Changing the Subject of the Formula Revision Notes for HSC SSCE Mathematics Standard

Changing the Subject of the Formula

What is the subject of a formula?

When working with equations and formulas, the subject is the variable (pronumeral) that appears alone on the left-hand side of the equals sign, with no numbers attached to it. For example, in the formula $y = 3x + 5$ , the variable $y$ is the subject because it stands by itself on the left side.

Understanding how to change the subject of a formula is an essential algebraic skill. Sometimes you need to rearrange a formula to make a different variable the subject, especially when you know the values of other variables and want to find a specific one. This process is also called making a variable the subject or rearranging a formula.

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Think of the subject as the "answer" variable—it's the one you're solving for. Just as in simple equations like $x = 5$ , where $x$ is clearly the subject, more complex formulas follow the same principle.

Method for changing the subject

The key principle when changing the subject of a formula is to isolate the variable you want by moving all other terms to the opposite side of the equation. You achieve this by using opposite operations (also called inverse operations) to undo what has been done to your chosen variable.

The process involves three main steps:

1. Identify the opposite operation

Every mathematical operation has an opposite that "undoes" it:

Addition $(+)$ is opposite to subtraction $(-)$
Multiplication $(\times)$ is opposite to division $(\div)$
Squaring (raising to power $2$ ) is opposite to taking the square root $(\sqrt{})$
Raising to power $n$ is opposite to taking the $n$ th root

2. Add or subtract to both sides

When a term is added or subtracted in your equation, you remove it by adding or subtracting the same value from both sides. This maintains the balance of the equation, which is the fundamental rule in algebra—whatever you do to one side must be done to the other.

3. Multiply or divide both sides

When your variable is multiplied or divided by a number, you use the opposite operation on both sides of the equation. If it's multiplied by $40$ , divide both sides by $40$ . If it's divided by $5$ , multiply both sides by $5$ .

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The golden rule of rearranging equations: Whatever you do to one side, you must do to the other side. This maintains the balance and equality of the equation. Think of it like a set of scales—to keep them balanced, any change on one side must be matched on the other.

The goal is to systematically undo each operation that has been applied to your variable, working backwards through the order of operations, until the variable stands alone on one side of the equation.

Worked examples

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Worked Example 1: Linear equation with addition and multiplication

Consider a formula for calculating the total cost of a child's birthday party:

$C = 40n + 75$

where $C$ is the total cost in dollars and $n$ is the number of children attending.

Let's make $n$ the subject of this equation. This would allow us to calculate how many children can attend for a given budget.

Solution:

We need to isolate $n$ by removing the operations applied to it. Currently, $n$ is multiplied by 40 and then 75 is added.

Step 1: Start with the original formula. $C = 40n + 75$

Step 2: Remove the added $75$ by subtracting 75 from both sides. $C - 75 = 40n + 75 - 75$ $C - 75 = 40n$

Step 3: Rearrange to put the subject on the left-hand side (this is conventional but not mathematically necessary). $40n = C - 75$

Step 4: Remove the multiplication by $40$ by dividing both sides by 40. $\frac{40n}{40} = \frac{C - 75}{40}$

Step 5: Simplify to get the final answer. $n = \frac{C - 75}{40}$

Now the formula is rearranged with $n$ as the subject. If you know the total cost $C$ , you can substitute it into this formula to find how many children can attend.

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Worked Example 2: Quadratic equation with powers

Pythagoras' theorem describes the relationship between the sides of a right-angled triangle:

$c^2 = a^2 + b^2$

where $a$ , $b$ , and $c$ are the lengths of the sides, with $c$ being the hypotenuse (longest side).

Part a: Make $a$ the subject of the formula.

Solution:

Step 1: Write the formula. $c^2 = a^2 + b^2$

Step 2: Remove $b^2$ by subtracting it from both sides. $c^2 - b^2 = a^2$

Step 3: Rearrange with the subject on the left. $a^2 = c^2 - b^2$

Step 4: Remove the square by taking the square root of both sides. $a = \sqrt{c^2 - b^2}$

The formula is now rearranged to find the length of side $a$ .

Part b: Find the length of $a$ given $c = 7$ and $b = 5$ . Give your answer correct to one decimal place.

Solution:

Step 5: Substitute the values into the rearranged formula. $a = \sqrt{c^2 - b^2}$ $a = \sqrt{7^2 - 5^2}$

Step 6: Evaluate the expression. $a = \sqrt{49 - 25}$ $a = \sqrt{24}$ $a = 4.89897949...$

Step 7: Round to one decimal place. $a \approx 4.9$

Therefore, the length of side $a$ is approximately 4.9 units.

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Worked Example 3: Exponential equation with powers and roots

The compound interest formula calculates the final amount in an investment:

$A = P(1 + r)^n$

where $A$ is the final amount, $P$ is the principal (initial amount), $r$ is the interest rate (as a decimal), and $n$ is the number of time periods.

Let's make $r$ the subject of this formula. This would allow us to calculate what interest rate is needed to reach a target amount.

Solution:

Step 1: Write the formula. $A = P(1 + r)^n$

Step 2: Remove the multiplication by $P$ by dividing both sides by $P$ . $\frac{A}{P} = (1 + r)^n$

Step 3: Remove the power of $n$ by taking the $n$ th root of both sides. $\sqrt[n]{\frac{A}{P}} = \sqrt[n]{(1 + r)^n}$ $\sqrt[n]{\frac{A}{P}} = 1 + r$

Step 4: Remove the added $1$ by subtracting 1 from both sides. $\sqrt[n]{\frac{A}{P}} - 1 = r$

Step 5: Rearrange with the subject on the left-hand side. $r = \sqrt[n]{\frac{A}{P}} - 1$

The formula now has $r$ as the subject, allowing you to find the required interest rate for a given investment scenario.

Exam tips

When working on changing the subject of a formula in exams, keep these practical strategies in mind to avoid common mistakes and work more efficiently:

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Always perform the same operation to both sides of the equation to maintain balance
Work through operations in reverse order—undo the last operation first
Check your rearranged formula by substituting simple values
Remember that taking a square root gives the positive value in most practical contexts
Write each step clearly on a new line to avoid errors

Remember!

These key concepts form the foundation of successfully changing the subject of any formula:

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Key Points to Remember:

The subject of a formula is the variable that appears alone on the left-hand side with no numbers attached
Use opposite operations to isolate your chosen variable: addition ↔ subtraction, multiplication ↔ division, squaring ↔ square root
Whatever operation you perform on one side of the equation, you must perform on the other side to maintain balance
Work systematically through the operations applied to your variable, undoing them one at a time
For complex formulas with powers or roots, remember that the opposite of raising to power $n$ is taking the $n$ th root

Changing the Subject of the Formula (HSC SSCE Mathematics Standard): Revision Notes