Changing the subject of a formula
Changing the subject of a formula
The subject of a formula is the variable that stands alone on one side, with no copy of it on the other side. In \(A=\pi r^2\), \(A\) is the subject. Changing the subject means rearranging the formula so that another variable stands alone. A formula is used for many values; an equation is solved once.
Why it works: undo the operations in reverse order. The formula \(y=\dfrac{3x-5}{2}\) builds \(y\) from \(x\) in three steps, so \(x\) is recovered by undoing them backwards, doing the same thing to both sides each time.
| Building \(y\) from \(x\) | Undoing, starting from \(y\) |
|---|---|
| multiply by \(3\): \(3x\) | multiply by \(2\): \(2y=3x-5\) |
| subtract \(5\): \(3x-5\) | add \(5\): \(2y+5=3x\) |
| divide by \(2\): \(y=\dfrac{3x-5}{2}\) | divide by \(3\): \(x=\dfrac{2y+5}{3}\) |
Method
- Clear any fraction by multiplying both sides by its denominator.
- Expand a bracket that contains the new subject.
- Move every term without the new subject to the other side.
- Divide both sides by the coefficient of the new subject.
Example
The sum of the interior angles of a polygon with \(n\) sides is \(S=180(n-2)\) degrees. Make \(n\) the subject, and find \(n\) when \(S=1080\).
\(\begin{aligned}S&=180(n-2) \\ S&=180n-360 \\ S+360&=180n \\ n&=\dfrac{S+360}{180}\end{aligned}\)
When \(S=1080\): \(n=\dfrac{1080+360}{180}=8\).
Check: divide the whole side by the coefficient. From \(2l=P-2b\), the result is \(l=\dfrac{P-2b}{2}\), not \(l=P-b\) or \(l=\dfrac{P}{2}-2b\).
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