Expressions using Letter-Numbers

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The language of algebra

Algebra has its own small vocabulary. Tap each word to see what it means — and how a tiny expression like 3n + 2 is built from these pieces.

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The three big ideas

In maths we often meet a number we do not know yet, or one that keeps changing. Instead of leaving a blank, we use a letter — called a letter-number or variable — to stand for it.

A letter such as n, x or a can stand for any number.
Once we choose the letter, one short rule can describe a whole pattern.
Example: a matchstick pattern needs 2n + 1 sticks to make n shapes. For 1 shape it is 3 sticks, for 2 shapes it is 5, for 3 shapes it is 7 — the rule 2n + 1 captures them all at once.

This is the whole power of algebra: write the rule once with a letter, and it works for every value of that letter.

Take the expression 3n + 2. We can name each part:

A term is a single part of the expression, separated by + or −. Here the terms are 3n and 2.
A coefficient is the number multiplying a variable. In the term 3n, the coefficient is 3.
A constant is a fixed number whose value never changes. Here the constant is 2.

Reading these parts correctly is the key to working with any expression — like learning the parts of a sentence before you write one.

To find the value of an expression, we substitute — that is, we put a number in place of the letter and then calculate (multiply before you add).

If n = 4, then 3n + 2 = 3 × 4 + 2 = 12 + 2 = 14.
If n = 1, the same expression gives 3 × 1 + 2 = 5.

Worked example. A pattern uses 2n + 1 matchsticks for n shapes. How many sticks are needed for n = 5?

Step 1. Substitute n = 5 into the expression: 2n + 1 → 2 × 5 + 1.

Step 2. Multiply first (coefficient × value): 2 × 5 = 10.

Step 3. Add the constant: 10 + 1 = 11. So 5 shapes need 11 matchsticks.

Common mistake: confusing an expression with an equation. An expression like 3n + 2 has no equals sign — it only describes a value. An equation like 3n + 2 = 11 has an equals sign and can be solved. Do not add an "=" where there is not one.

Where you'll meet it

Letter-numbers in the real world

Formulas everywhere

Every formula is an expression in letter-numbers. The area of a rectangle is area = length × breadth, and its perimeter is 2 × (length + breadth). Learn the letters once and the same formula works for any size of rectangle.

Spreadsheets & coding

In a spreadsheet a cell like = A2 × B2 uses letter-names that stand for changing numbers. In coding, a variable such as score = score + 1 works exactly like a letter-number in maths — a name that holds a value you can change.

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Expressions usingLetter-Numbers