# Substitution: Finding the Value

> Swap letters for numbers, work out the value, fill in tables and use made-up operations.

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

### Story time: The price board
At the fruit shop, the price board says "Cost $= 3a + 2$ dollars", where $a$ is the number of apples you buy. The 2 dollars is for the paper bag.

Noah buys 5 apples. He swaps $a$ for 5: $3 \times 5 + 2 = 17$. He pays 17 dollars.

His sister buys 10 apples: $3 \times 10 + 2 = 32$ dollars. Same rule, different number. Swapping a letter for a number is called **substitution**.

**In short:** Substitution means putting a number in place of each letter, then working out the answer with the order of operations. If $x = 3$, then $2x + 1 = 2 \times 3 + 1 = 7$.

## Before you start

- No sign means multiply: $3a = 3 \times a$, $ab = a \times b$.
- Powers: $x^2 = x \times x$.
- **Order of operations:** brackets first, then powers, then $\times$ and $\div$, then $+$ and $-$. (The full rules are in Order of Operations.)

## How to substitute

### Key idea: Three steps
1. Write the expression.
2. Replace each letter with its number. Put the number in **brackets** if that helps you see the multiplication.
3. Work it out using the order of operations.

### Worked example: One letter
Find $3x + y$ when $x = 2$ and $y = 5$.

$3 \times 2 + 5 = 6 + 5 = 11$

### Worked example: Powers
Find $3x^2 - y$ when $x = 2$ and $y = 4$.

$3 \times 2^2 - 4 = 3 \times 4 - 4 = 12 - 4 = 8$

The power is done first. Only the $x$ is squared, not the 3.

### Watch out: $2x^2$ is not $(2x)^2$
When $x = 3$: $2x^2 = 2 \times 9 = 18$. But $(2x)^2 = 6^2 = 36$. The power only belongs to what is right next to it, unless there are brackets.

### Worked example: Two letters multiplied
Find $ab + a^2$ when $a = 3$ and $b = 1$.

$3 \times 1 + 3^2 = 3 + 9 = 12$

### Worked example: Fractions and brackets
Find $(b + a)^2 \div 5$ when $a = 2$ and $b = 3$.

Brackets first: $3 + 2 = 5$. Then square: $25$. Then divide: $25 \div 5 = 5$.

Find $\frac{2ab}{4} - 1$ when $a = 2$ and $b = 3$.

Top: $2 \times 2 \times 3 = 12$. Then $12 \div 4 = 3$, and $3 - 1 = 2$.

## Rules and tables

A rule like $y = 4x - 3$ tells you how to get $y$ from $x$. Substitute each $x$ value to fill in a table.

| $x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $y = 4x - 3$ | 1 | 5 | 9 | 13 | 17 |

Notice $y$ goes up by 4 each time $x$ goes up by 1. The number in front of $x$ tells you the jump.

### Worked example: A rule with a square
$y = 2x^2 - 1$ when $x = 5$: $2 \times 25 - 1 = 49$.

$y = \frac{14 + x}{5}$ when $x = 6$: $\frac{20}{5} = 4$.

### Exam tip: Spot the jump
In $y = 40 - 2x$, every time $x$ goes up by 1, $y$ goes **down** by 2. Use this to check a whole table quickly.

## Made-up operations

Test writers love inventing a new symbol and telling you what it means. Just substitute.

### Worked example: A star rule
If $P * Q = \sqrt{P + Q} + P$, find $12 * 13$.

Here $P = 12$ and $Q = 13$.

$\sqrt{12 + 13} + 12 = \sqrt{25} + 12 = 5 + 12 = 17$

The square root $\sqrt{25}$ asks "which number times itself makes 25?" The answer is 5.

### Challenge: Selective level: careful substitution
If $A * B = A^2 + B - 10$, then $7 * 3 = 49 + 3 - 10 = 42$.

Watch the order: $3 * 7 = 9 + 7 - 10 = 6$. Swapping the numbers can change the answer.

Working backwards: if $y = 3x + 2$ and $y = 20$, then $3x + 2 = 20$, so $3x = 18$ and $x = 6$.

**Mind map**

```text
Substitution
  Steps
    Swap each letter for its number
    Use the order of operations
  Powers
    $2x^2$: square first, then times 2
    $(2x)^2$: brackets first
  Tables
    Use a rule like $y = 4x - 3$
    Look for the jump
  Made-up operations
    $P * Q$ tells you what to do
    Substitute carefully
```

## Quick summary

- Substitution means replacing each letter with its number.
- Then work it out with the order of operations: brackets, powers, $\times \div$, $+ -$.
- $2x^2$ squares only the $x$. $(2x)^2$ squares the whole bracket.
- A rule such as $y = 4x - 3$ lets you fill in a table of values.
- For a made-up operation like $P * Q$, put the first number in for $P$ and the second for $Q$.

## Quiz

Try these questions from the chapter quiz.

1. If $a = 2$ and $b = 3$, what is $2a + b$?
   - A. 5
   - B. 7
   - C. 8
   - D. 25
   - Answer: B. 7. $2 \times 2 + 3 = 4 + 3 = 7$.

2. If $x = 6$, what is $y$ when $y = 3x + 2$?
   - A. 11
   - B. 36
   - C. 20
   - D. 32
   - Answer: C. 20. $3 \times 6 + 2 = 18 + 2 = 20$.

3. If $a = 2$ and $b = 3$, what is $5a - 3b$?
   - A. 1
   - B. 2
   - C. 19
   - D. 7
   - Answer: A. 1. $5 \times 2 - 3 \times 3 = 10 - 9 = 1$.

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