Literal Equations and Formulas: notes and common mistakes

Key ideas

• A literal equation has two or more letters. A formula is a literal equation that describes a real relationship.
• To solve for a letter, get it alone on one side and treat every other letter like a number.
• Use the same moves as ordinary equations: do the same thing to both sides (never multiply or divide by zero) and undo what is attached to the letter, last thing first. For ax + b = c shapes, that means add or subtract first, then multiply or divide.
• If the letter appears in more than one term, factor it out first, then divide.
• Dividing by a letter assumes that letter is not zero.

Steps

1. Circle the letter you are solving for.
2. If it is in more than one term, get those terms together on one side and factor the letter out.
3. Undo what is attached to it, last-attached first, doing the same thing to both sides. For ax + b = c shapes, that means add or subtract first, then multiply or divide.
4. When you divide or multiply, do it to the whole side (use parentheses).
5. Check by picking numbers and testing both forms.

Formulas

• Distance: d = rt gives r = d ÷ t and t = d ÷ r (when the divisor is not 0)
• Perimeter of a rectangle: P = 2l + 2w gives w = (P − 2l) ÷ 2
• Temperature: F = (9/5)C + 32 gives C = (5/9)(F − 32)
• Simple interest: I = Prt gives P = I ÷ (rt) (when r ≠ 0 and t ≠ 0)
• Total with simple interest: A = P + Prt gives P = A ÷ (1 + rt) (when 1 + rt ≠ 0)
• Standard form of a line: Ax + By = C gives y = (C − Ax) ÷ B (when B ≠ 0)

Watch out

• Divide or multiply the whole side, not one term: (P − 2l) ÷ 2, not P − 2l ÷ 2.
• Before you divide by a letter, remember it must not be 0. Say “assume r ≠ 0.”
• Signs stay with their terms. In 5x − 2y = 10 the term is −2y, so you divide by −2, not by 2.
• If the letter appears twice, subtracting will not remove it. Factor first.
• The other letters do not disappear. They end up on the other side.

Why it works

• Why can you treat other letters like numbers? The balance rule works for any quantity, so the same legal moves work on letters.
• Why factor out a letter that appears twice? Factoring puts the letter in one place, so a single division can undo it.
• Why rearrange at all? One rearranged formula can answer many questions, without redoing the algebra each time.