Formulae and Rearrangement

Rearranging Formulae

A formula expresses a relationship between two or more variables. The subject of a formula is the isolated variable on its own on one side of the equals sign. Rearranging — or changing the subject — means rewriting the formula so that a different variable is isolated. The method mirrors solving an equation: apply inverse operations to both sides to move everything that is not the target variable to the other side.

The golden rule: whatever you do to one side, do to the other. Work through the operations in reverse order (reverse BIDMAS) — undo addition/subtraction first, then multiplication/division, then powers/roots.

v = u + at − u v − u = at ÷ a t = (v − u)/a Making t the subject of v = u + at

Harder Rearrangements: Squares and Roots

When the target variable is squared, isolate the squared term first, then take the square root (taking the positive root unless context requires otherwise). When it appears under a root, square both sides to eliminate the root.

\[ A = \pi r^2 \implies r^2 = \frac{A}{\pi} \implies r = \sqrt{\frac{A}{\pi}} \] \[ T = 2\pi\sqrt{\frac{l}{g}} \implies \frac{T}{2\pi} = \sqrt{\frac{l}{g}} \implies \frac{T^2}{4\pi^2} = \frac{l}{g} \implies l = \frac{gT^2}{4\pi^2} \]

Hardest Case: Subject Appears in Two Terms

If the target variable appears in two or more separate terms, collect all those terms on one side, then factorise out the target variable before dividing.

\[ y = \frac{3x+2}{x-1} \implies y(x-1) = 3x+2 \implies xy - y = 3x+2 \] \[ xy - 3x = 2 + y \implies x(y-3) = 2+y \implies x = \frac{y+2}{y-3} \]

Worked Examples

Make \( u \) the subject of \( v = u + at \). Make \( P \) the subject of \( P = \dfrac{F}{A} \).
\[ v = u + at \implies u = v - at \] \[ P = \frac{F}{A} \implies F = PA \quad \text{(multiply both sides by } A \text{)} \]

For a formula like \( S = \frac{n(a+l)}{2} \), making \( a \) the subject:

\[ 2S = n(a+l) \implies \frac{2S}{n} = a+l \implies a = \frac{2S}{n} - l \]
Make \( v \) the subject of \( E = \dfrac{1}{2}mv^2 \). Make \( g \) the subject of \( T = 2\pi\sqrt{\dfrac{l}{g}} \).
\[ 2E = mv^2 \implies v^2 = \frac{2E}{m} \implies v = \sqrt{\frac{2E}{m}} \] \[ \frac{T}{2\pi} = \sqrt{\frac{l}{g}} \implies \frac{T^2}{4\pi^2} = \frac{l}{g} \implies g = \frac{4\pi^2 l}{T^2} \]
Make \( u \) the subject of \( \dfrac{1}{f} = \dfrac{1}{u} + \dfrac{1}{v} \).

Target \( u \) appears in a fraction. Isolate \( \frac{1}{u} \) first, then take the reciprocal.

\[ \frac{1}{u} = \frac{1}{f} - \frac{1}{v} = \frac{v - f}{fv} \] \[ u = \frac{fv}{v - f} \]
Make \( x \) the subject of \( y = \dfrac{2x + 5}{x - 3} \).
\[ y(x-3) = 2x+5 \implies yx - 3y = 2x+5 \implies yx - 2x = 3y+5 \] \[ x(y-2) = 3y+5 \implies x = \frac{3y+5}{y-2} \]

 Key Takeaways

  • Changing the subject: apply inverse operations in reverse order to isolate the target variable on one side.
  • For squared targets: isolate the squared term first, then take the square root. For a target under a root: square both sides.
  • When the target appears in two or more terms: collect all those terms on one side, factorise out the target variable, then divide.
  • Always verify a rearrangement by substituting numbers: check that the original formula and the rearranged formula give consistent results.
  • The formula \( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \) (thin lens equation) is a classic harder rearrangement — isolate \( \frac{1}{u} \) as a single fraction before reciprocating.