EQN-03-12

Rearranging Formulas

Solve a formula for any one of its letters using the same inverse-operation moves as a numeric equation.

~5 min read•Difficulty 3/5

Treat the letter you want as the unknown and every other letter as a number. Use inverse operations to isolate it.

A = l \cdot w \to solve for w \to w = A / l

Unwind the operations applied to the target variable last-applied-first.

v = u + at \to solve for t

v - u = at \to t = (v - u) / a

Multiply both sides by the denominator first.

d = m / V \to V \cdot d = m \to V = m / d

Rearranging lets you plug numbers in once at the end, and it is the skill behind many "express y in terms of x" questions.

Worked examples

Example 1 (difficulty 2/5)

The perimeter of a rectangle is

P = 2 (l + w) .

Solve for w.

◎ Predict the answer first:

Example 2 (difficulty 3/5)

C = (5/9) (F - 32),

solve for F.

◎ Predict the answer first:

Distributing incorrectly when a factor wraps the target variable

Why: Parentheses are skipped under time pressure.

Fix: Either divide both sides by the factor or expand fully — consistently.

Only multiplying part of a side by the reciprocal

Why: The whole side must be scaled.

Fix: Apply the operation to every term on both sides.