Rearranging Formulae (Changing the Subject)

What rearranging a formula means

A formula links several quantities, for example A = l × w (area = length × width). The letter on its own — here A — is called the subject of the formula.

Rearranging (or transposing) a formula means rewriting it so a different letter becomes the subject. From A = lw we might want l = A/w, so we can find the length when we already know the area and the width.

The skill uses exactly the same ideas as solving linear equations: the balance rule and inverse operations. The difference is that the answer is a new formula, not a single number.

The two tools you need

1. The balance rule. A formula is an equation, so whatever you do to one side you must do to the other.

2. Inverse operations. To free a letter, undo each operation around it with its opposite:

Operation	Inverse
+	−
−	+
×	÷
÷	×
square (²)	square root (√)
square root (√)	square (²)

Undo operations in reverse order: peel away the outermost layer first, just like taking off a coat before a jumper.

Worked example 1: one step

Make x the subject of y = x − 7.

7 is subtracted from x, so add 7 to both sides:

y = x − 7
y + 7 = x

So x = y + 7.

Worked example 2: a division

Make r the subject of C = 2πr.

r is multiplied by 2π. Undo with division:

C = 2πr
C ÷ (2π) = r

So r = C / (2π).

Worked example 3: two steps

Make a the subject of v = u + at. (This is a real physics formula: final speed = start speed + acceleration × time.)

Peel from the outside in. First subtract u, then divide by t:

v = u + at
v − u = at          (subtract u from both sides)
(v − u) / t = a     (divide both sides by t)

So a = (v − u) / t.

Why this order? a is multiplied by t and then u is added. To reverse, undo the addition first, then the multiplication — the opposite order.

Worked example 4: a bracket

Make x the subject of y = 3(x + 2).

The whole bracket is multiplied by 3, so divide first:

y = 3(x + 2)
y / 3 = x + 2       (divide both sides by 3)
y / 3 − 2 = x       (subtract 2 from both sides)

So x = y/3 − 2. (You could also expand to y = 3x + 6 first; both routes give the same result.)

Worked example 5: a square

Make x the subject of A = x².

Take the square root of both sides:

A = x²
√A = x

So x = √A (taking the positive root, since lengths and areas are positive).

Worked example 6: a square root

Make x the subject of y = √x.

Square both sides to undo the root:

y = √x
y² = x

So x = y².

Worked example 7: the subject appears twice (factorising)

When the new subject appears in two terms, gather those terms on one side and factorise.

Make x the subject of ax = bx + c.

ax = bx + c
ax − bx = c          (gather x-terms on one side)
x(a − b) = c         (factorise: take out the common factor x)
x = c / (a − b)      (divide by the bracket)

Factorising is the key move here — it turns two x-terms into a single x. See factorising expressions for more on this.

A reliable method

1. Decide which letter you want as the subject.
2. Find every operation attached to it.
3. Undo them one at a time, from the outside in, doing the same to both sides.
4. If the subject appears more than once, collect those terms and factorise.
5. Write the final formula with the new subject on its own.

Activity: change the subject

Make the letter in brackets the subject.

1. P = 4s → (s)
2. y = mx + c → (x)
3. A = ½bh → (h)
4. s = d / t → (t)
5. y = √(x + 1) → (x)

Answers:

1. s = P/4 (divide by 4)
2. x = (y − c)/m (subtract c, then divide by m)
3. h = 2A/b (multiply by 2, then divide by b)
4. t = d/s (multiply by t, divide by s)
5. x = y² − 1 (square both sides, then subtract 1)

Where this leads

Rearranging formulae lets you reuse one relationship in many ways. You will need it for substitution into formulas and whenever you switch a real-world equation around to find the quantity you actually want.