This document provides instructions for changing the subject of a formula. It begins by explaining the concept of changing the subject, which means taking a formula with one subject and making a different term the subject. It then demonstrates the procedure through examples of changing the subject in formulas involving addition, subtraction, multiplication, division and combinations of operations. The key steps are to identify what mathematical operation was applied to the original subject and then apply the inverse operation to isolate the new subject. A shorter method is also introduced which involves flipping the sides of the formula and applying the inverse operation to the other side. Worked examples demonstrating both methods are provided.

1. Changing the Subject of a
Formula (Simple Formula)
Presented by Alona Hall
Substitute Numbers for Variables in Algebraic Expressions

2. Simple Formulae Involving the Four Basic
Operations
The objectives of this lesson serves to accomplish the following:
1. Provide an explanation of the concept ‘changing the subject of a formula’
2. Demonstrate the procedure related to ‘changing the subject’ in formulae
involving
Addition and Subtraction
Multiplication and Division
A combination of the aforementioned operations
3. Introduce a shorter method for ‘changing the subject of a formula’ or
‘transposing.’

3. The Concept
Given the formula Area of a rectangle = length x width (written A = lw),
we say that A is the subject of the formula and this is so because:
 It is on the left hand side (the subject usually is)
 The coefficient is 1
 The power is 1
 It is in the numerator
If, however, we are interested in finding the length of the rectangle (l),
then we would get 𝑙 =
𝐴
𝑤
. We say that the subject of the formula has
been changed to l.

4. The Concept
The topic ‘changing the subject of a formula’ therefore implies that:
• A formula will be given
• It will have a subject
• The subject must be changed to something else

5. Procedure Related to “changing the subject”
Think of the following process:
 I walk two steps forward, jump three steps left, make a 180° turn
clockwise, then throw a ball to Ralph.
Question: How do I undo that process to get back
to the starting point?
Consider this:
Ralph would throw the ball back to me, I would make a 180° turn
anti-clockwise, jump three steps right, step two steps backward.
(All things being equal, I should be back at the starting point)

6. Procedure Related to “changing the subject”
In order to get back to the starting point, we had to ‘reverse’ which
involved ‘undoing’ that is ‘doing the opposite of what was done’.
In the context of mathematics, we will be doing the inverse of what
was done. Importantly, we must identify what is done to the subject in
order to determine what must be ‘undone’.
THUS: THE SAME RULES USED TO SOLVE AN EQUATION WILL BE
APPLIED

7. Operation Inverse Operation
Addition/Positive Number Subtraction/Negative Number
Subtraction Addition
Multiplication Division
Division Multiplication
Operations and Their Inverses

8. Examples
Change the subject of the following formulae to the subject indicated
in brackets:
1. V = u + t (u)
2. 2𝜋𝑓 = 𝑤1 - 𝑤2 (𝑤1)
3. 𝐼 = 𝑛𝐴𝑣𝑒 (𝐴)
4. 𝑝 =
𝐹
𝐴
(F)
5. 𝑣2= ‫ݑ‬2+2ܽ‫ݏ‬ (a)

9. Solutions
1. v = u + t (u)
Identify what was done to the required subject, u, and ‘undo it’.
t was added to u so:
Subtract t from both sides
𝑣 − 𝑡 = 𝑢
rewrite the formula with subject, 𝑢, on the left
𝑢 = 𝑣 − 𝑡
NB: The rules used for solving equations were applied

10. Solutions
2. 2𝜋𝑓 = 𝑤1 - 𝑤2 (𝑤1)
𝑤2 𝑤𝑎𝑠 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑤1 so:
𝑎𝑑𝑑 𝑤2 𝑡𝑜 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠
2𝜋𝑓 + 𝑤2 = 𝑤1
Rewrite to get:
𝑤1 = 2𝜋𝑓 + 𝑤2

11. Solutions
3. 𝐼 = 𝑛𝐴𝑣𝑒 (𝐴)
𝐴 𝑤𝑎𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 𝑛𝑣𝑒 𝑠𝑜:
𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 𝑛𝑣𝑒
𝐼
𝑛𝑣𝑒
= 𝐴
𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑡𝑜 𝑔𝑒𝑡:
𝐴 =
𝐼
𝑛𝑣𝑒

12. Solutions
4. 𝑝 =
𝐹
𝐴
(F)
𝐹 𝑖𝑠 𝑏𝑒𝑖𝑛𝑔 𝑑𝑖𝑣𝑖𝑑𝑒𝑑 𝑏𝑦 𝐴 𝑠𝑜:
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 𝐴
𝐴𝑝 = 𝐹
𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑡𝑜 𝑔𝑒𝑡:
𝐹 = 𝐴𝑝

13. Solutions
5. 𝑣2
= ‫ݑ‬2
+2ܽ‫ݏ‬ (a)
𝑎 𝑤𝑎𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦 2𝑠 𝑡ℎ𝑒𝑛 𝑢2 𝑤𝑎𝑠 𝑎𝑑𝑑𝑒𝑑 𝑠𝑜:
(𝑟𝑒𝑣𝑒𝑟𝑠𝑒 𝑡ℎ𝑒 𝑜𝑟𝑑𝑒𝑟)
𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑢2 𝑓𝑟𝑜𝑚 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠
𝑣2 −𝑢2 = 2ܽ‫ݏ‬
𝑡ℎ𝑒𝑛 𝑑𝑖𝑣𝑖𝑑𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒𝑠 𝑏𝑦 2𝑠
𝑟𝑒𝑤𝑟𝑖𝑡𝑒 𝑡𝑜 𝑔𝑒𝑡:

14. A Shorter Method
Remember the principle:
In order to ‘undo’ what was done to the subject
we go in reverse order and use the inverse
operation.

15. Example
Make c the subject in the formula:
𝐹 −32
𝑐
=
9
5
Solution: 𝑐 𝑠ℎ𝑜𝑢𝑙𝑑 𝑏𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟. 𝑊𝑒 ℎ𝑎𝑣𝑒
÷ 𝑏𝑦 𝑐 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 ℎ𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 𝐿𝐻𝑆
Inverse 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 × 𝑐 𝑏𝑢𝑡 𝑠ℎ𝑜𝑤 𝑖𝑡 𝑜𝑛 𝑡ℎ𝑒 𝑟𝑖𝑔ℎ𝑡 ℎ𝑎𝑛𝑑 𝑠𝑖𝑑𝑒 (𝑅𝐻𝑆)
𝐹 − 32 =
9
5
× 𝑐
𝑁𝑜𝑤 𝑐 𝑖𝑠 𝑏𝑒𝑖𝑛𝑔 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑑 𝑏𝑦
9
5
so divide by
9
5
𝑜𝑛 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑠𝑖𝑑𝑒
Remember that dividing by a fraction is the same as multiplying by its reciprocal
(turn the fraction upside down
5
9
)

16. Solution (cont’d)
5
9
𝐹 − 32 = 𝑐
𝑐 =
5
9
𝐹 − 32
𝑖𝑛𝑡𝑟𝑜𝑑𝑢𝑐𝑒 𝑏𝑟𝑎𝑐𝑘𝑒𝑡𝑠, 𝑡ℎ𝑎𝑡 𝑖𝑠,
5
9
𝐹 − 32 𝑡𝑜 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
5
9
is multiplying everything on the other side.
F − 32 was on the other side.

17. Consider this alternative solution
Make c the subject in the formula:
𝐹 −32
𝑐
=
9
5
𝑐 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑖𝑛 𝑡ℎ𝑒 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑠𝑜:
𝑓𝑙𝑖𝑝 𝑡ℎ𝑒 𝐿𝐻𝑆 𝑤ℎ𝑖𝑐ℎ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑅𝐻𝑆 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑓𝑙𝑖𝑝𝑝𝑒𝑑 𝑎𝑠 𝑤𝑒𝑙𝑙
𝑐
𝐹 − 32
=
5
9
𝐹 − 32 𝑖𝑠 𝑑𝑖𝑣𝑖𝑑𝑖𝑛𝑔 𝑜𝑛 𝑡ℎ𝑒 𝐿𝐻𝑆 𝑠𝑜 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦
𝐹 − 32 𝑜𝑛 𝑡ℎ𝑒 𝑅𝐻𝑆
𝑐 =
5
9
(𝐹 − 32)
REMEMBER: 𝐵𝑟𝑎𝑐𝑘𝑒𝑡 𝑚𝑢𝑠𝑡 𝑒𝑛𝑐𝑙𝑜𝑠𝑒 𝐹 − 32 𝑡𝑜 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡
5
9
𝑖𝑠 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦𝑖𝑛𝑔 𝑏𝑜𝑡ℎ 𝑡𝑒𝑟𝑚𝑠.
Without brackets, we would have
5
9
F − 32 which has a DIFFERENT meaning.