Rearranging Formulae: AQA Level 2 Further Maths revision notes

Undo operations on both sides

The subject is the letter written alone on one side of a formula. To change the subject, perform the same operation on both sides. Keep fractions and brackets together until you can remove them safely.

Subtract additions, divide multipliers, and work from the outside of a grouped expression towards the subject. Any quantity you divide by must be non-zero.

Worked example 1

Make t the subject of v=u+at, where a\ne0

Subtract u from both sides: v-u=at
Your turn. Divide by a. Give the formula for t

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

t=\frac{v-u}{a}
Answer: t=\frac{v-u}{a}. Both terms in the numerator are divided by a

Worked example 2

The perimeter of a rectangle is P=2l+2w. Make l the subject.

Subtract 2w: P-2w=2l
Your turn. If P=30 cm and w=6 cm, what is l in centimetres?

Your answer

l=\frac{30-12}{2}=9 cm
Answer: l=\frac{P-2w}{2}=\frac P2-w. For a rectangle, P>2w>0

Explore the rearranged perimeter formula

Try it: keep P fixed and increase w. Each increase of 1 cm in the width decreases the length by 1 cm, so the perimeter stays the same.

Collect the subject terms and factorise

When the subject appears more than once, collect all its terms on one side, collect the other terms on the other side, then factorise. Dividing too early can leave the subject trapped in a denominator.

Worked example 3

Make x the subject of p=3x+qx, where q\ne-3

Factorise the right side: p=x(3+q)
Your turn. What expression must you divide by?

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

Divide by 3+q
Answer: x=\frac{p}{3+q}. The condition q\ne-3 ensures the divisor is non-zero.

Worked example 4

Make x the subject of y=\frac{3x+2}{x-4}, where x\ne4

Multiply by x-4: y(x-4)=3x+2. Expand: xy-4y=3x+2
Collect the x terms: xy-3x=4y+2. Factorise: x(y-3)=4y+2
Your turn. Give the formula for x

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

x=\frac{4y+2}{y-3}
Answer: x=\frac{4y+2}{y-3}, where y\ne3. The original formula cannot produce y=3: it would require 3x-12=3x+2

Undo powers and roots carefully

If x^2=k with k>0, then:

x=\pm\sqrt{k}

For a positive length, use the positive root. If a square root is already present, isolate it before squaring both sides.

Squaring a fraction squares its whole numerator and denominator:

\left(\frac{T}{2\pi}\right)^2=\frac{T^2}{4\pi^2}

Worked example 5

The formula T=2\pi\sqrt{\frac lg} relates positive quantities T, l and g. Make l the subject.

Divide by 2\pi: \frac{T}{2\pi}=\sqrt{\frac lg}
Your turn. What is (2\pi)^2?

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(2\pi)^2=4\pi^2
Square both sides: \frac{T^2}{4\pi^2}=\frac lg. Multiply by g
Answer: l=\frac{gT^2}{4\pi^2}. The positivity conditions make the original square-root formula consistent.

Worked example 6

Make v the subject of \frac1f=\frac1u+\frac1v, where f, u and v are non-zero.

Subtract \frac1u: \frac1v=\frac1f-\frac1u=\frac{u-f}{fu}
Your turn. Take the reciprocal of both sides to give v

Your answer

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

v=\frac{fu}{u-f}
Answer: v=\frac{fu}{u-f}, with u\ne f. If u=f, the original equation requires \frac1v=0, which no finite non-zero v can satisfy.

Common mistakes

Changing only one side. Apply the same operation to both sides of the equation.
Dividing only part of a sum. Bracket the whole sum and divide every term.
Leaving the subject on both sides. Collect its terms on one side, then factorise.
Forgetting to square a coefficient. Square the whole product: (3x)^2=9x^2.
Taking reciprocals term by term. Take the reciprocal of the whole expression, not each term in a sum.
Ignoring division or root restrictions. Check divisors are non-zero and choose roots consistent with the context.