Formulas are equations too
A literal equation has more than one variable: A = lw, d = vt, P = 2l + 2w. A formula is a literal equation that describes a real relationship.
The variable on its own is called the subject. In A = lw, A is the subject.
Why rearrange?
If you know the area and length of a rectangle and want the width, it’s easier to rearrange A = lw into w = A/l first, then substitute.
The method
Use inverse operations to isolate the new subject, exactly as you would to solve an equation. Treat the other variables like numbers.
Example: Make w the subject of A = lw.
- Divide both sides by l: w = A/l
Example: Make t the subject of d = vt.
- t = d/v
Two-step rearrangements
Example: P = 2l + 2w. Make l the subject.
- Subtract 2w: P − 2w = 2l
- Divide by 2: l = (P − 2w)/2
Example: F = 1.8C + 32. Make C the subject.
- Subtract 32: F − 32 = 1.8C
- Divide by 1.8: C = (F − 32)/1.8
Check with 212 °F: (212 − 32)/1.8 = 100 °C ✓
Formulas with powers and roots
Example: A = πr². Make r the subject.
- Divide by π: r² = A/π
- Square root: r = √(A/π) (r is a length, so take the positive root)
Example: v = √(2gh). Make h the subject.
- Square both sides: v² = 2gh
- Divide by 2g: h = v²/(2g)
When the variable appears twice
Factor it out.
Example: Make x the subject of ax + b = cx.
- ax − cx = −b
- x(a − c) = −b
- x = −b/(a − c) or b/(c − a)
Common formulas rearranged
| Formula | Rearranged |
|---|---|
| I = Prt | r = I/(Pt) |
| V = lwh | h = V/(lw) |
| y = mx + b | x = (y − b)/m |
| C = 2πr | r = C/(2π) |
| s = d/t | d = st |
Worked examples
Example 1: Make h the subject of V = (1/3)Bh → h = 3V/B.
Example 2: Make b the subject of y = mx + b → b = y − mx.
Example 3: A rectangle’s perimeter is 50 cm and width is 10 cm. Use l = (P − 2w)/2 → 15 cm.
Common mistakes
- Dividing only one term: (P − 2w)/2 is not P − w.
- Forgetting to undo operations in reverse order.
- Square-rooting each term separately: √(a + b) ≠ √a + √b.
Try it yourself
- Make m the subject of F = ma.
- Make r the subject of C = 2πr.
- Make x the subject of y = 3x − 7.
- Make h the subject of A = ½bh.
- Make v the subject of E = ½mv².
Answers: 1) m = F/a 2) r = C/(2π) 3) x = (y + 7)/3 4) h = 2A/b 5) v = √(2E/m)
Quick recap
- A literal equation has several variables.
- Rearrange with inverse operations to change the subject.
- Undo operations in reverse order, and factor if the variable appears twice.