Substituting and rearranging formulae
A formula links quantities, such as v = u + at. You need to put numbers into one (substitution) and to turn it round so a different letter is on its own (changing the subject). Rearranging uses the same balancing moves as solving equations.
Part 1 of 3: Learn it
In short
- Substitute with brackets round negative numbers, then follow the order of operations.
- To change the subject, undo what has been done to the new subject, in reverse order, to both sides.
- If the subject appears twice, collect those terms on one side and factorise it out.
Where this is in your specification
Spec points: DfE A2, A3 and A5 (a subject that appears twice is Higher tier)
| Board | Topic: Formulae and rearranging |
|---|---|
| DfE content | A2, A3, A5, A7 |
| AQA 8300 | Algebra |
| Edexcel 1MA1 | Algebra |
| OCR J560 | 6 Algebra |
| Eduqas C300 | HA2, HA5 |
| Cambridge IGCSE 0580 | E2.1, E2.5 |
| National 5 C847 75 | Algebraic skills: changing the subject of a formula |
Substitution
Replace each letter with its value, using brackets, then work it out with BIDMAS. If a = −3 and b = 4, then 2a² − b = 2 × (−3)² − 4 = 2 × 9 − 4 = 14.
Writing a formula
Turn the words into letters. A taxi charges £3 plus £1.80 per mile, so the cost C for m miles is C = 3 + 1.8m. Say which letter stands for what.
Changing the subject
- Make t the subject of v = u + at: subtract u, v − u = at, then divide by a, t = (v − u)/a.
- Make r the subject of A = πr²: divide by π, r² = A/π, then square root, r = √(A/π).
- Make x the subject of y = (x + 2)/5: multiply by 5, 5y = x + 2, then x = 5y − 2.
Subject appearing twice (Higher)
Get every term with the subject on one side and everything else on the other, factorise out the subject, then divide by the bracket.
Find P = 2(l + w) when l = 7.5 and w = 4.
Show the answer
2 × 11.5 = 23.
Part 2 of 3: See it worked
Worked examples
Example 1
Use s = ut + ½at² to find s when u = 3, t = 4 and a = −2.
- ut = 3 × 4 = 12
- ½at² = 0.5 × (−2) × 16 = −16
- s = 12 + (−16)
Answer: s = −4.
Example 2
Make x the subject of ax + 5 = 2x + b. (Higher)
- Collect x terms on the left: ax − 2x = b − 5
- Factorise: x(a − 2) = b − 5
- Divide by (a − 2)
Answer: x = (b − 5)/(a − 2).
Common mistakes
- Squaring a negative number without brackets on a calculator: −3² gives −9, but (−3)² is 9.
- Dividing only one term on a side by the number, such as v − u = at giving t = v − u/a.
- Forgetting that the square root of both sides is needed when the subject is squared.
- Leaving the new subject on both sides of the formula.
Make w the subject of P = 2l + 2w.
Show the answer
P − 2l = 2w, so w = (P − 2l)/2.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find P = 2(l + w) when l = 7.5 and w = 4.
2 × 11.5 = 23.
Make w the subject of P = 2l + 2w.
P − 2l = 2w, so w = (P − 2l)/2.
Make h the subject of V = (1/3)πr²h.
3V = πr²h, so h = 3V/(πr²).
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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