Maths bridge › Rearranging equations
Rearranging equations
Changing the subject of an equation, reliably, including through squares and roots.
One rule, applied in reverse order
An equation is a set of operations applied to the quantity you want. To free it, undo them: apply the inverse of each operation to both sides, in the reverse of the order they were applied. Adding undoes subtracting, dividing undoes multiplying, taking a square root undoes squaring.
Formula triangles stop at three symbols and cannot survive A-level equations. The both-sides method never runs out, which is why it is worth practising on easy cases until it is automatic.
Through brackets, squares and roots
WORKED EXAMPLE
Freeing a buried symbol
Solve ε = I(R + r) for r.
The r is inside a bracket that is multiplied by I. Undo the outermost layer first: divide both sides by I, giving ε/I = R + r.
Then subtract R: r = ε/I − R. Two moves, each applied to both sides, outermost first.
WORKED EXAMPLE
When the subject is under a root
The pendulum equation is T = 2π√(l/g). Solve it for g.
The g sits inside a root; a root is undone by squaring, so square both sides first, before any dividing: T2 = 4π2l/g.
Now g is a simple divisor: multiply both sides by g, divide by T2, and g = 4π2l/T2.
A pendulum of length 0.99 m with a period of 2.0 s gives g = 4π2 × 0.99/2.02 = 9.77 m s−2, which is the sanity check the algebra deserves.
YOUR TURN
The lens of the discharge equation
The time constant relation Q = Q0e−t/RC will meet you in the capacitance unit. Without touching the exponential yet, solve the simpler v2 = u2 + 2as for a, then check your result's units make sense before opening the working.
Show the working
Subtract u2 from both sides, then divide by 2s: a = (v2 − u2)/2s.
Units check: (m s−1)2 divided by metres is m2 s−2/m = m s−2, an acceleration. The algebra and the units agree.
Where physics leans on this: EMF and internal resistance · Motion graphs and SUVAT · Pendulums and springs. All eight skills: the maths bridge.