# Terms, Like Terms and Notation

> What a variable is, what a term is, how to spot like terms and how algebra is written.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/terms-and-like-terms/. Updated 3 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The mystery bag
Mia's teacher holds up a paper bag. "There are some marbles in here, but I am not telling you how many." She writes $m$ on the board. "This letter stands for the marbles in the bag."

Then she holds up three bags, all the same. "How many marbles now?" Mia thinks: one bag is $m$, so three bags are $m + m + m$. Her teacher writes $3m$. "Same thing, shorter."

That is algebra. A letter stands in for a number we do not know yet, and we still work with it.

**In short:** A variable is a letter that stands for a number, and a term is a number, a letter, or numbers and letters multiplied together, like $5y$. Like terms have exactly the same letters with the same powers, like $4ab$ and $9ab$.

## Before you start

You only need two things you already know:

- **Multiplication is repeated adding.** $3 \times 5$ means $5 + 5 + 5$.
- **Squares and cubes.** $5^2 = 5 \times 5 = 25$ and $2^3 = 2 \times 2 \times 2 = 8$. The small raised number is called the **power** (or **index**). It tells you how many times to multiply the number by itself.

## Variables

A **variable** (also called a **pronumeral**) is a letter or symbol that stands for a number we do not know yet.

Examples: $a$, $x$, $b^2$, $z^3$

The letter can change its value from question to question. That is why it is called a variable: it can vary.

## Terms

A **term** is one of these:

- a number on its own, such as $2$, $8$ or $27$
- a variable on its own, such as $x$
- numbers and variables multiplied (or divided) together, such as $2a$, $3xy$, $\frac{1}{2}b$ or $\frac{2pq}{z}$

Terms are separated by **plus** and **minus** signs.

### Worked example: Counting terms
How many terms are in $4x + 3y - 7$?

The plus and minus signs split it into $4x$, $3y$ and $7$. So there are **3 terms**.

The number in front of a variable is called the **coefficient**. In $7y$ the coefficient is $7$. In $x$ the coefficient is $1$, because $x$ means $1x$.

## Notation: how algebra is written

Mathematicians are lazy in a clever way. They leave out the $\times$ sign.

### Key idea: No sign means multiply
When a number sits right next to a letter, or two letters sit next to each other, they are **multiplied**.

$2x = 2 \times x$, which is 2 lots of $x$

$2a^2 = 2 \times a^2$

$3xy = 3 \times x \times y$

Two more rules for writing a term neatly:

1. Write the **number first**.
2. Write the letters in **alphabetical order**.

So $y \times 2 \times x$ is written $2xy$, not $y2x$.

### Watch out: $2x$ is not twenty-something
If $x = 4$, then $2x$ is $2 \times 4 = 8$. It is not "24". The $2$ and the $x$ are multiplied, not stuck together as digits.

## Like terms and unlike terms

**Like terms** have exactly the same letters with exactly the same powers. Only the number in front can be different.

| Like terms | Why |
|---|---|
| $2x$, $4x$, $11x$ | all are lots of $x$ |
| $x^2$, $2x^2$ | all are lots of $x^2$ |
| $3xy$, $5xy$, $10xy$ | all are lots of $xy$ |
| $x^6$, $7x^6$ | all are lots of $x^6$ |

**Unlike terms** have different letters, or the same letter with different powers.

- $x$, $y$ and $ab$ are unlike (different letters).
- $x$, $x^2$ and $x^5$ are unlike (same letter, different powers).

### Exam tip: Order of letters does not matter
$3xy$ and $5yx$ are like terms, because $x \times y$ is the same as $y \times x$. Just write the second one neatly as $5xy$.

Think of like terms as the same kind of fruit. $3$ apples and $5$ apples are like terms. $3$ apples and $5$ bananas are not. An apple is not a crate of apples either, so $x$ and $x^2$ are different too.

## Expressions and equations

An **expression** is a group of terms, like $3x + 4y$. It has no equals sign.

An **equation** says that two things have the **same value**. It always has an equals sign:

$$x + 3 = 2y - 4 \qquad 2x^2 - 3 = 15$$

The left side and the right side are balanced, like a see-saw that is level.

### Challenge: Selective level: reading tricky terms
- $\frac{x}{3}$ is one term. It means $x \div 3$, which is the same as $\frac{1}{3}x$.
- $-5ab$ is one term with coefficient $-5$. The minus sign belongs to the term after it.
- $2 \times 2 \times a \times a \times a$ is written $4a^3$. Multiply the numbers, then count each letter.
- $x^2y$ and $xy^2$ are **unlike**. In the first, $x$ is squared. In the second, $y$ is squared.

**Mind map**

```text
Algebra words
  Variable
    A letter for an unknown number
    Also called a pronumeral
  Term
    Number, letter, or both multiplied
    Split up by + and -
  Like terms
    Same letters, same powers
    $3x$ and $7x$
  Notation
    No sign means multiply
    Number first, letters in order
  Equation
    Has an equals sign
    Both sides balance
```

## Quick summary

- A **variable** (pronumeral) is a letter that stands for an unknown number.
- A **term** is a number, a variable, or numbers and variables multiplied together. Plus and minus signs separate terms.
- No sign between a number and a letter means **multiply**: $5y = 5 \times y$.
- Write the number first, then the letters in alphabetical order.
- **Like terms** have the same letters with the same powers: $4ab$ and $9ab$.
- $x$ and $x^2$ are **not** like terms.
- An **equation** has an equals sign and says both sides have the same value.

## Quiz

Try these questions from the chapter quiz.

1. Which of these is a variable?
   - A. $7$
   - B. $k$
   - C. $+$
   - D. $=$
   - Answer: B. $k$. A variable is a letter (or symbol) that stands for an unknown number.

2. What does $4p$ mean?
   - A. $4 + p$
   - B. $4 \times p$
   - C. $4 - p$
   - D. the number 4 followed by the digit p
   - Answer: B. $4 \times p$. No sign between a number and a letter means multiply.

3. Which pair are like terms?
   - A. $3x$ and $3y$
   - B. $5a$ and $5a^2$
   - C. $6m$ and $11m$
   - D. $2p$ and $2$
   - Answer: C. $6m$ and $11m$. Like terms have the same letters with the same powers. $6m$ and $11m$ are both lots of $m$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/terms-and-like-terms/#quiz