Rearranging formulae Revision Notes for AQA GCSE Further Maths

Rearranging formulae

What does it mean to rearrange a formula?

When we rearrange a formula, we're changing which variable appears on its own on one side of the equation. The variable that stands alone is called the subject of the formula. For example, in the formula $C = 2\pi r$ , $C$ is the subject because it's isolated on the left side.

Sometimes you need to calculate a different variable from the one that's currently the subject. To do this, you need to rearrange the formula using algebraic operations, always remembering the golden rule: whatever you do to one side of the equation, you must do to the other side too.

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The Golden Rule of Rearranging Formulae: Whatever mathematical operation you perform on one side of the equation, you must perform the exact same operation on the other side. This keeps the equation balanced and valid.

Simple rearrangements using basic operations

The most straightforward rearrangements involve using inverse operations like division and multiplication. Let's look at how this works with a familiar formula.

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Worked Example: Making r the subject of C = 2πr

Starting with the circumference formula $C = 2\pi r$ , we want to isolate $r$ . Since $r$ is being multiplied by $2\pi$ , we need to divide both sides by $2\pi$ :

Divide both sides by $2\pi$ : $\frac{C}{2\pi} = r$
Therefore: $r = \frac{C}{2\pi}$

This process involves undoing the multiplication by using division, which is the inverse operation.

Working with squares and square roots

When dealing with formulae involving squares and square roots, you need to be more careful about the steps involved and consider whether positive or negative values are possible.

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Worked Example: Making x the subject of h = √(x² + y²)

This formula comes from Pythagoras' theorem. To isolate $x$ , we need to work through several steps:

The process involves:

Square both sides to eliminate the square root: $h^2 = x^2 + y^2$
Subtract $y^2$ from both sides: $h^2 - y^2 = x^2$
Make $x^2$ the subject: $x^2 = h^2 - y^2$
Take the square root of both sides: $x = \pm\sqrt{h^2 - y^2}$

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Important note about square roots: When taking square roots, remember that there are usually both positive and negative solutions. However, in many real-world contexts (like measuring lengths), only the positive value makes sense.

More complex rearrangements

Some formulae require multiple steps and involve expanding brackets, collecting like terms, and factorising. Let's examine two increasingly complex examples.

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Worked Example: Rearranging v = u + at to make a the subject

This is a motion formula. To isolate $a$ :

Subtract $u$ from both sides: $v - u = at$
Divide both sides by $t$ : $\frac{v - u}{t} = a$
Therefore: $a = \frac{v - u}{t}$

Notice how we write the final answer with $a$ on the left-hand side for clarity.

Dealing with variables that appear multiple times

When the variable you want to make the subject appears more than once in the formula, you need to collect all terms containing that variable before factorising.

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Worked Example: Making t the subject of at = 3(t + 2)

The step-by-step process is:

Expand the brackets: $at = 3t + 6$
Collect all terms with $t$ on one side: $at - 3t = 6$
Factorise: $t(a - 3) = 6$
Divide both sides by $(a - 3)$ : $t = \frac{6}{a - 3}$

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Key tip: When you have brackets in the denominator of your final answer, you don't need to write additional brackets around them.

Working with fractions

Formulae involving fractions require careful manipulation, often starting by multiplying both sides by the denominator.

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Worked Example: Making x the subject of y = (x + 2)/(1 + 3x)

This requires several careful steps:

Multiply both sides by $(1 + 3x)$ : $y(1 + 3x) = x + 2$
Expand the brackets: $y + 3xy = x + 2$
Collect all $x$ terms on one side: $3xy - x = 2 - y$
Factorise the left side: $x(3y - 1) = 2 - y$
Divide both sides by $(3y - 1)$ : $x = \frac{2 - y}{3y - 1}$

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Key Strategies for Success

Work systematically: Always perform the same operation to both sides of the equation
Use inverse operations: Addition undoes subtraction, multiplication undoes division, and so on
Collect like terms: When your target variable appears multiple times, gather all instances together
Factorise when necessary: This allows you to isolate the variable when it appears in multiple terms
Check your work: Substitute values back into both the original and rearranged formulae to verify they give the same result

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Key Points to Remember:

The subject of a formula is the variable that stands alone on one side of the equation
Always perform the same operation to both sides when rearranging
Use inverse operations to "undo" what's being done to your target variable
When a variable appears multiple times, collect all terms containing it and then factorise
Square roots can have both positive and negative solutions, but context often determines which to use

Rearranging formulae (AQA GCSE Further Maths): Revision Notes