Changing the subject of a formula

What does "subject" mean?

In the formula $P = 2l + 2w$, the letter $P$ is on its own on the left. We say $P$ is the subject of the formula. Given $l$ and $w$, it gives you $P$ directly.

Sometimes you know $P$ and $w$ and need $l$. Then you want $l$ to be the subject. Changing the subject means rearranging the formula so a different letter stands alone, without changing what the formula says.

The idea is the same as solving an equation. Treat the letter you want as the unknown and treat all the other letters as if they were numbers.

Linear formulae

Think of the steps applied to $x$ in $y = 3x + 7$: first multiply by 3, then add 7. To get $x$ back, undo these in reverse order: subtract 7, then divide by 3.

Fractions and the letter in a bracket

If the formula has a fraction, multiply both sides by the denominator to clear it. Then continue as before.

The next example has the unknown inside a term with other letters. The same plan works.

Squares, roots and powers

To undo a square, take the square root. To undo a square root, square both sides. To undo a power such as $(1+i)^t$ with unknown base, take the matching root.

Look at the result: $l$ depends on $T^2$. If $T$ is doubled, $T^2$ becomes $4T^2$, so $l$ is multiplied by 4.

For a formula with a power such as $F = P(1+i)^t$, the unknown $P$ is just multiplied by a bracket, so divide. To find $i$ when $t = 4$, take the fourth root.

Formulae with the letter in two places

Sometimes your letter appears more than once, as in $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ from lenses. The plan: collect the terms with your letter, then combine into a single fraction, then flip.

Now notice what happens to $v = \frac{fu}{u - f}$ as $u$ gets close to $f$ from above. The bottom $u - f$ becomes a very small positive number, so $v$ becomes very large. Physically, the image moves further and further away.

Summary