Letters in maths
In algebra, we use letters to stand for numbers. A variable is a letter (like x, n or t) that represents a number that can change, or that we don’t know yet. A constant is a value that stays fixed.
In the expression 5x + 3:
| Part | Name |
|---|---|
| x | variable |
| 5 | coefficient (multiplies the variable) |
| 3 | constant |
Two jobs a variable can do
1. Stand for an unknown number. In x + 7 = 12, x is an unknown. Only one value works: x = 5.
2. Stand for a quantity that varies. In a formula like P = 4s (perimeter of a square), s can be any side length, and P changes with it.
Why use variables?
Variables let us write general rules:
- Instead of "3 + 5 = 5 + 3, 7 + 2 = 2 + 7, …" we write a + b = b + a.
- Instead of "pay is $15 times the hours worked" we write P = 15h.
Choosing letters
Any letter will do, but meaningful ones help:
- t for time, d for distance, C for cost, A for area.
- Avoid letters that look like numbers (O and 0, l and 1) when writing by hand.
Algebra writing rules
| We write | It means |
|---|---|
| 3x | 3 × x |
| xy | x × y |
| x/4 | x ÷ 4 |
| x² | x × x |
| x (alone) | 1x |
Constants in real life
In C = 2.5k + 4 (a taxi fare with k km):
- 2.5 is the rate per km (the coefficient)
- 4 is the fixed starting fee (the constant)
- k and C are variables
Worked examples
Example 1: In 7y − 2, name the variable, coefficient and constant.
Variable y, coefficient 7, constant −2.
Example 2: Write an expression for "a number increased by 9".
n + 9.
Example 3: A gym charges $30 to join plus $12 a month. Write a formula for cost C after m months.
C = 12m + 30.
Common mistakes
- Thinking a variable always means one secret number. In formulas, it varies.
- Reading 3x as "thirty-something" instead of 3 × x.
- Forgetting the hidden 1 in x (coefficient 1).
Try it yourself
- Identify the coefficient in 9m.
- Identify the constant in 4p + 11.
- Write: "6 less than a number".
- Write: "the product of a number and 8".
- In A = lw, which letters are variables?
Answers: 1) 9 2) 11 3) n − 6 4) 8n 5) A, l and w
Quick recap
- A variable stands for a number that is unknown or that changes.
- A constant is fixed; a coefficient multiplies a variable.
- Variables let us write general rules and formulas.