Mathematics (Core)Exam code: 0580
Rearranging Formulas
Algebra and Sequences
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
- A power of a different letter is just part of what multiplies the subject, so y = ax² gives a = y ÷ x²
| Operation to undo | What to do to both sides |
|---|---|
| A term added | Subtract it |
| A term subtracted | Add it |
| Multiplication | Divide |
| Division | Multiply |
- 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
Exam tip
A correct first step scores
Changing the subject was set in 7 of the 22 papers. On a two-mark question, one mark is for a correct first step, written as an equation. For v = 4t + 20 that is v − 20 = 4t, or v ÷ 4 = t + 5 if you divide every term first. So write that line down before the answer: if the last step slips, it still keeps one mark.
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