Rearranging a formula means changing its subject, the letter on its own on one side. Undo the operations acting on the new subject using inverse operations, in reverse order, doing the same to both sides. For y = 2x + 3, subtract 3 and then divide by 2 to get x = (y − 3)/2.
Inverse operations
- + is undone by −, and − by +
- × is undone by ÷, and ÷ by ×
- A square is undone by a square root, and a root by squaring
- Undo operations in the reverse order to how they were applied
- Always do the same thing to both sides
Inverse operations
The subject of a formula is the letter on its own. To change the subject, look at what has been done to the new subject and undo it with inverse operations. To make x the subject of y = x + 5, subtract 5 from both sides: x = y − 5.
Two-step rearranging
When several operations act on the letter, undo them in reverse order. In y = 2x + 3, x was multiplied by 2 and then 3 was added. So subtract 3 first, giving y − 3 = 2x, then divide by 2: x = (y − 3)/2. Use brackets to show that the whole of y − 3 is divided.
Fractions and brackets
If the subject is divided, multiply both sides to clear the fraction. In y = x/4 − 2, add 2 to get y + 2 = x/4, then multiply by 4: x = 4(y + 2). With a bracket, either divide both sides by the number outside or expand first.
Squares and square roots
If the subject is squared, rearrange to get the squared term on its own, then square-root both sides. From v² = u² + 2as, making s the subject gives s = (v² − u²)/(2a). Making u the subject gives u = ±√(v² − 2as); in many contexts only the positive root makes sense.
Checking a rearrangement
Substitute some numbers into the original formula, then check that the rearranged formula gives back the value you started with. For y = 2x + 3 with x = 5, y = 13; then (13 − 3)/2 = 5, which confirms the rearrangement.
Worked examples
Example 1. Make x the subject of y = x + 5.
- 5 has been added to x, so subtract 5 from both sides.
Answer: x = y − 5.
Example 2. Make x the subject of y = 2x + 3.
- Subtract 3: y − 3 = 2x.
- Divide by 2.
Answer: x = (y − 3)/2.
Example 3. Make x the subject of y = x/4 − 2.
- Add 2: y + 2 = x/4.
- Multiply by 4.
Answer: x = 4(y + 2).
Example 4. Make h the subject of A = ½bh.
- Multiply by 2: 2A = bh.
- Divide by b.
Answer: h = 2A/b.
Example 5. Make s the subject of v² = u² + 2as.
- Subtract u²: v² − u² = 2as.
- Divide by 2a.
Answer: s = (v² − u²)/(2a).
Example 6. Make a the subject of T = 3(a + k).
- Divide by 3: T/3 = a + k.
- Subtract k.
Answer: a = T/3 − k.
Common mistakes
- Undoing operations in the wrong order. Undo the last operation applied first.
- Doing something to only one side. Every step must be applied to both sides.
- Dividing only part of an expression. (y − 3)/2 needs brackets; y − 3/2 means something different.
- Forgetting the ± with a square root. Square-rooting can give a positive or negative value.
- Leaving the subject on both sides. The new subject must appear only once, on its own.
How rearranging formulae is examined
Changing the subject is a Foundation topic in the 1MA1 GCSE and the 4MA1 International GCSE, and Higher papers extend it to formulae where the subject appears twice. Questions ask you to:
- make a letter the subject of a one- or two-step formula
- rearrange a formula containing fractions or brackets
- rearrange a formula involving a square or square root
- rearrange a scientific formula, then substitute values
The rearranging formulae worksheet has questions with answers, and the GCSE algebra revision engine gives more.
Rearranging uses the same inverse operations as solving linear equations, and it is the reverse of substitution.
Frequently asked questions
What does it mean to change the subject of a formula?
To rearrange it so that a different letter is on its own on one side.
How do you rearrange a formula?
Use inverse operations to undo what has been done to the new subject, in reverse order, applying each step to both sides.
How do you deal with a square in a formula?
Get the squared term on its own, then take the square root of both sides.