n4
n over four
n ÷ 4; a fraction bar means divide
Key facts
An expression has no equal sign. Once two expressions are joined by =, you have an equation.
The parts of an expression separated by + or − are its terms. In 4x + 9, the terms are 4x and 9; 4 is the coefficient and 9 is a constant.
Words become expressions: “5 more than n” is n + 5; “5 less than n” is n − 5.
Expression examples, step by step
Example 1: Evaluate an expression
Warm-up
Evaluate 4x + 9 when x = 5.
Replace x with 5: 4(5) + 9.
Multiply first: 4 × 5 = 20.
Then add: 20 + 9 = 29.
Answer: When x = 5, the expression equals 29.
Example 2: Simplify an expression
Core
Simplify 3(a − 2) + 5a.
Distribute the 3 over both terms in the parentheses: 3a − 6 + 5a.
Group the like terms: 3a + 5a − 6.
Add the coefficients of a: 3 + 5 = 8, giving 8a − 6.
8a and −6 are not like terms, so this is as simple as it gets. There is nothing to solve: no equal sign.
Answer:3(a − 2) + 5a = 8a − 6
Example 3: Write an expression from words
Test-style
A gym charges a 25-dollar joining fee plus 12 dollars per month. Write an expression for the total cost of m months, then use it to find the cost of one year.
The fee is paid once: 25. The monthly charge repeats: 12m.
Total cost: 25 + 12m dollars. It is an expression, because it names a cost but states no equality.
One year is 12 months. Substitute m = 12: 25 + 12(12).
12 × 12 = 144, and 25 + 144 = 169.
Answer: The expression is 25 + 12m, and one year costs 169 dollars.
Common mistakes
✗ Mistake“Solve” 4x + 9 and get x = −94.
✓ Instead4x + 9 has no equal sign, so there is nothing to solve. You can evaluate it (at x = 1 it is 13) or leave it as it is.
Do not call an expression an equation unless it has an equal sign. Setting it equal to 0 invents an equation the problem never gave.
✗ Mistake3x + 4 = 7x
✓ Instead3x + 4 is already simplified; 3x and 4 cannot be combined.
Only like terms combine. Test with x = 2: 3(2) + 4 = 10, but 7(2) = 14, so the two are not equivalent.
✗ MistakeAt x = 3, 2x2 = 62 = 36.
✓ Instead2x2 = 2 × 32 = 2 × 9 = 18
The exponent belongs to x only. Square first, then multiply by the coefficient, following the order of operations.
✗ Mistake“5 less than n” is 5 − n.
✓ Instead“5 less than n” is n − 5.
“Less than” reverses the order of the words: start with n, then take 5 away. For n = 12, 5 less is 7, which is 12 − 5.
An equation states that two expressions are equal, so it has an equal sign and can be solved: 2x + 1 = 11 gives x = 5. The expression 2x + 1 alone has no single answer.
A formula is an equation that relates quantities, such as A = lw. The right side, lw, is an expression.
Quick check: 3 questions
Question 1 of 3
Which of these is an expression?
Show answer and explanation
Answer: C.5n − 2 has no equal sign or inequality symbol, so it is an expression. 5n − 2 = 8 and y = 5n are equations, and n > 2 is an inequality.
Question 2 of 3
What is the value of 3x2 − x when x = 2?
Show answer and explanation
Answer: A.3(2)2 − 2 = 3 × 4 − 2 = 10. 34 squares 3 × 2 instead of 2, 4 ignores the exponent, and 14 adds the last term instead of subtracting it.
Question 3 of 3
Which expression means “twice a number, decreased by 6”?
Show answer and explanation
Answer: D. Twice the number is 2n; decreased by 6 subtracts 6 from it: 2n − 6. 6 − 2n subtracts in the wrong order, 2(n − 6) doubles after subtracting, and n2 squares instead of doubling.
Expression: frequently asked questions
What is the difference between an expression and an equation?
An equation has an equal sign and an expression does not. 2x + 3 is an expression: a phrase that names a value. 2x + 3 = 11 is an equation: a sentence claiming two expressions are equal, which you can solve (x = 4). You simplify or evaluate expressions and solve equations.
Can a single number be an expression?
Yes. A single number such as 7, or a single variable such as x, is the simplest kind of expression. Longer expressions are built from these pieces with operations: 7x, 7 + x, x7. A numerical expression like 8 + 2 × 3 contains no variables and always simplifies to one number, here 14.
What does it mean to simplify an expression?
Simplifying rewrites an expression in a shorter form with the same value for every value of the variable. You remove parentheses with the distributive property and combine like terms. For example, 2(x + 4) + 3x simplifies to 5x + 8. Check by substituting a number: at x = 1, both forms equal 13.
CombinationA combination is a selection of items from a group in which the order does not matter. Picking Ana, Ben and Cy for a team is the same combination as Cy, Ana and Ben. The number of ways to choose r items from n is C(n, r) = n!r!(n − r)!; for example, C(5, 2) = 10.Probability · Algebra 2, Precalculus, StatisticsNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8RootsThe roots of an equation are the values of the variable that make it true; for a polynomial equation set equal to 0, they are the solutions. The roots of x2 + 3x − 10 = 0 are 2 and −5, because each makes the left side 0. Real roots appear as the x-intercepts of the graph.Algebra · Algebra 1, Algebra 2RelationA relation is any set of ordered pairs that links inputs to outputs, such as {(1, 4), (2, 5), (2, 7)}. It can be shown as a list, table, mapping diagram or graph. A relation in which every input has exactly one output is a function; this one is not, because 2 has two outputs.Functions · Grade 8, Algebra 1Equal SignThe equal sign (=) is the symbol that shows two expressions have the same value. In 3 + 5 = 2 + 6, both sides equal 8, so the statement is true. The equal sign does not mean “the answer comes next”; it works like a balanced scale, and a number sentence with = is called an equation.Symbols and Notation · Kindergarten, Grade 1, Grade 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8ConstantA constant is a fixed value that does not change. In an expression, a constant is a number without a variable, such as 8 in x + 8 or −3 in 4a − 3. Special numbers such as π are constants too. Changing the variable changes the other terms, but never the constant.Algebra · Grade 6, Grade 7, Grade 8EvaluateEvaluate means to find the single number value of an expression. Replace each variable with the number you are given, then calculate using the order of operations. For example, 2x + 7 evaluated at x = 3 is 2(3) + 7 = 13. The result is one number, not a new expression.Algebra · Grade 6, Grade 7, Algebra 1SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6Like TermsLike terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. 3x and 8x are like terms, and so are 5a2 and −2a2, but 4x and 4y are not. Only like terms can be combined by adding or subtracting their coefficients: 3x + 8x = 11x.Algebra · Grade 6, Grade 7, Grade 8Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8Equal ToEqual to means having exactly the same value or amount, and it is written with the symbol =. Numbers that look different can still be equal to each other: 12 is equal to 0.5. When two amounts are not the same, one is greater than or less than the other.Symbols and Notation · Kindergarten, Grade 1, Grade 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3TermA term is a single number, a variable, or numbers and variables multiplied together, such as 7, x or −4xy2. In an expression, terms are the parts separated by plus and minus signs, so 5x2 − 3x + 8 has three terms. The sign in front of a term belongs to it.Algebra · Grade 6, Grade 7, Algebra 1FormulaA formula is a rule written with symbols that shows how quantities are related. It is an equation with two or more variables, such as A = lw for the area of a rectangle. Formulas help solve problems the same way each time: substitute the values you know, then compute the one you need.Problem Solving · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Equivalent ExpressionsEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Algebra · Grade 6, Grade 7, Algebra 1PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2