Variables & Algebraic Expressions
A variable is a box that can hold any number. An expression is a recipe built from variables and numbers. Learn to translate English into algebra — watching the order traps — and to evaluate recipes by substitution.
1 Understand
The core idea in plain language.
A variable (usually x) stands for a number you do not know yet. An algebraic expression like 3x + 2 combines variables and numbers with operations. Unlike an equation, an expression has no equals sign — it is a phrase, not a sentence.
Translating words to algebra is mostly direct: “5 more than x” → x + 5, “twice a number” → 2n. But English has order traps: “5 less than x” means start at x and go down 5 → x − 5, not 5 − x. “Less than” reverses the order.
To evaluate an expression, substitute the value for the variable, then follow order of operations. For 2 + 3x with x = 4: substitute → 2 + 3(4), multiply first → 2 + 12 = 14. Substituting does not suspend PEMDAS.
Every expression is built from terms separated by + and −. In 4x² − 2x + 7 there are 3 terms. The coefficient is the number multiplying the variable (4 and −2); the constant is the term with no variable (7).
2 See it
Diagrams that make the idea visual.
The “less than” trap
“5 less than x”: start at x, then go down 5 → x − 5.
“5 minus x” would be 5 − x — a different trip entirely. The words “less than” flip the order; the word “minus” does not.
Substitute, then PEMDAS
Evaluate 2 + 3x at x = 4:
Substitute → 2 + 3(4). Multiply first → 2 + 12. Add → 14.
Adding first (2 + 3 = 5, then 5 × 4 = 20) ignores order of operations.
3 The key rule
Replace every variable with its value (in parentheses), then evaluate with parentheses, exponents, multiplication/division, addition/subtraction. Substitution never changes the order of operations.
4 Worked examples
Follow each step. The pattern is always the same.
- “5 less than x” — the phrase “less than” reverses the order.
- Start at x, go down 5: x − 5.
Check: Test with x = 10: “5 less than 10” is 5, and 10 − 5 = 5. Correct.
- “twice n” → 2n.
- “3 more than” that → 2n + 3.
Check: Test with n = 4: twice 4 is 8, 3 more is 11; 2(4) + 3 = 11. Correct.
- Substitute x = 4: 2 + 3(4).
- Multiply first: 2 + 12.
- Add: 14.
Check: Adding first would give 20 — the order-of-operations trap. 14 is correct.
- Substitute x = 3: 1(3)² − 2(3) + 5.
- Exponent first: 9 − 6 + 5.
- Left to right: 3 + 5 = 8.
Check: 9 − 6 = 3, 3 + 5 = 8. Correct.
5 Common mistakes
These errors show up on almost every quiz. Spot them now and they will never cost you points.
6 Key vocabulary
Say these words like you mean them.
- Variable
- A letter standing for an unknown number, like x.
- Algebraic expression
- A combination of variables, numbers, and operations with no equals sign.
- Term
- A part of an expression separated by + or −. 4x² − 2x + 7 has 3 terms.
- Coefficient
- The number multiplying a variable. In −2x, the coefficient is −2.
- Constant
- A term with no variable. In 4x + 7, the constant is 7.
- Evaluate
- Substitute values for the variables and compute.
- Substitute
- Replace each variable with its given value, in parentheses.
7 Quick check
Try each one on paper first, then reveal the answer.
Key points to remember
- An expression has no equals sign — it is a phrase, not a sentence.
- “Less than” reverses order: “a less than x” → x − a. Test with numbers.
- Evaluate by substitution, then PEMDAS — substitution never outranks ×/÷.
- Terms are split by + and −; coefficients multiply variables; constants stand alone.
- Parenthesize substituted negatives: x = −3 in 5x − 2 becomes 5(−3) − 2.