Rearranging a formula is solving an equation with letters instead of numbers. Undo the operations in reverse order, and (Extended) handle squares, square roots and a subject that appears twice.
Undo what has been done to the new subject, in reverse order, doing the same to both sides. If the subject appears twice (Extended), collect both terms on one side, factorise it out, then divide.
Section 01
Undo in reverse order.
In the lab choose a formula and press Next step. Each step undoes one operation, starting with the last thing that was done to the new subject. Try predicting each step before you press, then pick an Extended formula and watch the square root appear.
Worked example Two operations
Make \(t\) the subject of \(v = u + at\).
\(t\) was multiplied by \(a\), then \(u\) was added. Undo the adding first: \(v - u = at\).
From \(4y + 5 = 3x\), writing \(x = \dfrac{4y}{3} + 5\) divides only one term. The whole side must be divided: \(x = \dfrac{4y + 5}{3}\).
Section 02
Powers and roots (Extended).
A square is undone by a square root, and a square root is undone by squaring. Get the squared (or rooted) part on its own first, then apply the inverse to the whole of both sides.
Worked example Undo a square (Extended)
Make \(v\) the subject of \(E = \dfrac{1}{2}mv^{2}\).
Multiply both sides by 2: \(2E = mv^{2}\).
Divide by \(m\): \(\dfrac{2E}{m} = v^{2}\).
Square root both sides.
Answer: \(v = \sqrt{\dfrac{2E}{m}}\)
Worked example Undo a root (Extended)
Make \(l\) the subject of \(T = 2\pi\sqrt{\dfrac{l}{g}}\).
Divide by \(2\pi\): \(\dfrac{T}{2\pi} = \sqrt{\dfrac{l}{g}}\).
Square both sides: \(\dfrac{T^{2}}{4\pi^{2}} = \dfrac{l}{g}\).
Multiply by \(g\).
Answer: \(l = \dfrac{gT^{2}}{4\pi^{2}}\)
Your turn
(Extended) Make \(r\) the subject of \(V = \dfrac{1}{3}\pi r^{2}h\).
When you square \(\dfrac{T}{2\pi}\), the \(2\pi\) is squared too: \(4\pi^{2}\), not \(2\pi\).
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Make \(x\) the subject of \(y = x + 5\).
Quick check+10 XP
Make \(w\) the subject of \(A = lw\).
Quick check+10 XP
Make \(x\) the subject of \(y = 3x - 2\).
Quick check+10 XP
(Extended) Make \(r\) the subject of \(A = \pi r^{2}\).
Section 03
The subject twice (Extended).
If the new subject appears in two places, gather both terms on one side and everything else on the other. Then factorise the subject out and divide by the bracket.
Worked example Collect and factorise (Extended)
Make \(a\) the subject of \(3a + b = ca - 4\).
Terms in \(a\) to the right, others to the left: \(b + 4 = ca - 3a\).
Factorise out \(a\): \(b + 4 = a(c - 3)\).
Divide by \((c - 3)\).
Answer: \(a = \dfrac{b + 4}{c - 3}\)
Worked example Subject on top and bottom (Extended)
Make \(x\) the subject of \(y = \dfrac{x + 2}{x - 3}\).
Multiply by \((x - 3)\): \(y(x - 3) = x + 2\), so \(xy - 3y = x + 2\).
\(x\) terms to the left: \(xy - x = 3y + 2\).
Factorise: \(x(y - 1) = 3y + 2\).
Divide by \((y - 1)\).
Answer: \(x = \dfrac{3y + 2}{y - 1}\)
Your turn
(Extended) Make \(t\) the subject of \(s = \dfrac{2t + 1}{t - 1}\).
\(st - s = 2t + 1\), so \(st - 2t = s + 1\). Factorise: \(t(s - 2) = s + 1\), so \(t = \dfrac{s + 1}{s - 2}\).
Examiner trap
An answer such as \(x = \dfrac{3y + 2 + x}{y}\) still has \(x\) on the right, so it is not rearranged. The subject must appear on one side only, exactly once.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
Make \(t\) the subject of \(v = u + at\).
Exam style+20 XP
Make \(x\) the subject of \(y = \dfrac{3x - 5}{4}\).
Exam style+20 XP
(Extended) Make \(l\) the subject of \(T = 2\pi\sqrt{\dfrac{l}{g}}\).
Exam style+20 XP
(Extended) Make \(x\) the subject of \(y = \dfrac{x + 2}{x - 3}\).
Skill unlocked
Formulas rearranged.
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