Literal Equations
Solving a formula for one letter when everything else stays symbolic.
The explanation
A literal equation has several letters, and you solve for one of them.
Rearranging A = lw for w: divide both sides by l, giving w = A/l.
The method is identical to normal equations. Pretend every other letter is just a number you happen not to know.
For y = mx + b solved for x: subtract b, then divide by m, giving x = (y − b)/m.
The most common slip is dividing only part of a side. If you divide by m, everything on the other side gets divided, which is why the parentheses in (y − b)/m matter.
Solving a literal equation is inverting a formula with respect to one variable, and the algebra is unchanged from numerical equations. The additional demand is bookkeeping: several symbols must be manipulated without the feedback that arithmetic normally provides.
When the target variable appears in more than one term, factoring becomes necessary. To solve ab + c = ad for a, collect the a terms and factor: ab − ad = −c, so a(b − d) = −c and a = −c/(b − d) — with the standing restriction b ≠ d.
Every division introduces such a restriction, and stating it is part of a complete answer. Rearranging A = ½h(b₁ + b₂) for h gives h = 2A/(b₁ + b₂), valid provided the bases do not sum to zero.
This skill matters well beyond algebra class: it is exactly what is required to rearrange a physics formula or to convert a spreadsheet relationship into the quantity you actually want.
Worked example
Solve S = 2πr² + 2πrh for h.
- Isolate the h term: S − 2πr² = 2πrh.
- Divide both sides by 2πr.
- h = (S − 2πr²)/(2πr), for r ≠ 0.
Answer: h = (S − 2πr²)/(2πr)
Common mistakes
- Dividing one term of the numerator instead of the whole expression.
- Forgetting to factor when the target variable appears twice.