Mastering Rearranging Formulae and Changing the Subject

In GCSE maths, a formula is a mathematical rule that links variables together. Often, you will be asked to "change the subject" of a formula. The subject is simply the variable that stands alone on one side of the equals sign. Being able to rearrange these equations is a fundamental algebraic skill that allows you to solve for different unknowns in physics, geometry, and beyond.

Think of rearranging a formula 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 to keep the equation balanced. By applying inverse operations in the correct order, you can isolate any variable you need.

The Golden Rule: Inverse Operations

To change the subject, you must undo the operations applied to the target variable. You do this by working in reverse order of operations (BIDMAS/BODMAS). If a variable is being added, you subtract; if it is being multiplied, you divide; if it is squared, you take the square root.

Example 1: Simple Rearrangement Make $x$ the subject of $y = 3x + 5$.

1. Subtract 5 from both sides: $y - 5 = 3x$
2. Divide both sides by 3: $\frac{y - 5}{3} = x$
3. Final answer: $x = \frac{y - 5}{3}$

Handling Fractions

When the subject is trapped inside a fraction, your first priority should be to remove the fraction by multiplying both sides by the denominator. This clears the way to isolate your variable.

Example 2: Rearranging with Fractions Make $a$ the subject of $v = \frac{u + at}{2}$.

1. Multiply both sides by 2: $2v = u + at$
2. Subtract $u$ from both sides: $2v - u = at$
3. Divide by $t$: $\frac{2v - u}{t} = a$
4. Final answer: $a = \frac{2v - u}{t}$

Dealing with Brackets

If your target variable is inside a bracket, you have two choices: expand the brackets first or divide by the multiplier outside the bracket. Expanding is often safer if the variable appears in multiple terms.

For example, to make $x$ the subject of $A = 2(x + b)$, you can divide by 2 first to get $\frac{A}{2} = x + b$, then subtract $b$ to get $x = \frac{A}{2} - b$.

When the Subject Appears Twice

Sometimes, the variable you want to isolate appears on both sides of the equation or in two different terms. In these cases, you must gather all terms containing the target variable onto one side, factorise, and then divide.

Example 3: Factorising Make $m$ the subject of $2(2p + m) = 3 - 5m$.

1. Expand the brackets: $4p + 2m = 3 - 5m$
2. Add $5m$ to both sides: $4p + 7m = 3$
3. Subtract $4p$ from both sides: $7m = 3 - 4p$
4. Divide by 7: $m = \frac{3 - 4p}{7}$

Common Mistakes

• Forgetting to apply operations to everything: A common error is failing to apply an operation to every term on the other side of the equation. If you divide by 2, ensure every term is divided.
• Incorrect order of operations: Always undo addition and subtraction before multiplication and division, unless the terms are grouped in brackets or fractions.
• Sign errors: When moving terms across the equals sign, remember that positive terms become negative and vice versa.

Frequently Asked Questions

What does it mean to change the subject? It means rearranging the formula so that a specific variable is isolated on one side of the equals sign.

Do I have to write the subject on the left? By convention, the subject is usually written on the left, but $x = y + 2$ is mathematically identical to $y + 2 = x$.

What if the variable is squared? If you have $y = x^2$, you must take the square root of both sides to isolate $x$, resulting in $x = \sqrt{y}$.

Conclusion

Rearranging formulae is a core skill that builds your confidence in algebra. By systematically applying inverse operations and keeping your equations balanced, you can tackle even the most complex problems. Ready to see these steps in action? Head over to MathInstructor AI to generate a free, narrated animated lesson on this topic and master the art of changing the subject today.