KS3 · Maths

Changing the subject of a formula

A plane flies 6,000 miles at 400 mph. How long does it take? s × t = d gives distance, not time. Don't find a new formula: reshape it.

Maths · Rearranging formulae

One formula, a whole family

A plane has to fly 6,000 miles at 400 miles per hour. How long will that take? You know that speed × time = distance, but the formula isn't in the shape you need yet. Pick a formula, then pick the letter you want to find.

The formula you have → The letter you want to find

Each formula links some letters. The question you are asking decides which letter you need on its own.

12 ways to write these four relationships.

On Choose a formula. 4 branches to choose from.

Every family includes the formula itself: the letter already on its own is the subject, and needs no work. Being first doesn't make a letter the subject, and neither does being nearly alone. a² or 3a isn't a yet.

Watch out: Nothing ever 'jumps across' the equals sign. Each rearrangement comes from doing the same undo move to both sides.

Maths · Algebra

Same move, both sides

Think of the equals sign as a balance. Whatever you do to one side, you do to the other, and it stays level. Step through each example.

GoalMake x the subject of x − 2.7 = 8.1

x − 2.7 = 8.1

2.7 has been taken away from x. To get x on its own, undo that.

Step 1 of 3

2.7 has been taken away from x. To get x on its own, undo that.

Watch out: People say a number 'jumps across' the equals sign and changes its sign. What really happens is that you add or subtract the same thing on both sides. Say it that way and you won't lose track of the signs.

Maths · Your turn

Rearrange a science formula

In science, acceleration is a = (v − u)/t, where u is the starting speed, v is the final speed and t is the time. Make v the subject. Then use your answer to make u the subject.

a = (v − u)/tHow was v built? First u was subtracted, then the whole (v − u) was divided by t. So the ÷ t is undone first.
missing step

Which line is step 2?

a − t = v − uat = v − ua = v − u − t

Maths · Check the working

Where does it go wrong?

Make d the subject of d/2 + 5 = e.

A student's answer — which line goes wrong?

d/2 + 5 = e
d + 5 = 2e
d = 2e − 5

Maths · Powers and roots

Undoing a square

Powers use the same inverse-operation idea. For y = x² + 5, subtract 5 from both sides to get y − 5 = x². That's not finished, because x² isn't the subject; x is. So square-root both sides: x = √(y − 5). Now try y = x² − 9.

Make x the subject of y = x² − 9. Which is closest to what you think?

x = √y + √9. Add 9, then square-root each part.x = √(y − 9). Deal with the 9, then square-root.x = √y − 3. Square-root first, then sort out the 9.x = √(y + 9). Add 9 to both sides, then square-root the whole side.

How sure are you?

I'm sureI think soI'm guessing

x = √y + √9. Add 9, then square-root each part.

This is a common misconception

Adding 9 is exactly right. The slip is square-rooting the parts separately, and a root can't be split across a + sign. Test it: √16 + √9 = 4 + 3 = 7, but √(16 + 9) = √25 = 5. Different answers. After adding 9, square-root the whole side as one group: x = √(y + 9).

x = √(y − 9). Deal with the 9, then square-root.

This is a common misconception

Good instinct to keep it all under one root. But the 9 was subtracted from x², so undoing it means adding 9 to both sides, not subtracting it. Taking 9 away again moves you further from x², not back to it.

x = √y − 3. Square-root first, then sort out the 9.

This is a common misconception

This undoes things in the wrong order. x was squared first and then 9 was taken away, so the − 9 has to be undone first. Rooting first also splits the root across the subtraction, which never works. Add 9, then root the whole side.

x = √(y + 9). Add 9 to both sides, then square-root the whole side.

This matches the evidence

Yes. Add 9 to both sides to get y + 9 = x², then square-root both whole sides: x = √(y + 9). Check with x = 5: y = 25 − 9 = 16, and √(16 + 9) = √25 = 5.

Undo the last operation first, and when you take a root, root the whole side as one group.

Predict, then check

They look almost identical. Commit to an answer before you look.

y = 3x² and y = (3x)² differ only by a bracket. When each one is rearranged to make x the subject, what do you get?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Reshape a formula so the thing you want to find stands on its own.

What you need to know

The subject of a formula is the letter on its own, with a power of 1 and a coefficient of 1, written in terms of the other letters. It can be on either side of the equals sign.
One formula gives a family of rearrangements: from a × b = c you also get a = c ÷ b and b = c ÷ a. What you want to calculate decides which one you use.
Rearrange with inverse operations done to both whole sides, undoing in the reverse of the order the letter was built.
Keep groups together: multiply every term, keep brackets like (S − V) whole, and square-root the whole of the other side.

The big picture

A formula links letters, like s × t = d. Changing the subject means rearranging it so the letter you want to find stands on its own. You get there by doing inverse operations to both sides, undoing in the reverse order the letter was built, keeping groups and roots whole.

Key points

1A formula is a rule linking variables; the plural of formula is formulae.

2The subject stands alone with power 1 and coefficient 1. It isn't just 'the first letter', and a² or 3x is not a subject.

3Inverse pairs: + and −, × and ÷, squaring and square-rooting, cubing and cube-rooting.

4Nothing jumps across the equals sign. You do the same operation to both sides.

5Order matters: d/2 + 5 = e gives d = 2e − 10, but (d + 5)/2 = e gives d = 2e − 5.

6With harder formulae the answer is an expression, such as x = (15 − y)/3, not a number.

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Worked example

Problem

The formula F = 9C/5 + 32 converts a temperature in degrees Celsius (C) to degrees Fahrenheit (F). Make C the subject.

⚠ Watch out

Only multiplying or dividing part of a side. In d/2 + 5 = e, multiplying by 2 must turn the 5 into 10 as well. And when you square-root, the root covers the whole of the other side, never each term separately.

🧠

Memory hook

Last on, first off. The last thing done to your letter is the first thing you undo, just like taking off your coat before your jumper.

✓

Check yourself

Make x the subject of 7x − y = 28, then find x when y = 7. (Answer: x = (28 + y)/7, so x = 5.)

Flashcards

(14)

What is a formula?

A rule linking variables, such as s × t = d. The plural is formulae.

What makes a letter the subject of a formula?

It is on its own on one side, with power 1 and coefficient 1, and written in terms of the other letters.

In a + b = c, which letter is the subject?

c. The subject can be on the right-hand side, and it isn't simply the first letter.

Why is a not the subject of a² = c² − b²?

a has a power of 2. You still need to square-root both sides.

What is the golden rule for rearranging?

Do the same inverse operation to both whole sides, so they stay equal.

In what order do you undo operations?

In reverse: the last thing done to the letter is the first thing you undo.

What does 'move the +3 across and change its sign' really mean?

Adding −3 (subtracting 3) on both sides of the equation.

When should you rearrange a formula before using it?

When the thing you want to calculate isn't the subject. The letter you want decides the rearrangement.

Make a the subject of a + b = c.

a = c − b (subtract b from both sides).

Make t the subject of d = st.

t = d/s (divide both sides by s).

Make r the subject of C = 2πr.

r = C/(2π) (divide both sides by 2π).

Make h the subject of A = ½bh.

h = 2A/b (multiply both sides by 2, then divide by b).

When you square-root to finish a rearrangement, what must the root cover?

The whole of the other side as one group. For example, y = x² + 7 gives x = √(y − 7), with the root over all of y − 7.

Make x the subject of y = x³ − 7.

x = ∛(y + 7): add 7, then cube-root the whole side.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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One formula, a whole family

Same move, both sides

Rearrange a science formula

Where does it go wrong?

Undoing a square

More KS3 Maths topics