Simple formulae
Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable
What a learner can do afterwards
- Identify the subject of a given formula
- Use inverse operations to rearrange a formula to make a different variable the subject
- Rearrange formulae involving multiple steps including fractions and powers
1 · Read
A formula is a rule written with letters. It links real quantities, like speed, distance and time in v = d ÷ t, or length, width and perimeter in P = 2l + 2w. The letter standing alone on one side of the equals sign is called the subject. In v = d ÷ t the subject is v, and in A = l x w the subject is A. Rearranging a formula means making a different letter the subject, so you can find that quantity directly. It works exactly like solving an equation: whatever you do to one side, you must do to the other, and you undo each operation with its inverse. Added something? Subtract it. Multiplied? Divide. Divided? Multiply. Squared? Take the square root. When more than one thing was done to the letter, undo them in reverse order.
One undo is often enough. In A = l x w, w is multiplied by l, so divide both sides by l and get w = A ÷ l. In y = x + 3, x has 3 added on, so subtract 3 from both sides and get x = y - 3. The inverse takes you straight to the new subject.
Harder formulae need two undos, last one first. In m = 2n + 3, n is multiplied by 2 and then 3 is added. Undo the addition first: m - 3 = 2n. Then undo the multiplication: n = (m - 3) ÷ 2. With a square, the root comes last. In y = 2x², x is squared and then doubled. Divide both sides by 2 to get x² = y ÷ 2, then take the square root: x = √(y ÷ 2).
Rearranging first makes number questions easy. Zara knows a runner's speed and time and wants the distance. She turns v = d ÷ t into d = v x t, then puts her numbers in: d = 30 x 2 = 60. No guessing, just one multiply.
Check a rearrangement with easy numbers. If v = 10 and t = 3, then d = 30, and your new formula must give 30 too. If it does not, look for a spot where the inverse flipped, like adding where you should subtract.
The subject is the letter standing alone. To change it, undo each operation with its inverse on both sides, last one first, then put your numbers in.
Adapted from OpenStax College Algebra (CC-BY 4.0), openstax.org · license
More to read
2.6 Solve a Formula for a Specific Variable
The subject of a formula is the letter sitting alone on one side. In P = 4s the subject is P; in v = u + at the subject is v. Naming the subject is step zero, because rearranging means choosing a new letter for that starring role.
Changing the subject uses inverse operations in reverse order. To make x the subject of y = 3x + 5, first undo the + 5, then undo the x3. Working backwards through the build-up peels the wanted letter free.
9.7 Solve a Formula for a Specific Variable
Formulae with fractions need clearing before peeling. To make x the subject of y = (x + 3) / 2, multiply everything by 2 first, giving 2y = x + 3, then subtract 3. Clearing the fraction early keeps the working on one line per step.
Powers need their own inverse. Making r the subject of A = πr² means dividing by π and then taking the square root. Each operation in the formula is undone once, in reverse order, and skipping any single undo breaks the result.
2.3 Solve a Formula for a Specific Variable
Substitution turns a rearranged formula into an answer. Once y = 3x + 5 becomes x = (y - 5) / 3, putting y = 20 gives x = 5. Rearrange first with letters, substitute numbers last: that order avoids arithmetic tangles.
Real formulae reward the same routine. The Fahrenheit rule F = 9C / 5 + 32 takes C = 20 to F = 68 by multiply, divide, then add. Whether the formula comes from geometry or science, subject, inverse, substitute is the full method.
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