Algebraic Expressions

A letter can play two roles:

• A variable changes — like $t$ for the time on a journey.
• A constant stays fixed — like the number 3 in $n + 3$.

The best part is that letters obey exactly the same rules as numbers. So $n + n = 2n$, just as 4 apples plus 4 apples make 8 apples. Once you trust that, algebra stops feeling scary and starts feeling like counting.

• A letter stands for a number — known, unknown or changing.
• A variable changes; a constant stays the same.
• Letters follow the same rules as numbers.
• Picture real objects (apples, coins) to stay grounded.

These rules are not there to confuse you — they save writing and make every expression easier to read. Once you are used to them, $\tfrac{n}{4} + 5n$ reads as smoothly as a sentence.

• Hide the times sign: $3 \times n$ becomes $3n$.
• Number first, letter second: $5y$, not $y5$.
• Write $x$, not $1x$; write $\tfrac{n}{4}$ for $n \div 4$.
• $n \times n$ is written $n^2$.

Every term that has a letter also has a number multiplying it — that number is the coefficient.

A few useful points:

• The $+$ or $-$ sign in front belongs to its term, so in $4x + 7y - 2$ the last term is $-2$.
• A plain number with no letter, like $5$, is called a constant term.
• If a term is just $x$, its coefficient is 1 (because $x$ means $1x$).

Naming the parts of an expression carefully makes the next step — collecting like terms — much easier.

• An expression has terms but no equals sign.
• A term is one part of an expression.
• The coefficient is the number multiplying the letter.
• The coefficient of $x$ on its own is 1.

To simplify an expression, you collect the like terms together — just add or subtract their coefficients. Think of it as sorting fruit: you can add apples to apples, but you cannot add an apple to a banana.

So $3a + 2b + 5a$ simplifies to $8a + 2b$. The $a$-terms join to give $8a$; the $2b$ stays on its own because there is nothing else like it.

Watch the signs: in $7x - 3 + 2x + 5$, the $x$-terms give $7x + 2x = 9x$ and the numbers give $-3 + 5 = 2$, so the answer is $9x + 2$. A simplified expression is shorter but means exactly the same thing.

• Like terms have identical letter parts.
• Collect like terms by adding or subtracting coefficients.
• Unlike terms (e.g. $a$ and $b$) cannot be combined.
• Keep the sign attached to each term as you sort.

So $4(x + 2) = 4 \times x + 4 \times 2 = 4x + 8$.

A few things to be careful about:

• Multiply every term inside, not just the first one.
• A letter outside works too: $x(x + 3) = x^2 + 3x$.
• Mind the signs: $5(2y - 3) = 10y - 15$.

Why does this work? Because $4(x + 2)$ really is four copies of $(x + 2)$ added together: $(x+2)+(x+2)+(x+2)+(x+2)$, which collects to $4x + 8$. Expanding is just a fast way to do that.

• Expanding means multiplying out a bracket.
• The outside term multiplies every term inside.
• Keep track of $+$ and $-$ signs as you multiply.
• $x(x + 3) = x^2 + 3x$ — a letter can be outside too.

In everyday life, expressions describe patterns and pricing — like working out the cost of $n$ tickets at $8 each plus a $3 booking fee, which is $8n + 3$.

If algebra ever feels hard later, it is often a convention or a like-terms slip underneath. Come back to this guide whenever you need a refresher — there is no prize for rushing.

• Equations build directly on simplified expressions.
• Substitution needs you to read algebra shorthand correctly.
• Sequences and graphs are described using expressions.
• Expressions model real costs and patterns.

Only collect terms with the same letter part. $3x + 4y$ is already simplified — it cannot be made shorter.

How to avoid it

$6x + 2$