Rearranging Formulae
Rearranging a formula means changing which letter is on its own. You will use the same balancing rules as when solving an equation. Start with the simple examples, then build towards powers and repeated subjects. A final Higher extension combines a repeated subject with a fraction.
What does rearranging formulae mean?
A formula is a rule that connects quantities. Letters stand for numbers. In this lesson, x and y stand for two numbers connected by a rule.
For example: y = x + 5. This means “add 5 to x to get y”.
The subject is the letter on its own on one side of the equals sign. Here, y is the subject.
Make x the subject means rewrite the rule with x on its own. The answer will still contain y.
Find the value of x means work out a number. You need more information, such as y = 12.
Rearranging gives a rule you can use again whenever y changes. Examples 1–6 develop shared methods. For a subject appearing twice, OCR includes a Foundation route, while Pearson Edexcel assesses this at Higher. AQA lists changing the subject in shared A5 without specifying this case separately. Follow your board and teacher’s guidance on demand.
Example Question:
Make x the subject of:
y = x + 5
Which operation will undo adding 5? See Example 1.
Exam Tip
Keep both sides equal.
If you subtract 5 from one side, subtract 5 from the other side too.
Write the operation beside each step. Letters do not simply “move” across the equals sign.
Method — Rearranging Formulae
1. Find the letter you need on its own. This will be the new subject.
2. Choose the operation that undoes the last step. These pairs are called inverse operations: adding and subtracting; multiplying and dividing.
3. Do that operation to both sides. Write the new equation on the next line.
4. Repeat until the letter is on its own. The other letters stay in your answer.
If the letter appears twice, you need an extra step. Example 7 shows how to put those terms together.
Worked Examples
1. Undo adding — Foundation & Higher
Make x the subject.
y = x + 5
Subtract 5 from both sides.
y − 5 = x + 5 − 5
y − 5 = x
Write the same equality with x on the left.
x = y − 5
This is a rule, not a number. If we are then told y = 12, we can calculate x = 12 − 5 = 7.
2. Undo multiplying — Foundation & Higher
A rectangle is 5 cm long. Let w be its width in centimetres and A its area in square centimetres.
The area rule is A = 5w. Here, 5w means 5 × w.
Make w the subject.
Divide both sides by 5.
A ÷ 5 = 5w ÷ 5
A ÷ 5 = w
w = A ÷ 5
3. Undo dividing — Foundation & Higher
Make x the subject. Here, x and y are numbers.
y = x ÷ 3
Multiply both sides by 3.
3 × y = (x ÷ 3) × 3
3y = x
x = 3y
As with 5w, writing 3y means 3 × y.
4. Undo two operations — Foundation & Higher
Make x the subject.
y = 3x + 4
First subtract 4 from both sides.
y − 4 = 3x
Then divide both sides by 3.
(y − 4) ÷ 3 = x
x = (y − 4) ÷ 3
The brackets keep y − 4 together. Divide that whole amount by 3.
5. Undo brackets — Foundation & Higher
For a rectangle, P is the perimeter, l is the length and w is the width. Use the same length units for all three.
P = 2(l + w)
This means “add the length and width, then multiply by 2”. Make w the subject.
First divide both sides by 2.
P ÷ 2 = l + w
Then subtract l from both sides.
P ÷ 2 − l = w
w = P ÷ 2 − l
6. Undo squaring — Foundation & Higher
A square has area A and side length s. The rule is A = s². The symbol s² means s × s.
Make s the subject.
A square root undoes squaring. The symbol √ means “square root”.
Take the square root of both sides.
√A = s
s = √A
A side length cannot be negative, so use the non-negative square root.
For an area of 25 cm², s = √25 = 5 cm. Check: 5 × 5 = 25.
7. The letter appears twice — check your board
Make x the subject. Here, a is another number in the rule; ax means a × x.
y = x + ax
Both terms on the right contain x. Write x as 1 × x.
y = 1 × x + a × x
Take out the common factor x. This is called factorising.
y = x(1 + a)
Divide both sides by the whole bracket, 1 + a.
y ÷ (1 + a) = x
x = y ÷ (1 + a)
This works provided a is not −1. If a = −1, the bracket is zero and we cannot divide by it.
This extra factorising step is what lets us get x on its own.
Final answer (Example 4):
x = (y − 4) ÷ 3
We have made x the subject. We have not found a numerical value, because no value of y was given.
Check your rearrangement
Use Example 4 to see what a check tells us.
Choose x = 5 in the original rule:
y = 3 × 5 + 4 = 19
Now put y = 19 into the rearranged rule:
x = (19 − 4) ÷ 3 = 5
We get back the number we started with. This helps us spot mistakes.
One number check does not prove a rule always works. Doing the same valid operation to both sides is what keeps the rule correct.
When the subject is in a denominator
Worked example — Undo division by the subject
A journey follows v = d/t, where v is speed, d is distance and t is time. Make t the subject. Assume these quantities are positive.
Multiply both sides by t: vt = d.
Divide both sides by v: t = d/v.
The same balancing rule works even though t began underneath a fraction. We need v ≠ 0 to divide by v.
Higher extension — A repeated subject in a fraction
Make x the subject of y = (x + 3)/(x − 1). The original rule requires x ≠ 1.
Multiply by x − 1: y(x − 1) = x + 3.
Expand: xy − y = x + 3.
Subtract x from both sides: xy − x − y = 3.
Add y: xy − x = y + 3.
Factorise the left: x(y − 1) = y + 3.
Divide by y − 1: x = (y + 3)/(y − 1), where y ≠ 1.
Check with x = 3: the original gives y = 6/2 = 3; the rearranged rule gives x = 6/2 = 3. This is a useful check, while the balancing steps establish the rule.
Powers and roots in a multi-step formula
Worked example — Isolate the squared subject
Make x the subject of y = 3x² + 2. Subtract 2: y − 2 = 3x². Divide by 3: x² = (y − 2)/3.
Take both square roots: x = ±√((y − 2)/3), for y ≥ 2. If x represents a non-negative length, use only the positive-root branch (zero when y = 2). The whole quotient lies under the root.
Worked example — Undo a square root
Make x the subject of y = √(2x + 1), where y ≥ 0. Square both sides: y² = 2x + 1. Subtract 1, then divide by 2: x = (y² − 1)/2. Keep y ≥ 0; otherwise squaring would lose the original root’s sign condition.
Check your understanding — Make r the subject of A = 3r² + 6, where r is a non-negative length.
Subtract 6 and divide by 3: r² = (A − 6)/3.
r = √((A − 6)/3), requiring A ≥ 6. The length restriction selects the non-negative root.
Think Like a Mathematician
Think of y = 3x + 4 as instructions: multiply x by 3, then add 4.
To get x back, undo those instructions in reverse order: subtract 4, then divide by 3.
The letters have not changed what they mean. We have changed which quantity the formula helps us calculate.
Common mistakes to avoid
Changing only one side. Write what you do to both sides.
Stopping too soon. After y − 4 = 3x, x is still multiplied by 3. Divide both sides by 3 to finish.
Losing the brackets. (y − 4) ÷ 3 means divide the whole of y − 4 by 3.
Giving a number too soon. “Make x the subject” usually needs an answer containing other letters.
Do not divide by zero. When using a square root, check whether the letter represents a length.
Try it yourself
In Questions 1–4, x and y stand for numbers. Show one operation per line. Question 7 is shared square-root work; Question 8 follows the repeated-subject board guidance above.
1. Make x the subject: y = x + 9.
Subtract 9 from both sides.
y − 9 = x
x = y − 9
2. Make x the subject: y = 4x.
Divide both sides by 4.
y ÷ 4 = x
x = y ÷ 4
3. Make x the subject: y = 5x + 2.
Subtract 2 from both sides.
y − 2 = 5x
Divide both sides by 5.
(y − 2) ÷ 5 = x
x = (y − 2) ÷ 5
4. Make x the subject: y = (x − 4) ÷ 3.
Multiply both sides by 3.
3y = x − 4
Add 4 to both sides.
3y + 4 = x
x = 3y + 4
5. Make w the subject: P = 2(l + w). As before, P is a rectangle’s perimeter, l its length and w its width.
Divide both sides by 2.
P ÷ 2 = l + w
Subtract l from both sides.
P ÷ 2 − l = w
w = P ÷ 2 − l
6. A rectangle has area A, length l and width w. Make w the subject: A = lw. Here, lw means l × w, and l is positive.
Divide both sides by l.
A ÷ l = w
w = A ÷ l
7. A square has area A and side length s. Make s the subject: A = s².
Take the square root of both sides.
√A = s
s = √A
Use the non-negative root because s is a length.
8. Make x the subject of y = 2x + ax. Here, a is another number and ax means a × x.
Both terms on the right contain x. Factorise by taking x outside a bracket.
y = x(2 + a)
Divide both sides by the whole bracket, 2 + a.
y ÷ (2 + a) = x
x = y ÷ (2 + a)
This requires a not to be −2, because we cannot divide by zero.
Further practice — subjects in fractions
9. Make t the subject of v = d/t, where v, d and t are positive.
Multiply both sides by t: vt = d.
Divide by v: t = d/v.
10. Higher: make x the subject of y = (2x + 1)/(x − 3).
The original requires x ≠ 3. Multiply by x − 3: y(x − 3) = 2x + 1.
Expand: xy − 3y = 2x + 1.
Subtract 2x, then add 3y: xy − 2x = 3y + 1.
Factorise: x(y − 2) = 3y + 1.
x = (3y + 1)/(y − 2), with y ≠ 2.
For example, x = 4 gives y = 9; using y = 9 gives x = 28/7 = 4.
Frequently Asked Questions
What does “make x the subject” mean?
Rewrite the formula with x on its own on one side. There should be no x on the other side.
How is that different from finding x?
Rearranging gives a rule containing letters. Finding x gives a number once you know enough values.
What is an inverse operation?
It undoes another operation. Subtracting undoes adding. Dividing undoes multiplying.
Do I have to expand brackets first?
No. In P = 2(l + w), divide by 2 first. That removes the outside multiplication in one step.
What if x appears twice?
Put the terms containing x together. Factorise to write x outside one bracket, as in the Higher example. Then divide by that bracket if it is not zero.
Why do we choose only one square root for a length?
A length cannot be negative. For a number with no length restriction, x² = 9 has two answers: x = 3 and x = −3, because both square to 9.
Continue Learning
Previous: Quadratic Equations
Next (Higher): Algebraic Fractions
Foundation route: continue to Simultaneous Equations. Algebraic Fractions is a Higher lesson.