Mastering Rearranging Formulae at GCSE

In GCSE maths, you will frequently encounter equations where you need to isolate a specific variable. This process is known as changing the subject of a formula. The subject is the variable that stands alone on one side of the equals sign, representing the value you are trying to calculate.

Being able to rearrange formulae is a fundamental algebraic skill. It allows you to manipulate equations to solve for different unknowns, which is essential for topics ranging from coordinate geometry to physics kinematics. By mastering the use of inverse operations, you can confidently transform any formula to suit your needs.

The Golden Rule: Inverse Operations

To rearrange a formula, you must treat it exactly like solving a linear equation. The golden rule is that whatever you do to one side of the equals sign, you must do to the other. You achieve this by applying inverse operations in the reverse order of operations (BIDMAS/BODMAS).

Think of the subject as being trapped by other numbers and variables. Your goal is to peel away these layers one by one using the opposite operation:

• Addition $\leftrightarrow$ Subtraction
• Multiplication $\leftrightarrow$ Division
• Squaring $\leftrightarrow$ Square rooting

Step-by-Step: Changing the Subject

Let us look at a standard example. Suppose we have the formula $v = u + at$ and we want to make $a$ the subject.

1. Identify the term containing $a$: $at$.
2. Remove the $u$ by subtracting it from both sides: $v - u = at$.
3. Isolate $a$ by dividing both sides by $t$: $a = \frac{v - u}{t}$.

By following these logical steps, you ensure the equation remains balanced while isolating your target variable.

Handling Brackets and Fractions

When a formula contains brackets, it is often best to expand them first, unless the subject is trapped inside the bracket. For example, to make $w$ the subject of $P = 2(l + w)$:

1. Expand the brackets: $P = 2l + 2w$.
2. Subtract $2l$ from both sides: $P - 2l = 2w$.
3. Divide by 2: $w = \frac{P - 2l}{2}$.

If you have fractions, multiply every term by the denominator to clear them immediately. This simplifies the expression significantly before you begin isolating the subject.

Worked Example 1: Linear Rearrangement

Question: Make $x$ the subject of $3x - 5y = 10$.

Step 1: Add $5y$ to both sides to isolate the $x$ term. $$3x = 10 + 5y$$ Step 2: Divide the entire right side by 3. $$x = \frac{10 + 5y}{3}$$

Worked Example 2: Factoring

Sometimes the subject appears in two different terms. You must factorise to bring them together.

Question: Make $n$ the subject of $t = n + mn$.

Step 1: Factorise $n$ out on the right side. $$t = n(1 + m)$$ Step 2: Divide both sides by the bracket $(1 + m)$ to isolate $n$. $$n = \frac{t}{1 + m}$$

Common Mistakes

• Forgetting to apply operations to all terms: A common error is failing to divide or multiply every single term on the other side of the equation. Always ensure the entire side is treated equally.
• Incorrect order of operations: Students often try to divide before adding or subtracting. Remember to undo addition and subtraction first, then multiplication and division.
• Losing the negative sign: When moving terms across the equals sign, ensure you correctly track whether a term becomes positive or negative.

Frequently Asked Questions

What does it mean to make a variable the subject? It means to rearrange the formula so that the variable is isolated on one side of the equals sign, usually written as $x = \dots$.

Do I always have to expand brackets? Not always. If the subject is inside a bracket, you might be able to divide by the multiplier outside the bracket first to save time.

What if the subject appears twice? If the subject appears in multiple terms, you must factorise it out so that it appears only once, then divide by the remaining bracket.

Conclusion

Rearranging formulae is a core skill that unlocks success across the entire GCSE maths curriculum. By consistently applying inverse operations and keeping your working clear, you can tackle even the most complex algebraic manipulations. Ready to see these steps in action? Head over to MathInstructor AI to generate a free, narrated animated lesson on rearranging formulae tailored to your learning style.