Variables and Expressions
Learn what variables and algebraic expressions are, how to write them from word descriptions, and how to evaluate them by substituting values.
Every spreadsheet formula, every software calculation, every financial model starts with a variable — a placeholder for a number that changes. When your phone app calculates your monthly payment or your boss asks you to build a budget model, you're writing and evaluating algebraic expressions. This is where all of math transitions from arithmetic (fixed numbers) to algebra (flexible relationships).
- Define a variable and write an algebraic expression from a word description
- Identify the parts of an expression: constants, variables, coefficients, and terms
- Evaluate any expression by substituting a number for each variable
An algebraic expression is a mathematical phrase that contains at least one variable. Unlike an equation, it has no equals sign — it simply describes a quantity.
\[ \underbrace{3}_{\text{coefficient}} \underbrace{x}_{\text{variable}} + \underbrace{5}_{\text{constant}} \]To evaluate an expression, replace each variable with a given number and simplify using the order of operations. If \(x = 4\): \(3(4) + 5 = 12 + 5 = 17\).
Anatomy of an Expression
Expression: 5x² − 3x + 8
| Part | Example | What it is |
|---|---|---|
| Term 1 | 5x² | coefficient 5, variable x, exponent 2 |
| Term 2 | −3x | coefficient −3, variable x |
| Term 3 | 8 | constant term (no variable) |
3 terms total. The degree of the expression is 2 (highest exponent).
Write an algebraic expression for: "7 more than twice a number n."
- "twice a number n" → \(2n\)
- "7 more than" → add 7
The expression is \(2n + 7\). It has two terms: the variable term \(2n\) and the constant \(7\).
Evaluate \(4x^2 - 3x + 1\) when \(x = -2\).
- Substitute: \(4(-2)^2 - 3(-2) + 1\)
- Exponent first: \(4(4) - 3(-2) + 1\)
- Multiply: \(16 + 6 + 1\)
- Add: \(23\)
The expression evaluates to 23 when \(x = -2\).
A phone plan charges $25 per month plus $0.10 per text message. Write an expression for the monthly cost and evaluate it for 80 texts.
Expression: Let \(t\) = number of texts. Monthly cost = \(0.10t + 25\).
Evaluate at \(t = 80\):
\[ 0.10(80) + 25 = 8 + 25 = \$33 \]The monthly bill for 80 texts is $33.
Test yourself before moving on:
- In \(6m - 4\), identify the coefficient, variable, and constant.
- Write an expression for: "the product of 5 and a number, decreased by 3."
- Evaluate \(3a + 2b\) when \(a = 4\) and \(b = -1\).
▶ Show Answers
- Coefficient: 6, Variable: m, Constant: −4.
- \(5n - 3\) (or any letter for "a number").
- \(3(4) + 2(-1) = 12 - 2 =\) 10.
- Confusing expressions and equations: An expression has no equals sign — it represents a value. An equation has an equals sign and can be solved. \(3x + 2\) is an expression; \(3x + 2 = 8\) is an equation.
- Missing the negative sign on substitution: When substituting a negative number, always use parentheses: \(3(-2)^2\) not \(3-2^2\). The parentheses ensure the negative is handled correctly.
- Coefficient of 1 is invisible: The expression \(x + 4\) has an implied coefficient of 1 on \(x\) — it means \(1x + 4\). Don't forget it when combining like terms later.
- A variable is a letter representing an unknown — expressions describe relationships involving variables.
- Terms are separated by + or −; each term is a product of a coefficient and variables.
- To evaluate: substitute the given value(s) and simplify using order of operations.
- Always use parentheses when substituting negative numbers to avoid sign errors.
Every line of code that calculates something is an algebraic expression in disguise.
A developer writing totalCost = price * quantity + shippingFee is coding
the expression \(c = pq + s\). Data scientists write expressions to transform datasets —
normalizing a variable, applying a formula, building a prediction model. Variable fluency
is the bridge between arithmetic and programming logic.
Calculator Connection
The PEMDAS / Order of Operations Calculator evaluates any expression step by step — great for checking your substitution work and confirming the order of operations was applied correctly.
Try it with the Calculator
Apply what you've learned with this tool.
Variables and Expressions - Quiz
5 questions per attempt · Beginner · Pass at 70%
Start Quiz →