0.1Variables & Algebraic Expressions
Understand what variables represent, how to evaluate expressions by substituting values, and how to translate word problems into algebraic form.
Why This Matters
Variables are the language of algebra — every equation, formula, and function you'll ever use relies on them. From calculating speed in Physics to modeling populations in Biology, variables let you represent and solve real-world unknowns.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
▶ Watch Guided Practice Video
Opens on YouTube · Sang Real Math
Essential Question
How do variables allow us to write general rules that work for any number?
Lesson Overview
A variable is a letter used to represent an unknown or changing quantity. Algebra uses variables to write general rules and solve problems. An algebraic expression combines variables, numbers, and operations — but has no equals sign. When we substitute a number for a variable and simplify, we are evaluating the expression.
Worked Examples
Evaluate 3x + 7 when x = 4.
Substitute x = 4: 3(4) + 7
Multiply: 12 + 7
Add: 19
Evaluate 2a² − 5 when a = 3.
Substitute a = 3: 2(3)² − 5
Exponent first: 2(9) − 5
Multiply: 18 − 5
Subtract: 13
Evaluate (x + y) / 2 when x = 10 and y = 6.
Substitute: (10 + 6) / 2
Add numerator: 16 / 2
Divide: 8
Write an expression: 'five more than twice a number n'.
Twice a number n → 2n
Five more than → + 5
Evaluate 5m − 2n when m = 3 and n = 4.
Substitute: 5(3) − 2(4)
Multiply: 15 − 8
Subtract: 7
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Variables & Algebraic Expressions
Watch the guided practice walkthrough for variables and algebraic expressions, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Evaluate 4x − 3 when x = 5.
Hint: Substitute x = 5 first, then multiply, then subtract.
Evaluate x² + 2x when x = 3.
Hint: Apply the exponent before multiplying. Remember order of operations.
Write an expression for 'three less than four times a number k'.
Hint: 'Four times k' is 4k. 'Three less than' means subtract 3.
Evaluate (a + b) · c when a = 2, b = 3, c = 4.
Hint: Add inside the parentheses first, then multiply.
Identify the coefficient and constant in: 7y + 12.
Hint: The coefficient is the number in front of the variable. The constant stands alone.
Key Vocabulary
Variable
A letter (like x, y, or n) that stands for an unknown or changing number.
Example: In 3x + 5, x is the variable.
Constant
A fixed number in an expression that does not change.
Example: In 3x + 5, the constant is 5.
Algebraic Expression
A combination of variables, constants, and operations (no equals sign).
Example: 3x + 5 or 2a² − 7
Evaluate
Substitute a number for the variable and simplify to find the value.
Example: Evaluate 3x when x = 4: 3(4) = 12
Coefficient
The number multiplied by a variable.
Example: In 4x, the coefficient is 4.
Term
A single number, variable, or product of numbers and variables.
Example: 5x, 3, and 2xy are each one term.
Practice Questions
Interactive Practice — 5 Questions
Evaluate 3x + 7 when x = 4.
Which expression represents "five more than twice a number n"?
Evaluate 2a² − 5 when a = 3.
Evaluate (x + y) ÷ 2 when x = 10 and y = 6.
Evaluate 5m − 2n when m = 3 and n = 4.
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Evaluate 6x − 4 when x = 3.
Evaluate a² + 2b when a = 4 and b = 5.
Write an expression for "the product of 7 and a number k, decreased by 3".
Evaluate (p + q) × 2 when p = 5 and q = 3.
Evaluate 3m² − m + 2 when m = 2.
Common Mistakes
Forgetting to substitute the value for every occurrence of the variable.
Replace every instance of the variable with the given value before simplifying.
Ignoring order of operations after substituting — for example, adding before multiplying.
After substituting, apply PEMDAS: handle exponents, then multiply/divide, then add/subtract.
Confusing the coefficient with the constant — e.g., in 5x + 3, thinking 3 is the coefficient.
The coefficient multiplies the variable (5 in 5x). The constant stands alone (3).
Writing 'three less than twice n' as 3 − 2n instead of 2n − 3.
'Less than' reverses the order: 'three less than 2n' means 2n − 3, not 3 − 2n.
Math Tips
When evaluating, always write out the substitution step first — it prevents careless errors and makes your work easy to check.
Use parentheses when substituting negative numbers. For example, if x = −3, write 2(−3) not 2−3 to avoid sign errors.
On the SAT/ACT, "evaluate the expression" problems are common. Practice substituting quickly and accurately — they are usually 1-step questions worth easy points.
To translate word problems: "more than" → add, "less than" → subtract (and reverse order), "times" or "product" → multiply, "quotient" or "per" → divide.