Rearranging formulae

Apply the same operation to both sides of an equation. Fractions are easier to handle after multiplying by a common denominator, provided it is nonzero. When the desired variable appears in several terms, bring those terms together and factor it out. Dividing too early often leaves the variable on both sides.

Keep domain restrictions from the original formula. A new denominator must also be nonzero. Squaring an equation can introduce an extra sign or an extraneous solution, so state any positivity assumption before taking a square root. Check a rearrangement by substituting values into both the original formula and the new one.

Worked example

Rearrange 1/f = 1/u + 1/v to make v the subject.

1. Subtract 1/u: 1/v = (u − f)/(fu).
2. Invert the nonzero fraction.
3. The original formula needs f, u and v nonzero; the rearrangement also needs u ≠ f.

Answer: v = fu/(u − f)

Revise first: Algebraic fractions.