Rearranging Formulae (Edexcel IGCSE Maths): Revision notes

Section 1

What does it mean to rearrange a formula?

A formula links several letters (variables) together, e.g. $v = u + at$ . The letter on its own on the left-hand side is called the subject of the formula — here, $v$ is the subject.

Rearranging means using algebra to make a different letter the subject. You do this by applying the same operation to both sides of the formula (add, subtract, multiply, divide, square, square root...) until the letter you want is alone.

The golden rule: whatever you do to one side, you must do to the other side too, to keep the formula balanced.

Key termssubjectformulavariable

Think of it like this

Think of a formula like a set of scales that must stay balanced. If you add, remove, multiply or divide something on one side, you must do exactly the same on the other side, or the scales tip.

Section 2

How do I rearrange when the subject appears once?

Treat the formula like an equation you are solving, but leave the letters as they are instead of finding a number.

Work through the operations in reverse order (undo addition/subtraction first, then multiplication/division), keeping the new subject isolated.

Example: Make $u$ the subject of $v = u + at$ .

Subtract $at$ from both sides: $v - at = u$
Rewrite with the subject on the left: $u = v - at$

Example: Make $r$ the subject of $C = 2\pi r$ .

Divide both sides by $2\pi$ : $\dfrac{C}{2\pi} = r$
So $r = \dfrac{C}{2\pi}$

Key termsinverse operation

Exam tip

Ask yourself: 'What is being done to my target letter, and in what order?' Then undo those steps in reverse order, outside-in.

Common mistake

Do not divide only one term by an added/subtracted quantity. In $v = u + at$ , do not write $\dfrac{v}{a} = u + t$ — you must isolate the whole term $at$ first with subtraction, not division.

Section 3

What if the subject appears twice?

If the letter you want appears in more than one term, you must factorise it out before you can isolate it.

Example: Make $x$ the subject of $ax + b = cx + d$ .

Collect all $x$ terms on one side: $ax - cx = d - b$
Factorise: $x(a - c) = d - b$
Divide both sides by $(a - c)$ : $x = \dfrac{d - b}{a - c}$

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

Expand: $3p + 3q = p - 5$
Collect $p$ terms: $3p - p = -5 - 3q$
Simplify: $2p = -5 - 3q$
Divide: $p = \dfrac{-5 - 3q}{2}$

Key termsfactorisecollect like terms

Exam tip

Always move every term containing the target letter to the SAME side first, before you factorise. Move all other terms to the opposite side.

Common mistake

Forgetting to factorise is the most common error — you cannot divide by $a$ alone in $ax - cx = d - b$ because $x$ is attached to both $a$ and $c$ ; you must divide by the bracket $(a-c)$ .

Section 4

How do I rearrange when the subject is a power or under a root?

If the target letter is squared, cubed, or under a root sign, use the opposite operation to undo it last, after isolating the power/root term.

Example: Make $r$ the subject of $A = \pi r^2$ .

Divide both sides by $\pi$ : $\dfrac{A}{\pi} = r^2$
Square root both sides: $r = \sqrt{\dfrac{A}{\pi}}$

Example: Make $x$ the subject of $y = \sqrt{x + 3}$ .

Square both sides: $y^2 = x + 3$
Subtract 3: $x = y^2 - 3$

Example: Make $l$ the subject of $T = 2\pi\sqrt{\dfrac{l}{g}}$ .

Divide by $2\pi$ : $\dfrac{T}{2\pi} = \sqrt{\dfrac{l}{g}}$
Square both sides: $\left(\dfrac{T}{2\pi}\right)^2 = \dfrac{l}{g}$
Multiply by $g$ : $l = g\left(\dfrac{T}{2\pi}\right)^2$

Key termssquare rootsquaring

Common mistake

Do not square root or square only PART of an expression. $\sqrt{x+3}$ must be squared as a whole term: $(\sqrt{x+3})^2 = x + 3$ , not $x + \sqrt{3}$ .

Exam tip

Isolate the squared/rooted term completely (get it fully on its own on one side) before squaring or square-rooting both sides.

Must Know

The subject of a formula is the letter alone on one side; rearranging changes which letter is the subject.
Whatever operation you apply, apply it to both sides of the formula.
Undo operations in reverse order: additions/subtractions first, then multiplications/divisions, then powers/roots.
If the target letter appears twice, collect those terms on one side, factorise, then divide by the bracket.
If the target letter is squared, isolate the squared term first, then square root both sides (and vice versa for a root).
Always check your final answer by substituting numbers back into the original formula.

That's the notes covered.

Carry on to the next subtopic.