Rearranging Formulae

SPEC A5 • ALGEBRA • FOUNDATION + HIGHER

Change the subject of any formula. Balance both sides, undo operations in reverse BIDMAS, factorise when the variable appears twice.

Changing the Subject

Every formula has a subject — the single letter that sits alone on one side. In A = πr², the subject is A. Rearranging means swapping which letter is alone.

Subject of a Formula

The subject is the variable that appears on its own on one side of the equals sign. Rearranging means applying inverse operations to both sides so a different variable ends up alone.

Why Bother Rearranging?

When you know A and need to find r, you need r as the subject

Many exam questions ask you to "make x the subject" explicitly — it's a whole question type

The quadratic formula is itself a rearrangement of ax² + bx + c = 0 — you use rearranging every time you solve

In science questions (density, speed, pressure), you often need a different variable isolated

The Balance Rule

Rearranging follows the exact same principle as solving equations: whatever you do to one side, you must do to the other. The equation stays balanced.

The Golden Rule

Every step is an inverse operation applied to both sides at the same time. Add to one side → add to the other. Divide one side → divide the other. Square-root one side → square-root the other.

The 5-Step Procedure

For anything beyond the very simplest rearrangement, follow these five steps in order. Undo operations in reverse BIDMAS: addition/subtraction first, then multiplication/division, then powers/roots.

Step 1 — Clear Fractions

Multiply both sides by any denominator. This gets rid of division and gives you a flat equation.

Step 2 — Remove Powers and Roots

If your target variable is inside a square root, square both sides. If it is inside a cube root, cube both sides. (We'll reverse this at the end if the power is on the target variable itself.)

Step 3 — Expand Brackets, Collect Variable Terms

Expand any brackets. Move every term containing your target variable to one side; everything else to the other.

Step 4 — Factorise (if variable appears twice)

If the variable is in more than one term on your side, factorise it out. This is the step most students forget.

Step 5 — Divide / Root to Isolate

Divide both sides by whatever is multiplying your variable, or take the root of both sides if the variable is still trapped inside a power.

Worked Example 1 — Linear Rearrangement

A Foundation-level linear rearrangement. This is the simplest question type — just undo operations in reverse order.

01Make x the subject [2 marks]

Rearrange y = 3x + 7 to make x the subject.

y = 3x + 7

Undo the +7 first — subtract 7 from both sides.

y − 7 = 3x

← M1: −7 from both sides

Undo the ×3 — divide both sides by 3.

(y − 7) ÷ 3 = x

← A1: correct final form

x = (y − 7) / 3

Worked Example 2 — With Fractions

02Make x the subject [3 marks]

Rearrange y = 2(x + 3) / 5 to make x the subject.

y = 2(x + 3) / 5

Step 1 — Clear the fraction: multiply both sides by 5.

5y = 2(x + 3)

Step 5 — Divide both sides by 2.

5y / 2 = x + 3

Undo the +3 — subtract 3 from both sides.

x = 5y/2 − 3

Why Clear Fractions First?

Trying to "undo the +3 first" doesn't work here — the +3 is trapped inside the fraction. Clearing the denominator frees up the bracket first, then you can work through BIDMAS in reverse.

Inverse Operations Reference

Every operation has a matching inverse. When rearranging, you apply the inverse to both sides at once.

If the formula has…Apply to both sides…

+ a number (e.g. + 7)subtract that number (− 7)

− a numberadd that number

× by something (e.g. × 3)divide by that something (÷ 3)

÷ by somethingmultiply by that something

x² (on the target variable)take the square root (√) of both sides

x³ (on the target variable)take the cube root (∛) of both sides

√( ... containing x ... )square both sides to remove the root

y = k / x (x is dividing)multiply both sides by x, then divide by y

Worked Example 3 — Power on the Target

When the target variable is trapped inside a power (x², x³, etc.), the power is the last thing to undo. Isolate the power first, then root both sides.

03Make r the subject [3 marks]

The volume of a sphere is given by V = ⁴⁄₃ πr³. Make r the subject.

V = (4/3) × π × r³

Multiply both sides by 3 to clear the fraction.

3V = 4πr³

Divide both sides by 4π to isolate r³.

3V / (4π) = r³

Take the cube root of BOTH sides.

r = ∛[3V / (4π)]

Critical Point

The cube root acts on the whole right-hand side, not just the top or bottom. Write it as one cube-root symbol over the whole fraction.

Worked Example 4 — Root at the Start

This is the opposite pattern: the root is wrapped around the whole expression, and the target is inside. To free it, square both sides. This is based on AQA Nov 24 Paper 3 Q16.

04Make x the subject [3 marks]

Rearrange y = √((x/2) + 1) to make x the subject.

y = √((x/2) + 1)

Square both sides to remove the root.

y² = x/2 + 1

← M1: square both sides

Subtract 1 from both sides.

y² − 1 = x/2

← M1: isolate the x-term

Multiply both sides by 2.

2(y² − 1) = x

← A1: fully rearranged

x = 2(y² − 1) or equivalently x = 2y² − 2

Key Distinction

If the target variable is raised to a power (Example 3: r³) — take the root at the end. If the whole expression is under a root (Example 4: √ wraps everything) — square both sides at the start. Recognise the difference and you'll pick the right first move every time.

Roots: Apply to BOTH Sides

A classic trap on formulae with a power: students do the earlier steps correctly, then forget the final root. Compare these two attempts to make r the subject of A = πr².

❌ Weak Student

A = πr²A/π = r² ✓r = A/π ✗

Marks: 1 / 2

Divided correctly, but forgot to take the square root at the final step. Equation left with r² on one side, no root on the other.

✅ Strong Student

A = πr²A/π = r²r = √(A/π) ✓

Marks: 2 / 2

Divided by π, then square-rooted both sides. Full marks.

Examiner Alert

The error pattern: students isolate the squared term correctly (A/π = r²) but then drop the square when writing the final answer, getting r = A/π instead of r = √(A/π).

What to write instead: the final line must have the square root applied to the whole right-hand side. Check: is your target variable alone and not squared? If you see an r², you are not finished.

When the Variable Appears Twice

This is the most common rearranging question on Higher papers. It appears when the target variable is in both the numerator and denominator of a fraction, or multiplies two different terms.

The Key Insight

If your target variable appears in more than one term on one side, you can't divide it out — yet. You need to factorise first. Factorising bundles every instance of the variable into a single multiplied form, which you can then divide.

The Four-Step Pattern

(1) Clear the fraction → (2) Expand any brackets → (3) Collect all x-terms on one side, everything else on the other → (4) Factorise x out, then divide.

Worked MCE — Past Paper Question

This is based on AQA Jun 23 Paper 2, Q18 — a 3-mark rearranging question on the Higher paper.

05MCE Framework [3 marks]

Rearrange y = x / (x + 8) to make x the subject.

M — Method. Clear the fraction, collect x-terms on one side, factorise x out, divide.

C — Calculate.

y(x + 8) = x

← M1: clear the fraction

yx + 8y = x

← expand brackets

yx − x = −8y

← collect x-terms

x(y − 1) = −8y

← M1: factorise x

E — Evaluate. Divide both sides by (y − 1):

x = −8y / (y − 1)

← A1: fully rearranged

x = −8y / (y − 1) or equivalently x = 8y / (1 − y)

How the 3 Marks Break Down

M1 — clearing the fraction correctly (any valid method). M1 — factorising x out of two terms. A1 — any correct final rearranged form.

Diagnose the Pattern

Before you start any rearranging question, ask yourself: where does my target variable appear? The answer tells you which technique to use.

Where does the target appear?TechniqueExample

In one place, multiplied by a number5-step chain of inverse operationsy = 3x + 7

Inside a fraction (once), or with bracketsClear fractions first, then invert operationsy = 2(x + 3) / 5

Raised to a power (x², x³ on the target)Isolate the power, then take the root of both sidesA = πr² → r = √(A/π)

Wrapped inside a root (the target is inside √)Square both sides at the start to remove the rooty = √(x/2 + 1)

In two or more terms (variable twice)Clear fraction → expand → collect → factorise → dividey = x / (x + 8)

Quick Diagnostic

Scan for the target variable and count appearances. One appearance → work through inverse chain. Two or more → factorise. If the target is inside a root or under a power, that operation determines your first or last move.

Common Errors to Avoid

Students Often WriteCorrect VersionWhy It Matters

A/π = r², so r = A/πr = √(A/π) — take the rootFinal line must have target alone and un-squared

Divide by (y − 1) before factorising xFactorise x out first, then divideSkipping the factorise step loses M marks

y² = x + 2, so y = √x + 2y = √(x + 2)Square root of a sum, not term-by-term

y = 1/x, so x = −yx = 1/y — reciprocal, not negationReciprocal ≠ additive inverse

Subtract only from one sideSubtract from both sides simultaneouslyThe balance rule — always both sides

Stop at x(y − 1) = −8yDivide by (y − 1) to isolate x fullyTarget must be alone for the A1

Examiner Alerts — The Big Ones

Squaring / Rooting Only One Side

What students write: A/π = r², then r = A/π.

Why it loses marks: they divided by π correctly, but forgot the final square root. The right-hand side still has r², not r.

What to write instead: take the square root of both sides — r = √(A/π). If the final line has x², r², or any power on your target, you're not finished.

Variable Appearing Twice — Dividing Prematurely

What students write: on y = x/(x + 8), they get to yx + 8y = x and try to divide by (x + 8) or similar.

Why it loses marks: you can't divide by something containing your target variable — you'd just create another equation with x in it. You must collect terms and factorise first.

What to write instead: collect all x-terms on one side → factorise x out → then divide by the bracket that's left.

Sign Errors When Moving Terms

What students write: yx + 8y = x becomes yx + x = 8y (sign wrong on the 8y).

What to write instead: move the 8y to the other side as −8y, and move the x to the left as −x. Check each sign as you go.

The MCE Framework

Multi-mark rearranging questions follow the MCE framework. Method, Calculate, Evaluate — line-by-line is what scores.

MMethod. State your plan: clear fractions, expand, collect, factorise, divide (or root).

CCalculate. One inverse operation per line. Apply to both sides. Keep every line visible.

EEvaluate. State the rearranged formula with the required variable as the subject on its own.

Typical Mark Allocations (from AQA Mark Schemes)

Simple linear (2 marks): M1 correct inverse step, A1 fully rearranged form

With fraction or power/root (3 marks): M1 + M1 + A1 — two working steps and a final form

Variable appears twice (3–4 marks): M1 (clear fraction) + M1 (factorise) + A1, plus an extra M1 for expansion at 4 marks

How Often Does This Come Up?

Rearranging is a recurring 3–4 mark question on Higher papers. Here's the run of recent AQA Higher series — every series has had one, and most are the variable-appears-twice pattern.

PaperQuestionMarksPattern

Jun 22 P1H Q20y = x / (5x + 9)4Variable twice

Nov 22 P3H Q189m + 4(2m − 1) = p² + pm (make m subject)4Variable twice

Jun 23 P2H Q18y = x / (x + 8)3Variable twice

Nov 23 P2H Q20p = (m − 1) / (m + 1)²4Variable twice

Jun 24 P1H Q17y = (x + 3) / (7x)4Variable twice

Nov 24 P3H Q16y = √((x/2) + 1)3Root extraction

Pattern to Spot

Variable-appears-twice is the dominant pattern (five of the last six series). Nov 24 was the outlier — a root-extraction question instead. Prepare for both: spot the fraction-with-variable-in-numerator-and-denominator AND the whole-expression-inside-a-square-root.

Exam Technique

Top Exam Tip

Before you start rearranging, circle the variable you want as the subject. Every step is about isolating that one letter. If you ever find yourself doing something that doesn't move toward isolating the circled letter, stop and reconsider.

Strategy Paper 1 (Non-Calculator)

Write every line. Examiners award method marks per line of working.

One inverse operation per line — don't combine steps in your head.

If the variable appears twice, say out loud: "I need to factorise." It's the step students forget.

Keep fractions and signs tidy — sign errors when moving terms are the #1 arithmetic mistake.

Strategy Papers 2 & 3 (Calculator)

The method is identical to non-calculator — calculators don't rearrange for you.

Check your answer by substituting a specific value of x, computing y, then plugging y into your rearranged formula to see if you recover x.

If it doesn't recover, you have a sign or factorisation error.

Practice Question

Based on AQA Jun 24 Paper 1H Q17 [4 marks]

y = (x + 3) / (7x)

Rearrange to make x the subject. [4 marks]

M

Clear the fraction, expand, collect all x-terms on one side, factorise x, divide.

C

Multiply both sides by 7x: 7xy = x + 3 (M1 — clear fraction)

C

Move the x-terms to the same side: 7xy − x = 3 (M1 — collect x-terms)

C

Factorise x from the left-hand side: x(7y − 1) = 3 (M1 — factorise)

C

Divide both sides by (7y − 1): x = 3 / (7y − 1)

E

x = 3 / (7y − 1)(A1 — final form). The variable appears twice in the original, so factorisation was essential.

Quick-Fire Checklist

Before you move on, you should be able to do all of these without pausing:

Can You...

Rearrange y = 5x − 2 to make x the subject? (Answer: x = (y + 2)/5)

Rearrange A = πr² to make r the subject? (Answer: r = √(A/π))

Rearrange V = πr²h to make h the subject? (Answer: h = V/(πr²))

Rearrange y = √(x + 4) to make x the subject? (Answer: x = y² − 4)

Rearrange y = x/(x − 4) to make x the subject? (Answer: x = 4y/(y − 1))

Spot whether a question needs factorisation, a root, or squaring?

Check your rearranged formula by substituting a test value?

One-Line Takeaway

Diagnose where the target variable lives — then undo operations in reverse BIDMAS on both sides — and factorise when the variable appears twice. Get that sequence right and you'll handle every rearranging question on the paper.

Summary

Key Points

Subject — the variable that sits alone on one side of the equation.

Balance rule — every inverse operation goes on both sides at once.

5-step procedure — clear fractions → remove outer roots → expand and collect → factorise → divide (or root).

Diagnose first — identify where your target variable appears. That tells you which technique to use.

Power on the target — isolate the power, then take the root of both sides (e.g. A = πr² → r = √(A/π)).

Root on the whole expression — square both sides at the start (e.g. y = √(x+4) → x = y² − 4).

Variable appears twice — collect all instances, factorise out, then divide. The main Higher-tier pattern.

MCE — method (the plan), calculate (one inverse per line), evaluate (final form with target variable alone).

The Factorise Step — Why It Matters

If x appears in two or more terms, you cannot divide x out until you factorise.

Factorising bundles every copy of x into a single product: x × (bracket).

Then dividing by the bracket leaves x alone — the final answer.

Skipping the factorise step is the single biggest mark-loser on Higher rearranging questions.

Exam Tip

On every rearranging question, circle the target variable first and scan where it appears. Twice → factorise coming. Inside a root → square both sides. Under a power on itself → root both sides at the end. These three patterns cover almost every AQA rearranging question from Jun 22 to Nov 24.