# Adding and Subtracting Terms

> Collect like terms to make long expressions short, and leave unlike terms alone.

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

### Story time: Sorting the canteen order
Ravi helps at the school canteen. The order sheet says: 5 pies, 3 juices, 2 pies, 4 juices, 1 pie.

He does not read it out in that messy order. He groups it: pies $5 + 2 + 1 = 8$, juices $3 + 4 = 7$. Now the order is "8 pies and 7 juices".

If $p$ stands for a pie and $j$ for a juice, the sheet is $5p + 3j + 2p + 4j + p$, and Ravi's tidy version is $8p + 7j$. That tidying is called **collecting like terms**.

**In short:** You can only add or subtract like terms. Add or subtract the numbers in front and keep the letters the same, so $3x + 5x = 8x$, but $3x + 5y$ stays as it is.

## Before you start

- **Like terms** have the same letters with the same powers, such as $3x$ and $7x$. (See the chapter Terms, Like Terms and Notation.)
- $x$ on its own means $1x$.
- Adding and subtracting with negative numbers, for example $3 - 5 = -2$. (Covered fully in Negative and Positive Numbers.)

## Only like terms can be combined

### Key idea: The rule
You can add or subtract **like terms** by adding or subtracting the numbers in front. The letters stay the same.

$5x + 3x = 8x$

$14y - 3y = 11y$

$x^2 + 3x^2 = 4x^2$

$14y^2 - 10y^2 = 4y^2$

Why does it work? $5x$ is five lots of $x$ and $3x$ is three lots of $x$. Together that is eight lots of $x$, just like 5 apples and 3 apples make 8 apples.

### Watch out: Do not change the letters
$x^2 + 3x^2 = 4x^2$, not $4x^4$. When you add, you are counting how many $x^2$ you have. The power does not change.

Also, $7y + y = 8y$, not $7y^2$. Remember $y = 1y$.

## Unlike terms stay apart

$3x + 4y$ cannot be made shorter. Apples and bananas do not combine into "7 apple-bananas".

$$3x + 4y + 2x^2 - 2y = 3x + 2x^2 + 2y$$

Here only the $y$ terms combine, because $4y - 2y = 2y$. The $3x$ and $2x^2$ are unlike, so they stay as they are.

## Simplifying longer expressions

Follow these steps:

1. **Underline or circle** each group of like terms. Use a different mark for each group.
2. Keep the **sign in front** of each term with it. The sign belongs to the term on its right.
3. Add up each group.
4. Write the answer, usually letters in alphabetical order and the plain number last.

### Worked example: Collect like terms
Simplify $13a + 7b - 5a + 2b$.

$a$ terms: $13a - 5a = 8a$

$b$ terms: $7b + 2b = 9b$

Answer: $8a + 9b$

### Worked example: With a plain number
Simplify $8a + 4b - 3a + 2b + 5 - 2a$.

$a$ terms: $8a - 3a - 2a = 3a$

$b$ terms: $4b + 2b = 6b$

Number: $5$

Answer: $3a + 6b + 5$

### Worked example: When a group adds to zero
Simplify $9x + 2x - 6x - 5x + y$.

$x$ terms: $9 + 2 - 6 - 5 = 0$, so $0x = 0$.

Answer: $y$

### Exam tip: Check by substituting
Pick an easy number, like $a = 1$ and $b = 2$, and work out the long expression and your answer. They must give the same value. For the example above: $8 + 8 - 3 + 4 + 5 - 2 = 20$ and $3 + 12 + 5 = 20$. It matches.

### Challenge: Selective level: watch the signs and powers
Simplify $7x + 5x^2 - 3x + 3x^2 - 5 + 5x$.

$x$ terms: $7x - 3x + 5x = 9x$

$x^2$ terms: $5x^2 + 3x^2 = 8x^2$

Number: $-5$

Answer: $8x^2 + 9x - 5$

A favourite trap: $13m + 7n - 15 - 4m + 3n - 4mn$. The $4mn$ term has **both** letters, so it is unlike $4m$. Answer: $9m + 10n - 4mn - 15$.

**Mind map**

```text
Adding and subtracting terms
  Like terms
    Add or subtract the numbers
    Letters and powers stay the same
  Unlike terms
    Cannot combine
    $3x + 4y$ stays as it is
  Steps
    Mark each group
    Sign goes with the term after it
    Add each group
  Traps
    $x^2 + x^2 = 2x^2$
    $y = 1y$
    $mn$ is not like $m$
```

## Quick summary

- Only **like terms** can be added or subtracted.
- Add or subtract the numbers in front. The letters and powers do not change.
- Each term takes the sign in front of it.
- Unlike terms are left as they are.
- A group can add to zero and disappear.
- Check by putting in an easy number for each letter.

## Quiz

Try these questions from the chapter quiz.

1. $15a + 7a = $
   - A. $22a$
   - B. $22a^2$
   - C. $8a$
   - D. $105a$
   - Answer: A. $22a$. Both are lots of $a$, so add the numbers: $15 + 7 = 22$.

2. $28k - 17k = $
   - A. $45k$
   - B. $11k$
   - C. $11$
   - D. $11k^2$
   - Answer: B. $11k$. Subtract the numbers: $28 - 17 = 11$. The letter stays.

3. $4x^2 + 2x^2 = $
   - A. $6x^4$
   - B. $8x^2$
   - C. $6x^2$
   - D. $6x$
   - Answer: C. $6x^2$. Add the numbers: $4 + 2 = 6$. The power stays as 2.

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