Rearranging Formulas Revision Notes for Edexcel GCSE Maths

Rearranging formulas

What is rearranging formulas?

Rearranging formulas means changing a formula so that a different letter becomes the subject. The subject is the letter that appears on its own on one side of the equals sign. For example, if you have the formula $2x + z = 3(y + 2p)$ and you want to make $y$ the subject, you need to rearrange it to get $y = ...$ with everything else on the right-hand side.

infoNote

The subject of a formula is always the letter that stands alone on one side of the equals sign. Think of it as the "answer" that the formula is calculating.

The 7-step method

Rearranging formulas uses a very similar approach to solving equations. There's a systematic 7-step method that works for most formula rearrangement problems:

Here's how each step works:

Remove square root signs - If there are any square roots, eliminate them by squaring both sides of the equation
Remove fractions - Multiply every term by the denominator to clear fractions
Multiply out brackets - Expand any brackets using normal algebraic rules
Collect terms - Get all terms containing the subject letter on one side and everything else on the other side
Combine like terms - Reduce to the form $Ax = B$ where A and B can be numbers, letters, or a combination. You might need to factorise here
Divide by the coefficient - Divide both sides by A to get the subject on its own
Take square roots if needed - If you end up with $x^2 = ...$ , take the square root of both sides (don't forget the ± sign)

chatImportant

Key Point: You don't always need all 7 steps - just skip the ones that don't apply to your particular problem. This flexibility is what makes the method so powerful.

Working with fractions

When the subject appears in a fraction, you'll mainly use steps 2, 4, 5, and 6. Here's how it works:

lightbulbExample

Make $b$ the subject of the formula:

$a = \dfrac{5b + 3}{4}$

Step 1: There aren't any square roots, so ignore this step.

Step 2: Get rid of any fractions.

Multiply both sides by $4$ :

$4a = \dfrac{4(5b + 3)}{4}$

$4a = 5b + 3$

Step 3: There aren't any brackets, so ignore this step.

Step 4: Collect all the subject terms ( $b$ terms) on one side and non-subject terms on the other.

$5b = 4a - 3$

Step 5: It's now in the form $Ax = B$ .

Here, $A = 5$ and $B = 4a - 3$ .

Step 6: Divide both sides by $5$ .

$b = \dfrac{4a - 3}{5}$

The process involves:

Multiplying through by the denominator to eliminate the fraction
Collecting all subject terms on one side
Rearranging to get the subject on its own

infoNote

When dealing with fractions, always multiply through by the denominator early in the process. This clears the fraction and makes the remaining steps much simpler to handle.

Dealing with squares and square roots

When your subject is squared or under a square root, you'll need to use steps 1 and 7. This is where the method really shows its flexibility:

lightbulbExample

Example 1

Make $v$ the subject of the formula:

$u = 4v^2 + 5w$

Step 1–3: No square roots, fractions, or brackets, so skip these.

Step 4: Collect all subject terms on one side and non-subject terms on the other.

$4v^2 = u - 5w$

Step 5: It's now in the form $Ax^2 = B$ (where $A = 4$ , $B = u - 5w$ ).

Step 6: Divide both sides by $4$ .

$v^2 = \dfrac{u - 5w}{4}$

Step 7: Square root both sides.

$v = \pm \sqrt{\dfrac{u - 5w}{4}}$

(Don't forget the $\pm$ when square rooting.)

Example 2

Make $n$ the subject of the formula:

$m = \sqrt{n + 5}$

Step 1: Get rid of any square roots by squaring both sides.

$m^2 = n + 5$

Step 2–3: No fractions or brackets, so skip.

Step 4: Collect subject terms on one side and non-subject terms on the other.

$n = m^2 - 5$

This is already in the form $n = \dots$ , so steps 5–7 are not needed.

(Since $\sqrt{a}$ means the positive square root, you don't need a $\pm$ here.)

For square roots:

Square both sides at the beginning to eliminate the square root sign
Remember that when you take a square root at the end, you need both positive and negative solutions (±)

For squared terms:

Work through the normal steps until you have $x^2 = ...$
Then take the square root of both sides, remembering the ± sign

chatImportant

Critical reminder: When taking square roots, you must include both positive and negative solutions using the ± symbol. Forgetting this is one of the most common mistakes students make.

When the subject appears twice

Sometimes the letter you want to make the subject appears in multiple places in the formula. Don't panic - this just means you'll need to use factorising, usually in step 5:

lightbulbExample

Make $p$ the subject of the formula:

$q = \dfrac{p + 1}{p - 1}$

Step 1: There aren't any square roots, so ignore this step.

Step 2: Get rid of any fractions.

$q(p - 1) = p + 1$

Step 3: Multiply out any brackets.

$pq - q = p + 1$

Step 4: Collect all the subject terms ( $p$ terms) on one side and all non-subject terms on the other.

$pq - p = q + 1$

Step 5: Combine like terms on each side (factorise $p$ on the left).

$p(q - 1) = q + 1$

Step 6: Divide both sides by $(q - 1)$ .

$p = \dfrac{q + 1}{q - 1}$

(Since $p$ isn't squared, you don't need step 7.)

The key steps are:

Collect all subject terms on one side as usual
Factorise to take out the common factor (your subject letter)
Divide both sides by what's left in the brackets

infoNote

This technique is particularly useful when dealing with rational expressions where the subject appears in both the numerator and denominator. The factorisation step is what brings everything together.

Top tips for success

infoNote

Essential Tips for Mastering Formula Rearrangement:

Don't rush - Work through each step carefully and methodically
Skip irrelevant steps - Not every problem needs all 7 steps
Check your work - Substitute your answer back into the original formula to verify it's correct
Watch the signs - Pay careful attention to positive and negative signs throughout
Remember the ± - When taking square roots, don't forget both solutions

bookmarkSummary

Key Points to Remember:

Rearranging formulas is just like solving equations - use the same systematic approach
The 7-step method provides a reliable framework, but you can skip steps that don't apply
When dealing with fractions, multiply through by the denominator early on
Squares and square roots require special attention at the beginning and end of the process
If the subject appears twice, you'll need to factorise to collect the terms together
Always check your final answer by substituting back into the original formula

Rearranging Formulas (Edexcel GCSE Maths): Revision Notes