Changing the subject

A formula gives one letter from the others. To get a different letter, undo the steps in reverse order, doing the same to both sides: y = 3x + 5 becomes x = (y − 5)/3. Check by substituting.

P = 2l + 2w gives the fence around a garden. But you have 20 m of fence and a 6 m side. How wide can it be? Turn the formula around.

A formula like $P = 2l + 2w$ gives $P$ when you know $l$ and $w$ . $P$ is the subject: the letter on its own on one side.

Sometimes you know $P$ and want $w$ . Changing the subject rearranges the formula so $w$ is on its own. You undo the steps, as when you solved equations, but with letters.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Which formula gives w?

Test each option with numbers: if $l = 6$ and $w = 4$ , then $P = 20$ . Which option gives back 4?

Which formula gives

w

from

P = 2l + 2w

w = P \div 2 - 2l

w = P - 2l \div 2

w = (P - 2l) \div 2

w = (P + 2l) \div 2

TryTry it on the board and compare the result with your prediction.

Check with numbers

The board works the formula forwards. Your rearranged formula should give the same $w$ without trying.

Try it yourselfYour edits are saved in this browser.

Set $l = 6$ , then find the $w$ that makes $P = 20$ . Does your formula from Predict give the same $w$ ?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Undo in reverse order

$x$ was multiplied by 3, then 5 was added. Undo in reverse order: take away 5, then divide by 3.

Worked exampleBefore each step, say what you think comes next.

$y = 3x + 5$ $x$ was multiplied by 3, then 5 was added.
$y - 5 = 3x$ Undo the last step first: take 5 from both sides.
$\frac{y - 5}{3} = x$ Undo $\times 3$ : divide the whole of both sides by 3.
$x = \frac{y - 5}{3}$ Write the subject on the left.
$y = 11: x = \frac{11 - 5}{3} = 2$ Check by substituting: $3 \times 2 + 5 = 11$ . ✓

Try one: make $x$ the subject of $y = 2x + 7$ .

Read Jo's working one line at a time. Which step did she do first?

Write each step on its own line. The checker marks the first line that does not follow.

Make $x$ the subject of $y = 2x + 7$ .

Jo makes $x$ the subject of $y = 4x - 3$ . Click the line where her working goes wrong.

Make $x$ the subject of $y = 3x + 5$ . Write one step per line.

Start from $y = 3x + 5$

ModifyChange the problem on purpose and see what happens.

One change at a time

Each task changes one thing.

Need a hint?

w = (P − 2l) ÷ 2.

Another hint

(26 − 16) ÷ 2.

No board from here: undo the steps, then check with numbers.

Try it yourselfYour edits are saved in this browser.

Use $w = (P - 2l) \div 2$ to find $w$ when $P = 26$ and $l = 8$ . Set the board to check.

Use the board. This ticks when you get there.

One step: make $x$ the subject of $y = x + 9$ .

A minus sign: make $x$ the subject of $y = 5x - 4$ .

Undo a divide: make $d$ the subject of $v = \dfrac{d}{t}$ (speed = distance ÷ time).

A fraction: make $h$ the subject of $A = \dfrac{bh}{2}$ (area of a triangle).

Edge case: make $x$ the subject of $y = 7 - x$ .

PractiseA few on your own, to check you have it.

Rearrange and check

A plumber charges $c = 50h + 80$ dollars for $h$ hours. Make $h$ the subject.

Make $w$ the subject of $P = 2l + 2w$ . Write one step per line.

Start from $P = 2l + 2w$

Use your formula: what is $w$ when $P = 30$ and $l = 9$ ?

Review: factorise $6x - 15$ fully.

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Work it backwards

Write your formula for $n$ first. Then use it, and only then move the slider.

Stretch: make $l$ the subject of $P = 2l + 2w$ , then make $r$ the subject of $C = 2\pi r$ .

Try it yourselfYour edits are saved in this browser.

Need a hint?

Undo + 6, then undo × 4: n = (T − 6)/4.

Another hint

(30 − 6) ÷ 4.

Hiring a bike costs $T = 4n + 6$ dollars for $n$ hours. Make $n$ the subject, use it to find how many hours 30 dollars buys, then set the board to check.

Use the board. This ticks when you get there.

For teachers

Students rearrange formulas to change the subject by undoing the steps in reverse order on both sides, and check each rearrangement by substituting numbers, on a formula board that works the original forwards (AC9M8A01, node 3). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can make a given letter the subject of a formula by undoing the steps in reverse order.
I can do the same to the whole of both sides, so I divide every term.
I can check a rearranged formula by substituting numbers.

Before class

Do the Run board and the working check yourself first. Students need solving equations by backtracking and the balance method (Year 7 equations unit), substituting into formulas, and the two lessons before (expanding and factorising).

The one idea: changing the subject is solving an equation with letters. Undo what was done to the letter, last step first, doing the same to both sides. The common mistake is dividing only one term (y = 4x − 3 becomes y ÷ 4 = x − 3); it is planted after the method, in Jo's working. Undoing the − 3 first avoids it.

Answers are marked by form and value: the subject must be on its own on the left, and the formula must agree with the original for every value. An equal formula in another shape, such as x = y/2 − 7/2 for (y − 7)/2, is marked right.

Running the lesson

Predict: take a show of hands, then test each option with l = 6 and w = 4 (P = 20). Only one gives back 4.
Run: ask students to say why the board needed trial and error and the rearranged formula did not.
Investigate: model the worked example live as a flow: x → × 3 → + 5 → y, then backwards: y → − 5 → ÷ 3 → x.
Modify: before the edge case y = 7 − x, ask: what is x if y = 5? Most find 2 mentally; then ask how they did it.
Make: ask students to show the formula they wrote before the board answer, and how they checked it.

When students get stuck

A student divides only one term (Jo's mistake). Ask: Put in x = 2. Does your formula give back the y you started with? Fix: Undo the add or subtract first, so there is only one term left to divide.
A student undoes the steps in the wrong order (x = y/3 − 5 from y = 3x + 5). Ask: What was done to x first? What was done last? Fix: Draw the flow: x → × 3 → + 5 → y. Go back along it from y.
A student gets lost with letters. Ask: What would you do if the formula were 11 = 3x + 5? Fix: Solve it with the number, then do exactly the same steps with y.

Stretch and support

Stretch: Make l the subject of P = 2l + 2w, then r the subject of C = 2πr, then x the subject of ax + b = c.

Support: Give a number version first (11 = 3x + 5), then the letter version beside it, step for step.

Where it leads

This completes AC9M8A01: expanding, factorising and changing the subject.

Where it leads: formulas in science (v = d/t, F = ma, ρ = m/V), linear relations (y = mx + c rearranged for x), and in Years 9 and 10 rearranging formulas with the subject in more than one term (factorise it out).

What each step is for

which formula gives w?: Committing to a rearrangement of P = 2l + 2w before testing it with numbers tests whether they undo the steps in reverse order and divide the whole side.
check with numbers: Finding w on the board for P = 20 and l = 6, then working out the predicted formula, shows that a rearranged formula gives the same numbers as the original: substitution is the check.
undo in reverse order: Changing the subject undoes the steps applied to the letter in reverse order, doing the same to the whole of both sides. Jo's mistake divides only one term.
one change at a time: One change at a time: a context checked on the board, one step, a minus sign, undoing a divide, a fraction in the formula, then the edge case of a subject that is subtracted.
rearrange and check: Rearranging real formulas with full working, mixed; dividing only one term is the planted slip, and the review keeps factorising fresh.
work it backwards: Rearranging a cost formula and using it to answer a question, then checking on the board, puts the lesson's two moves together: change the subject, then substitute.

Now you can

Make a given letter the subject of a formula by undoing the steps in reverse order.
Do the same to the whole of both sides, so I divide every term.
Check a rearranged formula by substituting numbers.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson13. Solving harder equationsGet the x terms on one side and the numbers on the other, doing the same to both sides; expand any brackets first. Then check by substituting into the first equation.

Source: content/maths/algebra/rearranging/lesson.md