Rearranging Formulae and Changing the Subject

“Make x the subject” means rewrite the formula so it starts x = …. The technique is the same as solving an equation — you just carry letters instead of numbers, which removes the arithmetic that usually tells you when you’ve gone wrong.

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1. The principle

Do the same thing to both sides, undoing the operations in reverse order.

Think of what is being done to the subject, then reverse it — last operation first.

Example: make x the subject of y = 3x + 5

The formula does: ×3, then +5. Undo in reverse:

1. Subtract 5: y − 5 = 3x
2. Divide by 3: x = (y − 5)/3

Reverse order matters. Dividing by 3 before subtracting the 5 requires dividing every term — a common source of error. Deal with the outermost operation first.

Divide the WHOLE side, not just one term. From y − 5 = 3x you get (y − 5)/3, with the bracket. Writing y − 5/3 is a different expression.

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2. Order — which operation to undo first

Example: make x the subject of y = (x + 2)/5

The formula does: +2, then ÷5. Undo the division first:

1. Multiply both sides by 5: 5y = x + 2
2. Subtract 2: x = 5y − 2

Work from the outside in. Ask “what happens to x LAST?” and undo that first. Moving the wrong term first was a documented error, and it usually creates unnecessary brackets.

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3. Squares and square roots

These undo each other — but watch the ± and what exactly is being squared.

Example: make x the subject of y = x² + 3

1. Subtract 3: y − 3 = x²
2. Square root: x = ±√(y − 3)

Include ± when you take a square root, unless the context rules out negatives (a length, for instance). Tutors flagged this repeatedly — it was one of the most-repeated pieces of advice in the topic.

Example: make x the subject of y = √(x − 1)

1. Square both sides: y² = x − 1
2. Add 1: x = y² + 1

Square the WHOLE side. From y = √(x − 1), squaring gives y², not y — and the left side is squared as a whole, so if it were y + 2 it would become (y + 2)², not y² + 4.

Square the correct quantity. Squaring the result instead of the variable, or vice versa, was recorded as an error. Track which letter you are isolating.

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4. Brackets

Sometimes you should expand; sometimes the bracket is already helping you.

Example: make p the subject of R = T(4 − p)

Option A — divide first (usually cleaner):

1. R/T = 4 − p
2. p = 4 − R/T

Option B — expand first:

1. R = 4T − Tp
2. Tp = 4T − R
3. p = (4T − R)/T

Both are correct and equivalent.

Don’t expand a bracket unnecessarily. If the whole bracket is multiplied by something, dividing by that something removes it in one step. Expanding when it wasn’t needed was a specific recorded error and adds work.

Watch the signs. In 4 − p, the p is negative. Isolating it will require a sign change.

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5. Fractions

Multiply both sides by the denominator to clear the fraction.

Example: make x the subject of y = a/x

1. Multiply both sides by x: xy = a
2. Divide by y: x = a/y

When the subject is on the bottom, multiply it up first. You cannot isolate x while it is a denominator.

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

1. 3y = 2x + 1
2. 3y − 1 = 2x
3. x = (3y − 1)/2

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6. When the subject appears twice

This is the hardest type, and the method is always the same:

1. Get all terms containing the subject on one side. 2. FACTORISE it out. 3. Divide.

Example: make x the subject of ax + b = cx + d

1. Collect x terms: ax − cx = d − b
2. Factorise: x(a − c) = d − b
3. x = (d − b)/(a − c)

Example: make x the subject of y = (x + 1)/(x − 3)

1. Multiply out: y(x − 3) = x + 1
2. Expand: xy − 3y = x + 1
3. Collect x terms: xy − x = 3y + 1
4. Factorise: x(y − 1) = 3y + 1
5. x = (3y + 1)/(y − 1)

Factorising is the step that makes these possible. Without it you cannot isolate a letter that appears in two places — and it is the step students most often don’t think of.

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7. Checking your answer

Substitute numbers. Pick easy values, evaluate the original formula, then check your rearranged version gives back the starting value.

For y = 3x + 5 with x = 2: y = 11. Then x = (11 − 5)/3 = 2 ✓

Show every step. Tutors were explicit that marks are given for the steps — a bare final answer risks scoring nothing if it’s wrong, and jumping steps is where sign errors creep in.

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8. Mistakes that cost marks

Undoing operations in the wrong order.

Dividing only one term instead of the whole side.

Sign errors when moving terms.

Forgetting ± when square-rooting.

Squaring only part of a side.

Squaring the wrong quantity.

Expanding brackets unnecessarily.

Trying to isolate the subject while it’s a denominator.

Not factorising when the subject appears twice.

Leaving the answer not in the form “x = …”.

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Frequently asked questions

What does “make x the subject” mean? Rearrange the formula so it reads x = …, with x alone on one side.

What order do I undo operations in? Reverse order — undo the last thing done to the subject first.

Do I need ± when square rooting? Yes, unless the context (a length, for example) rules out the negative.

What if the subject is inside a square root? Square both sides — the whole side, not term by term.

What if the subject is on the bottom of a fraction? Multiply both sides by that denominator first.

What if the subject appears twice? Collect those terms on one side, factorise the subject out, then divide.

Should I expand brackets? Only if it helps. If the whole bracket is multiplied by something, divide by it instead.

How do I check my answer? Substitute numbers into the original and the rearranged versions.

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Quick revision checklist

• I undo operations in reverse order
• I do the same thing to both sides
• I divide the whole side, using brackets
• I handle signs carefully when moving terms
• I include ± when taking a square root
• I square the whole side
• I can clear fractions by multiplying up
• I can isolate a subject that starts in a denominator
• I factorise when the subject appears twice
• I decide whether expanding a bracket actually helps
• I check by substituting numbers
• I write the final answer as x = …

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These notes cover rearranging formulae and changing the subject in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.