Algebra · Formulae & rearranging
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Rearranging formulae
Why changing the subject is solving-an-equation in disguise, why operations must be undone in reverse order (onion peeling), and how fractions, squares and roots come apart.
Algebra · Formulae & rearranging
Rearranging formulae
Why changing the subject is solving-an-equation in disguise, why operations must be undone in reverse order (onion peeling), and how fractions, squares and roots come apart.
Why it works
"Make the subject" means: rewrite the formula as with appearing exactly once, alone. It's the same balance game as solving an equation — the only difference is that letters, not numbers, ride along. Every move is an inverse operation applied to both whole sides.Peel the onion: last operation on, first operation off. In , the was multiplied by 4 then had 7 subtracted. Undo in reverse: add 7 (), then divide by 4:
The WHOLE side gets divided — divides only one term and breaks the balance. The fraction bar is a bracket: .
Letters obey the same moves as numbers. : subtract , divide by — . That is unknown changes nothing; the balance doesn't care what's riding on it.
Squares and roots are inverses of each other. : divide by first, THEN take the square root of the whole side:
The root must swallow everything — roots only the top, a different (wrong) formula. Going the other way, a root is undone by squaring the whole side: from , isolate the root FIRST (), then square:
Squaring before the root is alone drags the and the 2 into the square and wrecks it — isolate, then invert.
Check by substituting a number. Put an easy value through the original and your rearrangement — if gives at , then must come back. One number catches most slips.