20.5 Rearranging Formulae

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: substituting, writing formulae and changing the subject. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Substitute positive and negative numbers into formulae
• Write a formula from a description in words
• Change the subject of a formula using inverse operations
• Rearrange formulae that involve brackets, fractions, squares and square roots

Key words

• Formula: A rule linking two or more quantities, such as v = u + at. The letters stand for quantities.
• Subject: The letter on its own on one side of a formula. In A = lw, A is the subject.
• Substitute: Replace each letter with its value, then work out the answer.
• Inverse operation: The operation that undoes another: + and −, × and ÷, squaring and square-rooting.

Substituting into formulae

Write the formula, replace each letter with its value in brackets, then use the order of operations. Brackets help with negative numbers.

Worked example: v = u + at when u = 12, a = −3, t = 5

v = 12 + (−3) × 5 = 12 − 15 = −3

Answer: v = −3

Worked example: s = ut + 1/2 at² when u = 4, t = 3, a = 2

s = 4 × 3 + 1/2 × 2 × 3² = 12 + 9 = 21 (square t before multiplying)

Answer: s = 21

Writing formulae

Choose a letter for each quantity, then turn the words into algebra. A formula always has an equals sign and a subject.

Worked example: a plumber's charge

A plumber charges a £45 call-out fee plus £30 per hour. Write a formula for the total charge £C for h hours.
Fixed part: 45. Part that depends on hours: 30h.

Answer: C = 45 + 30h

Changing the subject

To make a different letter the subject, undo what has been done to it, in reverse order. Do the same to both sides each time, just like solving an equation.

Worked example: make x the subject of y = 3x − 4

Add 4 to both sides: y + 4 = 3x
Divide both sides by 3: (y + 4)/3 = x

Answer: x = (y + 4)/3

Worked example: make t the subject of v = u + at

Subtract u: v − u = at
Divide by a: t = (v − u)/a

Answer: t = (v − u)/a

Key fact: Use brackets to show what is divided: t = (v − u)/a is not the same as v − u/a.

Brackets, fractions, squares and roots

• ÷ 4 (a fraction); do … × 4 both sides; Example y = x/4 → x = 4y
• 2( … ) (a bracket); do … ÷ 2, or expand first; Example P = 2(a + b) → a = P/2 − b
• squaring; do … square root; Example A = r² → r = √A
• square root; do … square; Example y = √x → x = y²

Worked example: make r the subject of A = πr², then find r when A = 50

Divide by π: A/π = r². Square root: r = √(A/π)
r = √(50 ÷ π) = √15.915… = 3.99 (to 2 d.p.)

Answer: r = √(A/π); r ≈ 3.99

Worked example: make C the subject of F = 9C/5 + 32

Subtract 32: F − 32 = 9C/5. Multiply by 5: 5(F − 32) = 9C
Divide by 9: C = 5(F − 32)/9. Check with F = 212: 5 × 180 ÷ 9 = 100 ✓

Answer: C = 5(F − 32)/9

Common mistakes

• Undoing steps in the wrong order. Undo the last thing that was done first (usually + or − before × or ÷).
• Only changing one side of the formula. Whatever you do, do it to both sides.
• Leaving out brackets, e.g. writing t = v − u/a instead of (v − u)/a.
• Squaring a negative wrongly when substituting: (−3)² = 9.

Revision questions

Try these, then check your answers below each one.

Question 1. v = u + at. Work out v when u = 10, a = -4 and t = 3.

Answer: -2

Working: 10 + (-4) × 3 = 10 - 12 = -2

Question 2. Make x the subject of y = 4x + 7. (Choose from: x = (y - 7)/(4), x = y/4 - 7, x = (y + 7)/(4), x = 4y - 7.)

Answer: x = (y - 7)/(4)

Working: y - 7 = 4x, so x = (y - 7)/(4).

Question 3. P = 2(l + w). Work out w when P = 36 and l = 11.

Answer: 7

Working: 36 ÷ 2 = 18, 18 - 11 = 7.

Related lessons

• Previous lesson: 20.4 Solving Simultaneous Equations Algebraically
• Next lesson: 20.6 Proof
• In this chapter: 20.1 Graphs of Cubic and Reciprocal Functions
• In this chapter: 20.2 Non-linear Graphs
• In this chapter: 20.3 Solving Simultaneous Equations Graphically

Continue in the full course

This lesson is section 20.5 (EGCMF2005) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.