Summary — What's In This Chapter
- A formula is a rule linking quantities using letters, always containing an "=" sign — like y = mx + c.
- The subject of a formula is the letter sitting completely alone on one side.
- To change the subject, remove fractions first, then undo operations in reverse order using inverse operations — exactly like solving an equation.
- If the target letter is trapped inside a bracket, you either expand the bracket or divide by the bracket's coefficient.
- Watch out for fractions within fractions and dividing by negatives — these are the two biggest sources of careless errors.
- If the subject appears twice, collect all the subject-terms on one side, then factorise to pull the subject out as a single common factor.
- If the subject appears twice with the same power (like x² and px²), collect and factorise those powers together, then take the root at the very end.
1. Formulas Where the Subject Appears Once
What actually is a "formula", and what's a "subject"?
Think of a formula like a recipe written in code. Instead of saying "add the two parallel sides, multiply by the height, then halve it" to find the area of a trapezium, we compress that whole sentence into:
Every formula has an equals sign, and one letter usually sits completely by itself on one side — that's the subject. In y = mx + c, the subject is y, because it's alone on the left and everything else is bundled up on the right.
Plain words: the subject is "the letter the formula is currently solved for." Changing the subject just means solving the same equation for a different letter instead.
The Method — Step by Step
Rearranging a formula uses exactly the same logic as solving an equation: whatever you do to one side, you must do to the other, to keep the "balance" true. Here's the general order of attack:
- Remove any fractions first. Multiply both sides by the lowest common denominator — this clears the messiest part of the problem immediately.
- Expand any brackets that contain the letter you're solving for (more on this below).
- Use inverse operations to peel away everything that isn't the target letter — undo addition with subtraction, undo multiplication with division, undo squaring with square-rooting, and so on. Undo the operations in the reverse order to how they were "built up" (this is the exact same logic used when solving normal equations).
Worked Example
Make x the subject of (5x + 6)/2 = y
Step 1 — clear the fraction: multiply both sides by 2 → 5x + 6 = 2y
Step 2 — undo the +6: subtract 6 from both sides → 5x = 2y − 6
Step 3 — undo the ×5: divide both sides by 5 → x = (2y − 6) / 5
Note: (2y−6)/5, 2y/5 − 6/5, 2(y−3)/5, 0.4(y−3), and 0.4y − 1.2 are all the same answer written differently. Examiners accept any fully simplified equivalent form.
Should I expand brackets, or divide instead?