# Solving Equations

> Keep the balance, undo each step and find the value of the unknown, including brackets and squares.

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

### Story time: The balance scales
In the science room, Ella puts 5 identical bags of sand and a 3 kg weight on one side of the scales. On the other side she puts 18 kg of weights. The scales are perfectly level.

"How heavy is one bag?" asks her friend.

Ella takes the 3 kg weight off the left side. To keep it level she also takes 3 kg off the right. Now 5 bags balance 15 kg. So one bag is $15 \div 5 = 3$ kg.

She has just solved the equation $5s + 3 = 18$.

**In short:** Solving an equation means finding the number the letter stands for. Do the same thing to both sides to keep it balanced, and undo the steps in reverse order: if $3x + 2 = 14$, take away 2 and then divide by 3 to get $x = 4$.

## Before you start

- An **equation** has an equals sign, and both sides have the same value.
- **Inverse operations** undo each other: $+$ undoes $-$, and $\times$ undoes $\div$. Squaring is undone by a square root.
- Square numbers: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100$. Cube numbers: $1, 8, 27, 64, 125$.

## Keep it balanced

### Key idea: The golden rule
Whatever you do to one side of an equation, do **exactly the same** to the other side. The goal is to get the letter on its own.

*Diagram: Balance scales: five bags and a 3 kilogram weight on the left balance 18 kilograms on the right*
*Five bags plus 3 kg balance 18 kg. Take 3 kg off both sides to keep it level.*

### Worked example: Two steps
$5s + 3 = 18$

Subtract 3 from both sides: $5s = 15$

Divide both sides by 5: $s = 3$

### Worked example: Undo a minus
$5a - 2 = 23$

Add 2 to both sides: $5a = 25$

Divide both sides by 5: $a = 5$

### Exam tip: Undo in reverse order
To build $5a - 2$, you take $a$, multiply by 5, then subtract 2. To undo it, go backwards: add 2 first, then divide by 5. It is like taking off your shoes before your socks.

### Exam tip: Always check
Put your answer back in. For $a = 5$: $5 \times 5 - 2 = 23$. Correct!

## Equations with brackets

You can expand first, or divide both sides by the number outside. Dividing is often quicker.

### Worked example: Divide first
$3(x - 4) = 33$

Divide both sides by 3: $x - 4 = 11$

Add 4: $x = 15$

### Worked example: Brackets with two steps inside
$9(2a - 10) = 36$

Divide by 9: $2a - 10 = 4$

Add 10: $2a = 14$

Divide by 2: $a = 7$

## Fractions, decimals and squares

### Worked example: A fraction
$\frac{m + 38}{9} = 6$

Multiply both sides by 9: $m + 38 = 54$

Subtract 38: $m = 16$

### Worked example: Decimals
$0.25n = 3$. Since $0.25 = \frac{1}{4}$, a quarter of $n$ is 3, so $n = 12$.

$3x = 4.2$. Divide by 3: $x = 1.4$.

### Worked example: Squares and cubes
$a^2 + 5 = 21$, so $a^2 = 16$ and $a = 4$, because $4 \times 4 = 16$.

$2x^2 = 50$, so $x^2 = 25$ and $x = 5$.

$b^3 = 125$, so $b = 5$, because $5 \times 5 \times 5 = 125$.

In this book the answers to square equations are positive numbers.

### Challenge: Selective level: "I am thinking of a number"
"If you multiply me by 6 and subtract 7, you get 59. What am I?"

Write it: $6n - 7 = 59$. Add 7: $6n = 66$. Divide by 6: $n = 11$.

A longer one: Joshua multiplies a number by $\frac{2}{5}$, subtracts 6, then divides by 0.75. He gets 40. Work backwards: $40 \times 0.75 = 30$, then $30 + 6 = 36$, then $36 \div \frac{2}{5} = 36 \times \frac{5}{2} = 90$. The number was 90.

**Mind map**

```text
Solving equations
  Balance
    Same thing to both sides
    Get the letter alone
  Undo
    Reverse order
    + undoes -, x undoes ÷
  Brackets
    Divide by the outside number
    Or expand first
  Squares and cubes
    $a^2 = 16$ gives $a = 4$
    $b^3 = 125$ gives $b = 5$
  Check
    Put the answer back in
```

## Quick summary

- Do the same thing to both sides to keep an equation balanced.
- Undo operations in reverse order: deal with $+$ and $-$ first, then $\times$ and $\div$.
- With brackets, divide by the outside number first, or expand.
- For a fraction, multiply both sides by the bottom number.
- If $a^2 = 16$ then $a = 4$. If $b^3 = 125$ then $b = 5$.
- Always check by putting your answer back in.

## Quiz

Try these questions from the chapter quiz.

1. Solve $3a - 1 = 14$.
   - A. $a = 4$
   - B. $a = 5$
   - C. $a = 13$
   - D. $a = 15$
   - Answer: B. $a = 5$. Add 1: $3a = 15$. Divide by 3: $a = 5$.

2. Solve $15 = 4a + 3$.
   - A. $a = 4.5$
   - B. $a = 3$
   - C. $a = 12$
   - D. $a = 2$
   - Answer: B. $a = 3$. Subtract 3: $12 = 4a$. Divide by 4: $a = 3$.

3. Solve $4 + 9x = 31$.
   - A. $x = 3$
   - B. $x = 4$
   - C. $x = 27$
   - D. $x = 35$
   - Answer: A. $x = 3$. Subtract 4: $9x = 27$. Divide by 9: $x = 3$.

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