# Rearrange Formulae

In this worksheet, students will rearrange formulae (in which subject appears once only) to make different terms the subject using inverse operations.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation, Algebraic Formulae

Difficulty level:

### QUESTION 1 of 10

If you rearrange a formula, you change the variable which is its subject.

In A = b × h (the formula for area of a rectangle), A is the subject, because it is on its own on one side of the equals sign.

You can change the subject of the formula by using inverse operations.

Have a look at an example below.

e.g. v = u + at (formula for final velocity). Make a the subject of the formula.

We need to inverse the operations.

Let's first move u to the other side by subtracting it from both sides: v - u = at

at means a × t, so to reverse this, we need to divide both sides by t: (v - u) ÷ t = a

Now a is the subject of the formula, as it is on its own on one side of the equation.

See how (v - u) is in brackets?

This is because we want to divide the entire left-hand side by t, not just v or u.

Now it's your turn to inverse those operations!

In this activity, we are going to rearrange formulae to make different terms the subject using inverse operations.

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

5x - y = 7
x = (7 - 5y) ÷ 2
5x + y = 7
x = (7 + y) ÷ 5
x + 5y = 7
x = (7 - y) ÷ 5
2x + 5y = 7
x = 7 - 5y

Below some formulae have been rearranged to make the subject, but not all of these are correct.

Select if each rearrangement has been completed correctly or not.

 Correct Incorrect P = 3m - 5 > m = (P + 5) ÷ 3 P = m + 5t > m = (P - 5t) ÷ 5 P= 5m + 1 > m = P - 6 P = 5m + 3t > m = (P - t) ÷ 15

Make t the subject of this formula:

v = u + at

t = v - u ÷ a

t = (v - a) ÷ u

t = (v - u) ÷ a

t = (v + a) ÷ u

Make s the subject of this formula:

t = sx - r

s = t + r - x

s = (t + r) ÷ x

s = (t - x) ÷ r

s = t + r ÷ x

Make p the subject of this formula:

T = (6p - 7) ÷ 2

p = (2T + 7) ÷ 6

p = 2T + 7 ÷ 6

p = 2(T + 7 ÷ 6)

p = T + 7 × 2 ÷ 6

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

a(x - b) = 2
x = 3y - 2
(x + 2) ÷ 3 = y
x = (2 ÷ a) + b
(x ÷ 3) + b = y
x = 4a ÷ 3
3x ÷ 4 = a
x = 3(y - b)

Make a the subject of this formula:

3(2a - 5) = b

a = (b ÷ 3) + 5 ÷ 2

a = 3b + 5 ÷ 2

a = (b ÷ 3 + 5) ÷ 2

a = 3b - 5 × 2

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

y = x - 4
x = y + 4
y = x + 4
x = 2y - 4
y = 2x - 4
x = y -4
y = (x + 4) ÷ 2
x = (y + 4) ÷ 2

What is the correct first step to make m the subject of the formula y = mx + c?

+ c

× x

- c

÷ c

What is the correct second step to make t the subject of the formula s = (2t + 3) ÷ r?

× r

÷ 2

× 2

− 3

• Question 1

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

5x - y = 7
x = (7 + y) ÷ 5
5x + y = 7
x = (7 - y) ÷ 5
x + 5y = 7
x = 7 - 5y
2x + 5y = 7
x = (7 - 5y) ÷ 2
EDDIE SAYS
Did you remember to use inverse operations? Let's look at the first formula together: 5x - y = 7 We need to get x on its own so that it becomes the subject. We can add y to both sides to start to isolate the x: 5x = 7 + y Then we can divide both sides by 5 to get x on its own: x = (7 + y) ÷ 5 Can you use this process and example to help you find the other three pairs?
• Question 2

Below some formulae have been rearranged to make the subject, but not all of these are correct.

Select if each rearrangement has been completed correctly or not.

 Correct Incorrect P = 3m - 5 > m = (P + 5) ÷ 3 P = m + 5t > m = (P - 5t) ÷ 5 P= 5m + 1 > m = P - 6 P = 5m + 3t > m = (P - t) ÷ 15
EDDIE SAYS
Only the first question has been rearranged correctly. In the remaining three, the inverse operations were not calculated correctly. Let's look at the second rearrangement together: P = m + 5t To isolate m, we can take 5t from both sides: P - 5t = m It's as simple as that; we have got m on its own! The answer given was m = (P - 5t) ÷ 5 which does not match m = P - 5t. Have a look at the others again. Can you spot where the mistakes have been made?
• Question 3

Make t the subject of this formula:

v = u + at

t = (v - u) ÷ a
EDDIE SAYS
To make t the subject, we need to get it on its own on one side of the equals sign. v = u + at Let's start by subtracting u from both sides: v - u = at at means a × t so we need to divide by a to get t on its own: (v - u) ÷ a = t The first option looks very similar to this, but it is missing brackets. Always use brackets if you want to divide the entire expression, rather than just part of it.
• Question 4

Make s the subject of this formula:

t = sx - r

s = (t + r) ÷ x
EDDIE SAYS
Did you use inverse operations to change the subject of t = sx - r? Add r first: t + r = sx Divide by x: (t + r) ÷ x = s Now s is the subject of the formula. Don't forget those all-important brackets! They show that we need to divide the answer to (t + r) by x, not t and r separately.
• Question 5

Make p the subject of this formula:

T = (6p - 7) ÷ 2

p = (2T + 7) ÷ 6
EDDIE SAYS
This is a little trickier! Firstly, let's deal with ÷ 2. We need to do the inverse, which is × 2: 2T = 6p - 7 Now let's add the 7 to both sides: 2T + 7 = 6p Finally, let's divide by 6: (2T + 7) ÷ 6 = p Don't forget to use brackets to show that the whole of the sum on the left is being divided by 6.
• Question 6

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

a(x - b) = 2
x = (2 ÷ a) + b
(x + 2) ÷ 3 = y
x = 3y - 2
(x ÷ 3) + b = y
x = 3(y - b)
3x ÷ 4 = a
x = 4a ÷ 3
EDDIE SAYS
Remember that multiplication is the inverse of division. If the formula to be rearranged has brackets, first deal with everything outside of the brackets and then move inside the brackets. Let's look at an example together: (x + 2) ÷ 3 = y First, let's do the inverse of ÷3 which is ×3: (x + 2) = 3y Now we can move inside the brackets, and do the inverse of adding two: x = 3y - 2 Done! Follow this process to rearrange then match the other three formulae.
• Question 7

Make a the subject of this formula:

3(2a - 5) = b

a = (b ÷ 3 + 5) ÷ 2
EDDIE SAYS
This is quite a complex formula to work with, so be careful! First, divide by 3: (2a - 5) = b ÷ 3 Now add 5: 2a = b ÷ 3 + 5 Finally, divide by 2. You will need brackets here to catch the whole expression: a = (b ÷ 3 + 5) ÷ 2 Did you follow each of these steps?
• Question 8

Make x the subject of each formula below, then match each original formula to its new version.

## Column B

y = x - 4
x = y + 4
y = x + 4
x = y -4
y = 2x - 4
x = (y + 4) ÷ 2
y = (x + 4) ÷ 2
x = 2y - 4
EDDIE SAYS
To change the subject of a formula, you need to use inverse operations. The last two formulae are the most challenging here. Let's have a look at: y = 2x - 4 First + 4, this will give us: y + 4 = 2x Now ÷ 2. The answer will be x = (y + 4) ÷ 2 Can you rearrange the others and find the matches independently?
• Question 9

What is the correct first step to make m the subject of the formula y = mx + c?

- c
EDDIE SAYS
To make m the subject of y=mx + c, the most sensible first step is to move c. To do this, we need to subtract c from both sides. Can you follow the rest of the steps through to rearrange this fully?
• Question 10

What is the correct second step to make t the subject of the formula s = (2t + 3) ÷ r?

− 3
EDDIE SAYS
The most sensible first step is to multiply by r, so that we can then work inside the brackets. If we do this, we reach: s × r = 2t + 3 Now the second step will be to move + 3, by subtracting it. So − 3 is the correct second step. Can you work out the final step? Well done, if you thought it was ÷ 2!
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