1. Formulae and the subject
What a formula is
- A formula is a rule connecting quantities, written with letters and containing an equals sign
- The letter standing alone on one side is the subject of the formula
- In A = lw, the subject is A
- Changing the subject means rearranging so that a different letter stands alone
Why the order matters
- Rearranging undoes the operations applied to the subject, in reverse order
- Whatever is done to one side must be done to the other, which keeps the two sides equal
2. When the subject appears once
The order of operations
- Identify what has been done to the letter you want, then undo each step in reverse
- Fractions go first: scale the whole equation up by whatever sits underneath, so nothing is left divided
- Move terms that are added or subtracted, then deal with multiplication and division, and take roots or powers last; the exception is a root or power over a whole expression containing the subject, such as y = √(x + 3), which is undone as soon as it stands alone, before anything inside it
| Operation to undo | What to do to both sides |
|---|---|
| A term added | Subtract it |
| A term subtracted | Add it |
| Multiplication | Divide |
| Division | Multiply |
| A square | Take the square root |
| A square root | Square |
- The new subject may be left in a fraction, which is a perfectly acceptable final form
- C = 2πr rearranges to r = C ÷ (2π)
- v = u + at rearranges to u = v − at
- A fraction inside a fraction is cleared the same way, by multiplying the whole equation by the inner denominator before doing anything else
- Where the division at the end is by a negative quantity, multiply the top and the bottom by −1 to tidy the signs
- 4 − 3x = y gives x = (y − 4) ÷ (−3), which is written as (4 − y) ÷ 3
Worked example
Clearing a fraction first
Make x the subject of y = (3x − 4) ÷ 5.
Solution:
- Rearranging undoes the operations that were done to x, in the reverse of the order they were applied
- Reading the right-hand side as it was built: x was multiplied by 3, then 4 was subtracted, then the whole thing was divided by 5
- So undo the division first, then the subtraction, then the multiplication
- Multiply both sides by 5, which clears the fraction: 5y = 3x − 4
- Add 4 to both sides: 5y + 4 = 3x
- Divide both sides by 3, remembering it divides the whole left-hand side, not just the 4
- The rearranged formula is x = (5y + 4) ÷ 3
- Check with a number: x = 3 gives y = (9 − 4) ÷ 5 = 1, and putting y = 1 into the new formula gives (5 + 4) ÷ 3 = 3
- Writing 5y + 4 ÷ 3 without the bracket would divide only the 4, which is why the bracket is part of the answer
3. When the subject appears twice
Gathering and factorising
- Where the wanted letter appears more than once, it must end up appearing once before it can be isolated
- Expand any brackets containing it first
- Collect every term containing that letter on one side, and everything else on the other
- Factorise to bring the letter outside a bracket, which leaves it appearing once
- Divide by the bracket to finish
Worked example
The subject appearing on both sides
Make x the subject of k(x − 3) = 2x + 1.
Solution:
- x appears twice, so it cannot simply be isolated by undoing operations: it has to be collected first
- Expand the bracket so nothing is hidden inside it: kx − 3k = 2x + 1
- Gather every term containing x on one side and everything else on the other
- Subtract 2x from both sides and add 3k to both sides: kx − 2x = 1 + 3k
- The left-hand side now has x in both terms, so factorise it out: this is the step the marks concentrate on
- x(k − 2) = 1 + 3k
- x is now multiplied by a single bracket, so divide both sides by it
- The rearranged formula is x = (1 + 3k) ÷ (k − 2)
- Check with k = 4: the original reads 4(x − 3) = 2x + 1, giving 4x − 12 = 2x + 1 and x = 6.5, and the formula gives (1 + 12) ÷ 2 = 6.5
- The answer keeps k in it, which is expected: a formula rearranged for one letter is still a formula
- The same method handles a formula such as at = 5 − bt, which becomes t(a + b) = 5 and then t = 5 ÷ (a + b)
- A squared subject appearing twice factorises in exactly the same way, and the root is taken at the very end
- ax² = b − cx² gathers to x²(a + c) = b, so x² = b ÷ (a + c) and x = ±√(b ÷ (a + c))
- The factorising step is what the question is testing, so it must be shown
Exam tip
Two easy-to-skip lines carry marks
Examiners mark a rearrangement move by move, and a wrong final answer still keeps the earlier marks: up to 3 of 4, or 2 of 3. Two lines that are easy to do in your head carry a mark each, so write them down: the expanded line after clearing a fraction, and the square on its own, such as x² = b ÷ (a + c) above, before taking the root.
Worked example
The subject above and below a fraction line
Make x the subject of y = (2x + 1) ÷ (x + 3).
Solution:
- x is on the top and on the bottom, so it appears twice: clear the fraction first, then gather as above
- Multiply both sides by the whole denominator, kept in a bracket: y(x + 3) = 2x + 1
- Expand the bracket: xy + 3y = 2x + 1
- Gather the x terms on the left and everything else on the right: xy − 2x = 1 − 3y
- Factorise: x(y − 2) = 1 − 3y
- Divide both sides by the bracket
- The rearranged formula is x = (1 − 3y) ÷ (y − 2)
- Check with x = 2: the original gives y = 5 ÷ 5 = 1, and the formula gives (1 − 3) ÷ (1 − 2) = 2
- Multiplying the top and the bottom by −1 gives (3y − 1) ÷ (2 − y), the same formula written another way