Using and rearranging formulae — changing the subject
Rearranging is solving in disguise. Instead of finding a numerical value, you isolate a chosen letter. Treat both sides equally and undo operations in reverse BIDMAS order.
The general strategy
- Identify the letter you want as the subject.
- Move every other term off that letter using inverse operations: + ↔ −, × ↔ ÷, square ↔ √.
- Work outwards, peeling layers in reverse BIDMAS order: undo addition/subtraction first, then multiplication/division, then powers/roots, then brackets.
- Whatever you do to one side, do to the other.
✦Worked example— Worked example 1 — single step
Make x the subject of y = x + 7.
Subtract 7 from both sides: y - 7 = x. So x = y - 7.
✦Worked example— Worked example 2 — two steps
Make x the subject of y = 4x - 3.
Add 3: y + 3 = 4x. Divide by 4: x = (y + 3)/4.
✦Worked example— Worked example 3 — fraction
Make r the subject of A = πr².
Divide by π: A/π = r². Square root: r = √(A/π) (positive root because radius is positive).
✦Worked example— Worked example 4 — letter in two places
Make x the subject of y = 3x + ax.
Factor x out of the right: y = x(3 + a). Divide: x = y/(3 + a).
This factor-and-divide trick is essential whenever the target appears more than once.
✦Worked example— Worked example 5 — fraction with target in denominator
Make x the subject of y = 5/(x - 2).
Multiply both sides by (x - 2): y(x - 2) = 5. Divide by y: x - 2 = 5/y. Add 2: x = 5/y + 2.
✦Worked example— Worked example 6 — root containing the target
Make x the subject of y = √(x + 3) - 1.
Add 1: y + 1 = √(x + 3). Square: (y + 1)² = x + 3. Subtract 3: x = (y + 1)² - 3.
⚠Common mistakes— Common mistakes (examiner traps)
- Wrong order of operations. Reverse BIDMAS — undo addition before division, division before squaring. Doing the inside layers first is the most common slip.
- Forgetting the ± when square-rooting. In
x² = 16, x = ±4. Many GCSE problems assume positive roots from context (lengths, radii); state the assumption. - Cancelling across an addition.
(x + a)/adoes not equalx. You can only cancel common factors. - Losing a sign when moving terms across the equals sign. Each move flips the sign.
- Ignoring the case where the target appears twice. Always factor first, then divide.
➜Try this— Quick check
Make c the subject of E = mc².
Divide by m: c² = E/m. Square root: c = √(E/m) (taking the positive root for physical mass).
AI-generated · claude-opus-4-7 · v3-deep-algebra