Changing the subject

Rearranging linear and non-linear formulas, including y = ax² + c, by undoing operations in reverse order, then using the new formula to answer questions.

The subject of a formula is the letter on its own, usually on the left. In $C = 450 + 38n$, the subject is $C$: give it $n$ and it hands you the cost. But what if you know the budget and want the number of guests? Then you need $n$ on its own. That is changing the subject.

Undo in reverse order

Think about getting dressed: socks first, then shoes. To undo it you take the shoes off first. Formulas work the same way. List what was done to your letter, in order, then undo those steps from the last one back to the first.

Example 1●

Make $a$ the subject of $v = u + at$.

$v - u = at$

Building v from a: first multiply by t, then add u. So undo the + u first.

$a = \dfrac{v - u}{t}$

Then undo the multiplication by t.

Example 2●●

Make $C$ the subject of $F = \dfrac{9C}{5} + 32$, then convert $77^\circ$F to Celsius.

$F - 32 = \dfrac{9C}{5}$

Done to C: times 9, divide by 5, add 32. Undo the add 32 first.

$5(F - 32) = 9C$

Undo the divide by 5 by multiplying both sides by 5.

$C = \dfrac{5(F - 32)}{9}$

Undo the times 9.

$C = \dfrac{5(77 - 32)}{9} = 25$, so $77^\circ\text{F} = 25^\circ\text{C}$

Now substitute F = 77: 77 minus 32 is 45, times 5 is 225, divided by 9 is 25.

InteractiveBacktracking: undo the steps in reverse

Try this

Choose $F = \dfrac{9C}{5} + 32$. The top row shows what happens to $C$. Tap the undo steps in the right order to build the new formula. Then try $y = ax^2 + c$. Which step only works if you remember the $\pm$?

Formulas with a square

The syllabus asks you to rearrange formulas like $y = ax^2 + c$. The last thing done to $x$ was squaring it, so the last thing you undo is a square root. A square root has two answers, one positive and one negative, so write $\pm$.

Rearranging y = ax² + c

$$x = \pm\sqrt{\frac{y - c}{a}}$$

In real problems a length, a time or a speed can't be negative, so you keep only the positive root. Say so in your answer.

Example 3●●

A ball dropped from an $80$ m high building is at height $h = 80 - 4.9t^2$ metres after $t$ seconds. Make $t$ the subject, then find when it hits the ground.

$4.9t^2 = 80 - h$

Get the t squared term on its own: add 4.9t² to both sides and subtract h.

$t^2 = \dfrac{80 - h}{4.9}$

Divide by 4.9.

$t = \sqrt{\dfrac{80 - h}{4.9}}$

Square root. Time can't be negative, so keep the positive root.

$t = \sqrt{\dfrac{80}{4.9}} \approx 4.04$ seconds

The ground is h = 0.

Example 4●●●

The power used by a heater is $P = I^2R$. Find the current $I$ when $P = 2400$ watts and $R = 24$ ohms.

$I^2 = \dfrac{P}{R}$

Undo the times R first.

$I = \sqrt{\dfrac{P}{R}} = \sqrt{\dfrac{2400}{24}} = \sqrt{100} = 10$ amps

Then the square. Current here is a size, so take the positive root.

Your turn

The volume of a cylinder is $V = \pi r^2 h$. Make $r$ the subject, then find $r$ when $V = 1000$ cm$^3$ and $h = 12$ cm.

Show the working

Divide by $\pi h$: $r^2 = \dfrac{V}{\pi h}$. Square root (a radius is positive): $r = \sqrt{\dfrac{V}{\pi h}}$. Then $r = \sqrt{\dfrac{1000}{12\pi}} = \sqrt{26.52\ldots} \approx 5.15$ cm.

Flashcards

Changing the subject

List what was done to the letter, then undo it in reverse order

If $y = ax^2 + c$, then $x =$

$\pm\sqrt{\dfrac{y - c}{a}}$

When to drop the $pm$

When the letter is a length, time, speed or other quantity that can't be negative

Practice

Make $t$ the subject of $d = st$.
Make $W$ the subject of $P = 2L + 2W$.
Make $x$ the subject of $y = 3x^2 - 5$.
To make $x$ the subject of $y = \dfrac{x + 4}{7}$, the first step is to:
Use $C = \dfrac{5(F - 32)}{9}$ to convert $104^\circ$F to Celsius.
A circle has area $50$ cm$^2$. Use $A = \pi r^2$ to find its radius, to two decimal places.
Use $E = \tfrac12 mv^2$ to find $v$ when $E = 360\,000$ and $m = 1800$.
Use $h = 80 - 4.9t^2$ to find, to two decimal places, the time in seconds when $h = 0$.
Exam style: A pizza shop prices pizzas with $C = 0.04d^2 + 5$, where $C$ is the price in dollars and $d$ is the diameter in cm. A pizza costs $\$21$. Find its diameter.
Exam style: A cone has volume $V = \dfrac13\pi r^2 h$. Make $h$ the subject and hence find $h$, to two decimal places, when $V = 300$ cm$^3$ and $r = 5$ cm.