Algebraic Expressions

Imagine a bag with some apples inside, but you cannot see how many. You could call that number $n$. If you add 3 more apples, the bag now holds $n + 3$ apples — even though you still do not know the exact total.

That is the whole idea of algebra: a letter stands for a number. We use letters when a number is unknown, or when it can change.

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.

Algebra has its own neat way of writing things. These conventions keep expressions short and clear, and everyone who learns them reads them the same way.

The main rules are:

• We hide the multiplication sign. $3 \times n$ is written $3n$, and $a \times b$ is written $ab$.
• The number goes before the letter: write $5y$, never $y5$.
• $1 \times x$ is just written $x$ — we do not write $1x$.
• Division is written as a fraction: $n \div 4$ becomes $\tfrac{n}{4}$.
• A letter multiplied by itself uses a power: $n \times n = n^2$.

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.

An expression is a mix of letters and numbers joined by $+$ and $-$ signs, with no equals sign. For example, $3n + 5$ is an expression.

Each separate part of an expression is a term. In $3n + 5$ there are two terms: $3n$ and $5$.

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.

Like terms have exactly the same letter part. $4a$ and $7a$ are like terms. So are $3xy$ and $9xy$. But $4a$ and $4b$ are not like terms, and neither are $a$ and $a^2$.

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.

Sometimes an expression has a bracket, like $4(x + 2)$. This means "4 lots of $(x + 2)$". To expand it, you multiply everything inside the bracket by the number outside.

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.

Getting comfortable with expressions now opens up everything else in algebra:

• Equations are just expressions set equal to something — you will simplify before you solve.
• Substitution and formulae ask you to swap a number in for a letter, which only makes sense once you read $3n$ correctly.
• Sequences are described by expressions like $2n + 1$ for the $n$th term.
• Graphs turn expressions such as $y = 2x + 1$ into straight lines you can draw.

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.