Literal Equations

Solve a formula for any one of its letters.

The idea

Definition (Literal equation).

A literal equation is an equation relating several letters — a formula. Solving it for a chosen letter means rearranging until that letter stands alone on one side, expressed in terms of the others.

The rearrangement turns a formula around: $C = 2\pi r$ computes the circumference from the radius, while the same relation solved for $r$, namely $r = \frac{C}{2\pi}$, computes the radius from the circumference.

The steps are the ones used on an equation with numbers: perform the same operation on both sides, undoing each operation with its inverse — subtraction undoes addition, division undoes multiplication — in the reverse of the order they were applied, while the other letters are carried along as constants.

Two differences appear in practice. The answer is an expression rather than a number, so no arithmetic remains at the end. And each division by a letter assumes that letter is not zero, an assumption worth stating, since the finished formula no longer shows it.

Ways to work on it

• Walkthrough. Rearrange three familiar formulas for a chosen letter.
• Practice. Solve a random formula for one of its letters.
• Hardest. A two-step rearrangement with a fraction and a grouped sum.

Not sure where to start? Take the ten-question placement test.