Evaluate 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.
In Spanish
evaluar
Also called
evaluate an expression, find the value, plug in, evaluating
Evaluating turns an expression into one number: substitute first, then follow the order of operations.
Formal definition and notation
To evaluate an expression for given values of its variables is to substitute each value for its variable everywhere it appears and carry out the operations, in the conventional order, until a single number remains. Evaluating a functionf at an input a in its domain means computing the output f(a).
Symbol
Read it as
Meaning
x = 4
x equals 4
the value to use for the variable; it tells you what to substitute
f(4)
f of 4
evaluate the function f when the input is 4
3(4)
3 times 4
parentheses keep the substituted number together and show multiplication
Key facts
Evaluating gives a number. Simplifying gives a shorter expression. Solving finds the value of a variable that makes an equation true.
The same expression has different values for different inputs: 5n is 10 when n = 2 and 35 when n = 7.
Wrap a negative input in parentheses: x2 at x = −5 is (−5)2 = 25, not −25.
Formulas are evaluated the same way: the area of a rectangle A = lw with l = 8 and w = 3 is 24 square units.
When there are several variables, use every given value: 2a − b at a = 6, b = 4 is 12 − 4 = 8.
How to evaluate an expression
How to evaluate an expression
substitute, then simplify with the order of operations
1. Substitute
rewrite the expression with each variable replaced by its number, in parentheses: x2 at x = −3 becomes (−3)2
2. Grouping and exponents
work inside parentheses, then powers
3. Multiply and divide
left to right
4. Add and subtract
left to right; what is left is the value
Write each step on a new line. Most errors happen when the substituted number is not put in parentheses.
Try it: evaluate 3x squared
Interactive
Type a value for x and press Run. The machine evaluates 3x2 by squaring first, then multiplying by 3. Try x = −2 and x = 2: both give 12.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Simplifying rewrites an expression in a shorter form without a value for the variable: 3x + 2x = 5x. Evaluating needs a number for x and ends with a number.
Evaluatevs.Substitute
Substituting is only the first step of evaluating: replacing x with 4. Evaluating also finishes the arithmetic.
Quick check: 3 questions
Question 1 of 3
What is the value of 4a − 3b when a = 5 and b = 2?
Show answer and explanation
Answer: A.4(5) − 3(2) = 20 − 6 = 14. 17 computed 20 − 3 and dropped b, 26 added the products instead of subtracting, and 20 − 3b stopped halfway: b was never replaced.
Question 2 of 3
Evaluate 2x2 when x = −3.
Show answer and explanation
Answer: C. Square first: (−3)2 = 9, then multiply by 2 to get 18. Choice -18 forgot that a negative squared is positive, 36 squared 2 × (−3) instead of only x, and -36 made both errors.
Question 3 of 3
The formula F = 1.8C + 32 converts Celsius to Fahrenheit. What is F when C = 25?
Show answer and explanation
Answer: D.1.8(25) = 45, and 45 + 32 = 77 degrees Fahrenheit. 58.8 added 25 and 32 before multiplying, 45 stopped before adding 32, and 84.6 computed 1.8 × 47.
Evaluate: frequently asked questions
What is the difference between evaluate and solve?
You evaluate an expression and solve an equation. To evaluate, you already know the variable’s value and compute the result: 3x at x = 4 is 12. To solve, you know the result and find the variable: 3x = 12 means x = 4. Evaluating ends with a number; solving ends with a value of the variable that makes the equation true.
Do I need parentheses when I substitute?
Yes, use them every time. Parentheses keep the number in one piece so the operations apply to all of it. They matter most for negatives and exponents: x2 at x = −2 is (−2)2 = 4, but writing −22 gives −4. They also show multiplication, as in 5(3) rather than 53.
Can an expression be evaluated without values for every variable?
Not completely. If any variable is still unknown, you can substitute the values you have and simplify, but the result is an expression, not a number. For example, 2a + b with a = 3 becomes 6 + b. You need a value for b to finish evaluating it.
VariableA 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 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 7ExpressionAn 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.Algebra · Grade 6, Grade 7, Grade 8SubstituteSubstitute 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 8NumberA 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 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4SquareA 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 3ExponentAn 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 8MileA mile (mi) is a US customary unit of length for long distances, equal to 5,280 feet or 1,760 yards. One mile is about 1.61 kilometers, so it is longer than a kilometer. Road distances and speed limits in the United States use miles and miles per hour.Measurement · Grade 4, Grade 5, Grade 6SideA 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 2AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8Simplest FormSimplest form is the way of writing a fraction so its numerator and denominator have no common factor other than 1. To get there, divide the top and bottom by their greatest common factor: 68 = 6 ÷ 28 ÷ 2 = 34. The value stays the same; only the numbers get smaller. It is also called lowest terms.Number System · Grade 4, Grade 5, Grade 6SimplifySimplify 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 6Function NotationFunction notation is the way of writing a function as f(x), read “f of x”: f names the function, x is the input, and f(x) is the output. Writing f(x) = 2x + 5 means the same as y = 2x + 5, and f(3) = 11 says that the input 3 gives the output 11.Functions · Grade 8, Algebra 1, Algebra 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 8FormulaA 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 7Equivalent 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 1