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.
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.
Practice Questions
- If y = 6, find 2y + 3.
- If p = 3 and q = 4, find p squared plus q.
- Write an expression for the total cost of buying k books at 8 each.
- If n = -2, find 5n - 1.
- A car travels at speed v km/h for t hours. Write an expression for the distance covered.