Variables – The Letters of Algebra
A variable is a letter that stands in place of a number that is unknown or that can change. Variables are the most fundamental idea in algebra — without them, every problem would have to be solved from scratch.
The choice of x as the classic symbol for an unknown is often traced to René Descartes, who in his 1637 La Géométrie used letters from the end of the alphabet (x, y, z) for unknowns and letters from the start (a, b, c) for known quantities — a convention still followed in maths classrooms today. Variables are what let a single formula work for every case: the same rule d = st (distance = speed × time) applies whether you are walking, driving, or launching a rocket, because s, t, and d can stand for any values at all.
What Is a Variable?
A variable is a symbol, usually a letter, that represents an unknown or changing value. Common letters used are x, y, z, a, b, n, and t.
Why Use Letters?
Suppose you want to write a rule for the total cost of buying pencils. If each pencil costs 30p and you buy n pencils, the total is 30n pence. Here n can be any number — 1, 5, 20, or 100. The variable n makes the rule work for every possible quantity.
Variables in Different Contexts
| Context | Variable | Meaning |
|---|---|---|
| Distance = Speed times Time | d, s, t | Any distance, speed, or time |
| Area of a rectangle | l, w | Any length or width |
| Temperature conversion | C, F | Any Celsius or Fahrenheit value |
| An unknown age | a | A specific but unknown number |
Variables vs Specific Unknowns
Sometimes a variable represents a specific unknown value that we are trying to find. In 2x = 10, x is unknown but fixed — it must be 5. In y = 2x + 1, both x and y are variables because their values change together.
Reading Algebraic Notation
| Written | Meaning |
|---|---|
| 3x | 3 times x (the multiplication sign is omitted) |
| x + 5 | x plus 5 |
| x / 4 or x over 4 | x divided by 4 |
| x squared | x times x |
| 2x + 3y | twice x added to three times y |
Substituting Values
Substitution means replacing a variable with a specific number and calculating the result.
3(4) + 7 = 12 + 7 = 19.
4(2) - 3(5) = 8 - 15 = -7.
Key Takeaways
- A variable is a letter representing an unknown or changing number.
- The multiplication sign between a number and a variable is omitted: 3x means 3 times x.
- Substitution replaces variables with numbers to evaluate an expression.
- Variables allow one rule to work for infinitely many situations.