Rearranging Formulas and Literal Equations
Rearranging a formula means isolating one letter using the same moves you use on numerical equations: undo addition, then multiplication, in reverse order of operations — while every other letter just rides along like a number.
The reason this feels harder than solving 2x + 3 = 11 is that the equation is crowded with letters, and your eye doesn't know which one is the star. Fix that by picking the target letter and mentally graying out every other letter — treat each one as if it were just some number, even though you never learn its value. Then undo the operations attached to your target in reverse order: undo addition and subtraction first, then undo multiplication and division. The steps are identical to solving for x in a plain equation; only the bookkeeping looks scarier because the 'numbers' have names like P and t.
Order matters because operations stack on your target letter like layers. In A = P(1 + rt), r is buried inside a sum, which is inside a multiplication by P. You peel from the outside in: divide by P first to strip that layer, then subtract 1 to clear the sum, then divide by t last, since t is the final layer wrapping r. Same idea in V = π r2 h: h is only multiplied by π r2, so one division finishes the job, giving h = V / (π r2).
This connects to solving equations with variables on both sides, which is next on the ladder. When the target letter appears twice, say in ax + b = cx - d, you can't isolate it with one division yet. First gather every term with x onto one side and every other term onto the other side, then factor x out of what's left before you divide. That factoring step is the one genuinely new move literal equations add.
Key terms
- Literal equation
- An equation whose terms are mostly letters standing for known quantities, like formulas from geometry or finance.
- Isolating a variable
- Performing inverse operations until the target letter sits alone on one side of the equation.
- Order of operations in reverse
- Undoing addition and subtraction before undoing multiplication and division when peeling layers off a variable.
- Factoring out the variable
- Pulling a common variable out of multiple terms so the equation has only one copy of it, needed when the target appears on both sides.
Watch it explained
- How it worksHow do you treat every other letter as an ordinary number while isolating the one you want?
- Worked exampleWalk me through solving A = P(1 + rt) for r.
- Worked exampleFind h in terms of V and r starting from the cylinder volume V = π r2 h.
- Going furtherWhat changes about the steps when the letter you are solving for sits on both sides?
Check yourself
End of topic.