Topic 2.2

Changing the Subject of a Formula

1. Undoing Addition and Subtraction

Changing the subject means rewriting a formula so that a different letter stands alone on one side.
Undo the operations one layer at a time, in reverse order, doing the same to both sides so the formula stays balanced.
Find the term containing the subject first. If something is added to it, subtract it; if something is subtracted, add it: $y = x + c \Rightarrow x = y - c$ .
Only after the subject term stands alone should you remove any multiplier or denominator.

Operation	Undo it by
$+ c$	$- c$
$\times a$	$\div a$
square	square root
square root	square
$\div b$	$\times b$

Make

x

the subject of

y = x + 7

Subtract 7 from both sides:

y - 7 = x

x = y - 7

Make

p

the subject of

p - 6 = 2 q

Add 6 to both sides:

p = 2 q + 6

Make

v

the subject of

s = v t + u

Subtract u from both sides:

s - u = v t

Divide both sides by t:

v = \frac{s - u}{t}

Make

b

the subject of

L = 2 h + 2 b

Subtract 2h from both sides:

L - 2 h = 2 b

Divide both sides by 2:

b = \frac{L - 2 h}{2}

Make

q

the subject of

4 p + q = 3 r - 2

Subtract 4p from both sides:

q = 3 r - 4 p - 2

Make

b

the subject of

P = 2 a + 3 b

Subtract 2a from both sides:

P - 2 a = 3 b

Divide both sides by 3:

b = \frac{P - 2 a}{3}

2. Undoing Multiplication and Division

Once the subject term stands alone, remove whatever multiplies or divides it: $a x = b \Rightarrow x = \frac{b}{a}$ .
Many formulae need two stages: first remove the added terms, then the multiplier.
If the subject is divided by something, multiply both sides by it.
When dividing, keep the whole of the other side on top, e.g. $\frac{v - u}{a}$ , not $v - \frac{u}{a}$ .

Make

x

the subject of

5 x = 2 y

Divide both sides by 5:

x = \frac{2 y}{5}

Make

r

the subject of

\frac{r}{3} = k

Multiply both sides by 3:

r = 3 k

Make

m

the subject of

2 m - 4 = p

Add 4 to both sides:

2 m = p + 4

Divide both sides by 2:

m = \frac{p + 4}{2}

Make

t

the subject of

v = u + a t

Subtract u from both sides:

v - u = a t

Divide both sides by a, keeping the whole of v − u on top:

t = \frac{v - u}{a}

Make

P

the subject of

F = \frac{P}{A}

Multiply both sides by A:

P = F A

Make

r

the subject of

C = 2 π r

Divide both sides by 2π:

r = \frac{C}{2 π}

3. Powers and Roots

When the subject is squared, cubed or under a root, first get that term on its own.
Then apply the inverse to both sides: square root for a square, cube root for a cube, square for a square root.
Do not take a root too early, or other terms will still be mixed in with the subject: $A = π r^{2} \Rightarrow r = \sqrt{\frac{A}{π}}$ .

Make

r

the subject of

A = π r^{2}

Divide both sides by π:

\frac{A}{π} = r^{2}

Take the square root of both sides:

r = \sqrt{\frac{A}{π}}

Make

h

the subject of

V = \frac{1}{3} B h

Multiply both sides by 3:

3 V = B h

Divide both sides by B:

h = \frac{3 V}{B}

Make

L

the subject of

T = 2 π \sqrt{\frac{L}{g}}

Divide both sides by 2π:

\frac{T}{2 π} = \sqrt{\frac{L}{g}}

Square both sides:

\frac{T^{2}}{4 π^{2}} = \frac{L}{g}

Multiply both sides by g:

L = \frac{g T^{2}}{4 π^{2}}

Make

y

the subject of

P = (3 y + 1)^{2}

Take the square root of both sides:

\sqrt{P} = 3 y + 1

Subtract 1:

\sqrt{P} - 1 = 3 y

Divide by 3:

y = \frac{\sqrt{P} - 1}{3}

Make

t

the subject of

\frac{t - 4}{5} = y

Multiply both sides by 5:

t - 4 = 5 y

Add 4:

t = 5 y + 4

Make

n

the subject of

P V = n R T

Divide both sides by RT:

n = \frac{P V}{R T}

4. Fractions and Proportions

If the subject is in a denominator, multiply both sides by that denominator first, so it is no longer trapped in a fraction.
For one fraction equal to another, cross-multiply, then simplify before isolating the subject.
If this creates brackets, expand them only when it helps to collect the subject.

\frac{a}{b} = \frac{c}{d} \Rightarrow a d = b c \Rightarrow b = \frac{a d}{c}

Make

x

the subject of

\frac{3}{x} = k

Multiply both sides by x to bring it out of the denominator:

3 = k x

Divide both sides by k:

x = \frac{3}{k}

Make

R

the subject of

V = \frac{I R}{1 + R}

Multiply both sides by 1 + R:

V (1 + R) = I R

Expand:

V + V R = I R

Collect the R terms on one side:

V = I R - V R

Factorise:

V = R (I - V)

Divide by I − V:

R = \frac{V}{I - V}

Make

t

the subject of

\frac{d}{t} + c = f

Subtract c:

\frac{d}{t} = f - c

Multiply both sides by t:

d = t (f - c)

Divide by f − c:

t = \frac{d}{f - c}

Make

b

the subject of

\frac{a}{b} = \frac{c}{d}

Cross-multiply:

a d = b c

Divide both sides by c:

b = \frac{a d}{c}

Make

x

the subject of

\frac{a}{x} + \frac{b}{c} = d

Subtract b/c:

\frac{a}{x} = d - \frac{b}{c}

Write the right-hand side as one fraction:

\frac{a}{x} = \frac{c d - b}{c}

Cross-multiply:

a c = x (c d - b)

Divide by cd − b:

x = \frac{a c}{c d - b}

Make

b

the subject of

A = \frac{1}{2} (a + b) h

Multiply both sides by 2:

2 A = (a + b) h

Divide both sides by h:

\frac{2 A}{h} = a + b

Subtract a:

b = \frac{2 A}{h} - a

5. The Subject in More Than One Term

When the subject appears more than once, collect every term containing it on one side and everything else on the other.
Factorise the subject out of those terms.
Divide once by the bracket that multiplies the subject.
Look for patterns such as $p x + q x = x (p + q)$ and $a t - b t = t (a - b)$ .

\begin{aligned} P & = k + m k \\ P & = k (1 + m) \\ k & = \frac{P}{1 + m} \end{aligned}

Make

x

the subject of

y = a x + b

Subtract b:

y - b = a x

Divide by a:

x = \frac{y - b}{a}

Make

x

the subject of

A = p x + q x + r

Subtract r:

A - r = p x + q x

Both terms contain x, so factorise:

A - r = x (p + q)

Divide by p + q:

x = \frac{A - r}{p + q}

Make

k

the subject of

P = k + m k

Factorise k out of both terms:

P = k (1 + m)

Divide by 1 + m:

k = \frac{P}{1 + m}

Make

t

the subject of

s = a t - b t

Both terms contain t, so factorise:

s = t (a - b)

Divide by a − b:

t = \frac{s}{a - b}

Make

f

the subject of

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

Add the fractions on the right over the common denominator uv:

\frac{1}{f} = \frac{v + u}{u v}

Take the reciprocal of both sides:

f = \frac{u v}{u + v}

Make

x

the subject of

y = 3 x + a x

Both terms contain x, so factorise:

y = x (3 + a)

Divide by 3 + a:

x = \frac{y}{3 + a}

Make

n

the subject of

C = n R + (n - 1) L

Expand the bracket:

C = n R + n L - L

Add L to both sides:

C + L = n R + n L

Factorise:

C + L = n (R + L)

Divide by R + L:

n = \frac{C + L}{R + L}

Make

p

the subject of

S = \frac{2 p}{p + r}

Multiply both sides by p + r:

S (p + r) = 2 p

Expand:

S p + S r = 2 p

Collect the p terms on one side:

S r = 2 p - S p

Factorise:

S r = p (2 - S)

Divide by 2 − S:

p = \frac{S r}{2 - S}

Make

k

the subject of

y = \frac{k - 2}{k + 2}

Multiply both sides by k + 2:

y (k + 2) = k - 2

Expand:

y k + 2 y = k - 2

Collect the k terms on one side and the rest on the other:

2 y + 2 = k - y k

Factorise:

2 (y + 1) = k (1 - y)

Divide by 1 − y:

k = \frac{2 (1 + y)}{1 - y}