Evaluating Expressions

Learn how to substitute values for variables and calculate the result of algebraic expressions.

Intermediate25 minLesson
Printable worksheet

Definition

Evaluating an expression means finding its numerical value by replacing each variable with a given number and then performing the operations.
To evaluate an expression:
  1. 1.Substitute the given value(s) for each variable
  2. 2.Follow order of operations (PEMDAS/BODMAS)
  3. 3.Calculate the final result
For example, to evaluate 3x+53x + 5 when x=4x = 4:
3x+5=3(4)+5=12+5=173x + 5 = 3(4) + 5 = 12 + 5 = 17

Try it now

Evaluate x+4x + 4 when x=3x = 3

Worked Examples

Evaluate 2x+72x + 7 when x=5x = 5
1

Write the original expression

2x+72x + 7 → Ready to substitute
2

Replace xx with 55

2(5)+72(5) + 7 → Substitution complete
3

Multiply first (order of operations)

10+710 + 7 → Multiplication done
4

Add to get the final answer

10+7=1710 + 7 = 17

Common Mistakes

Forgetting to multiply: Writing 2x2x as 2525 when x=5x = 5 instead of 2×5=102 \times 5 = 10

Why it's wrong: The coefficient and variable are being multiplied, even though the multiplication sign is not written.

Correct: 2x2x means 2×x2 \times x. When x=5x = 5: 2x=2×5=102x = 2 \times 5 = 10

Wrong order of operations: Doing 3+2×4=203 + 2 \times 4 = 20 instead of 3+8=113 + 8 = 11

Why it's wrong: Addition was done before multiplication. Remember PEMDAS: Multiplication comes before Addition.

Correct: Multiply first: 3+2×4=3+8=113 + 2 \times 4 = 3 + 8 = 11

Substituting in the wrong place: For 2x+y2x + y with x=3x = 3, y=4y = 4, writing 23+423 + 4

Why it's wrong: The number was placed next to the variable as digits rather than multiplying.

Correct: 2x+y=2(3)+4=6+4=102x + y = 2(3) + 4 = 6 + 4 = 10

Interactive Visual

Function graph

024680510152025xy(5, 17)

Tap or hover over the graph to read off values.

2x + 7

Table of values

x012345
y7911131517

Order of Operations

  1. 1. Parentheses
  2. 2. Exponents
  3. 3. × and ÷
  4. 4. + and −
2×(7+5)−32 \times \left(\colorbox{#fde68a}{$\color{#1f2937}7 + 5$}\right) - 3

Parentheses first: 7+5=127 + 5 = 12

Step 1 of 3

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Practice Problems

16 problems
Problem 1 of 16
Easy
Evaluate x+4x + 4 when x=3x = 3

Why It Matters

Evaluating expressions is fundamental to all of mathematics and science:
  • Formulas: Using A=lwA = lw to find the area of a rectangle with given dimensions
  • Physics: Calculating speed using v=d÷tv = d \div t with actual distances and times
  • Finance: Computing costs using expressions like c=5n+20c = 5n + 20 where nn is the number of items
  • Programming: Every computer program evaluates expressions millions of times
This skill lets you turn abstract formulas into concrete answers!

Real World Applications

Calculating Costs

Businesses use expressions to calculate prices based on quantity.

Example:

A taxi charges a 3 euro base fare plus 2 euros per kilometer. The expression 3+2k3 + 2k gives the total cost for kk kilometers.

1Try It Yourself

A taxi ride is k=8k = 8 kilometers long.

What is the total fare?

Step 1: Write the mathematical expression

Evaluate 3+2k3 + 2k when k=8k = 8:

Temperature Conversion

Scientists convert between Celsius and Fahrenheit using formulas.

Example:

To convert Celsius to Fahrenheit, use F=95C+32F = \frac{9}{5}C + 32. For C=20°C = 20°: F=95(20)+32=36+32=68°F = \frac{9}{5}(20) + 32 = 36 + 32 = 68°

2Try It Yourself

The temperature is 25°C25°C.

What is this in Fahrenheit?

Step 1: Write the mathematical expression

Evaluate 95C+32\frac{9}{5}C + 32 when C=25C = 25:

Key Takeaways

  • 1Evaluating an expression means substituting values for variables and calculating the result
  • 2Always use parentheses when substituting to keep the work clear
  • 3Follow order of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
  • 4Check your answer by confirming each step was done correctly

Frequently Asked Questions

Parentheses make the substitution clear and prevent mistakes. For 3x3x with x=5x = 5, writing 3(5)3(5) clearly shows multiplication, whereas 3535 looks like thirty-five.
Parentheses make the substitution clear and prevent mistakes. For 3x3x with x=5x = 5, writing 3(5)3(5) clearly shows multiplication, whereas 3535 looks like thirty-five.
Negative answers are completely valid! For example, x−10x - 10 when x=3x = 3 gives 3−10=−73 - 10 = -7. Just follow the operations correctly.
No, you can substitute variables in any order. Just make sure to replace ALL occurrences of each variable with its given value.

Glossary

Evaluate
To find the numerical value of an expression by substituting values and calculating
Substitute
To replace a variable with a given number
Variable
A letter that represents an unknown or changeable value (e.g., xx, yy, nn)
Coefficient
The number multiplied by a variable (e.g., in 3x3x, the coefficient is 33)
Expression
A mathematical phrase containing numbers, variables, and operations (e.g., 2x+52x + 5)
Today's challengeOne problem, three tries

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