1/15
00:00

HH.2 Evaluate variable expressions

Loading questions...

What are variable expressions?

Variable expressions are mathematical phrases that contain numbers, operations, and variables. A variable is a symbol (usually a letter) that represents an unknown value.

Examples:
  • 3x + 5 (where x is the variable)
  • 2y - 7 (where y is the variable)
  • a ÷ 4 + 2 (where a is the variable)
Note

Think of variables as empty boxes that can hold different values. The expression changes its value depending on what number you put in the box.

How to evaluate variable expressions

To evaluate a variable expression means to substitute a given number for the variable and then calculate the value of the expression using the correct order of operations.

Steps:
  • Replace the variable with the given number
  • Follow the order of operations: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)
  • Calculate step by step to find the final value
Note

Always rewrite the expression after substituting the value for the variable. This helps prevent calculation errors.

Evaluating expressions with different operations

Variable expressions can include all four basic operations: addition, subtraction, multiplication, and division. The process for evaluation remains the same regardless of which operations are used.

Examples with Different Operations:
  • Addition: Evaluate x + 9 when x = 5 → 5 + 9 = 14
  • Subtraction: Evaluate 12 - y when y = 7 → 12 - 7 = 5
  • Multiplication: Evaluate 4n when n = 6 → 4 × 6 = 24
  • Division: Evaluate k ÷ 3 when k = 15 → 15 ÷ 3 = 5
Note

In algebra, we often write multiplication without the × symbol. For example, 4n means 4 × n, and 2(5) means 2 × 5.

Expressions with multiple operations

When expressions contain more than one operation, you must follow the order of operations to evaluate them correctly.

Examples with Multiple Operations:
  • Evaluate 2x + 5 when x = 4 → 2(4) + 5 = 8 + 5 = 13
  • Evaluate 3(y - 2) when y = 7 → 3(7 - 2) = 3(5) = 15
  • Evaluate a ÷ 2 + 10 when a = 8 → 8 ÷ 2 + 10 = 4 + 10 = 14
  • Evaluate 20 - 3m when m = 4 → 20 - 3(4) = 20 - 12 = 8
Note

Remember PEMDAS: Parentheses first, then Exponents, then Multiplication and Division (left to right), and finally Addition and Subtraction (left to right).

Expressions with multiple variables

Some expressions contain more than one variable. When evaluating these expressions, substitute the appropriate value for each variable.

Examples with Multiple Variables:
  • Evaluate x + y when x = 3 and y = 8 → 3 + 8 = 11
  • Evaluate 2a - b when a = 5 and b = 3 → 2(5) - 3 = 10 - 3 = 7
  • Evaluate p × q ÷ 2 when p = 6 and q = 4 → 6 × 4 ÷ 2 = 24 ÷ 2 = 12
  • Evaluate (m + n) × 3 when m = 2 and n = 5 → (2 + 5) × 3 = 7 × 3 = 21
Note

When an expression has multiple variables, make sure you substitute the correct value for each variable. Different variables can have different values.

Real world applications

Variable expressions are used to represent real-world situations where quantities can change. Evaluating these expressions helps solve practical problems.

Real-World Examples:
  • Pizza party: If pizza costs $12 and you buy p pizzas, the cost is 12p. For 3 pizzas: 12 × 3 = $36
  • Road trip: If you drive at 55 miles per hour for h hours, distance = 55h. For 4 hours: 55 × 4 = 220 miles
  • Baking: A recipe needs 2 cups of flour per batch. For b batches: flour needed = 2b. For 5 batches: 2 × 5 = 10 cups
Note

When solving word problems, identify what the variable represents and what operation(s) to use before evaluating the expression.