WJEC MATHEMATICS
INTERMEDIATE
ALGEBRA
REARRANGING
(CHANGING THE SUBJECT)
1
Contents
Keeping the Balance
Changing the Subject
Expanding and Rearranging
Multiple subjects
Factorising the subject
Credits
WJEC Question bank
http://www.wjec.co.uk/question-bank/question-search.html
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Keeping the Balance
When rearranging equations you need to keep the balance. That
means, whatever you do to one side of the equation you must do it
to the other side. Hence, keeping it balanced.
This topic is very similar to solving. These booklets should be used
together.
Changing the Subject
Making something 'the subject' of an equation means you want to get
that letter on its own on one side of the equation.
If you are asked to make 𝑥 the subject of the formula, get 𝑥 =
If you are asked to make 𝑎 the subject of the formula, get 𝑎 =
If you are asked to make 𝑦 the subject of the formula, get 𝑦 =
etc.
Example 1
Make 𝑥 the subject of the formula
𝑥 − 3𝑦 = 7
To get the 𝑥 on its own we need to remove the ′ − 3𝑦′ term. To
eliminate this, we need to add 3𝑦 to both sides.
𝑥 − 3𝑦 = 7
+3𝑦
+3𝑦
𝑥 = 7 + 3𝑦
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Example 2
Make 𝑥 the subject of the formula
4𝑥 + 2𝑦 = 6
Common mistake here is to try and get rid of the 4 first - Leave this
until last! Get the term with 𝑥 on its own first by subtracting 2𝑦
4𝑥 + 2𝑦 = 6
−2𝑦
−2𝑦
4𝑥 = 6 − 2𝑦
Now we can eliminate the 4. Remember 4𝑥 = 4×𝑥 which means
to eliminate the 4 we need to divide both sides of the equation by 4.
4𝑥 = 6 − 2𝑦
÷4
𝑥=
÷4
6 − 2𝑦
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Example 3
Make 𝑥 the subject of the formula
𝑥 2 + 2𝑦 = 6
−2𝑦
Inverse operations
+/×/÷
2
−2𝑦
𝑥 = 6 − 2𝑦
𝑥 = √6 − 2𝑦
square / square
root
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Exercise A12
Make 𝑥 the subject of each of the following equations.
1.
𝑥+𝑦 =𝑧
2.
𝑥 − 4𝑦 = 8
3.
3𝑥 + 7𝑦 = 12
4.
5𝑦 + 2𝑥 = 25
5.
6𝑥 − 5𝑦 = 13
6.
𝑥2 − 𝑦 = 𝑧
7.
𝑥 2 + 5𝑦 = 4
8.
−4𝑦 + 𝑥 2 = 𝑦
9.
1
2
𝑥 − =𝑦
2
10.
4𝑥 2 + 2𝑦 = 6
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Expanding and Rearranging
For more help on expanding, complete the 'Expanding' booklet.
Some questions require you to expand a single bracket (or two as
we'll see later) before we rearrange.
Example 4
Make 𝑥 the subject of the formula
Santa's hat
4(5𝑥 + 𝑦) = 𝑧
This equation becomes:
20𝑥 + 4𝑦 = 𝑧
−𝟒𝒚
−𝟒𝒚
20𝑥 = 𝑧 − 4𝑦
÷ 𝟐𝟎
÷ 𝟐𝟎
𝑧 − 4𝑦
𝑥=
20
Exercise A13
1.
2(𝑥 + 𝑦) = 𝑏
2.
3(𝑥 − 5𝑦) = 𝑟 2
3.
5(3𝑥 + 𝑦) = 6𝑦
4.
−(4𝑥 + 3𝑦) = 𝑝𝑞
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Multiple subjects
The more difficult rearranging questions have the subject on both
sides.
Example 5
Make 𝑥 the subject of the formula
5𝑥 + 𝑦 = 2𝑥 − 𝑧
As you can see, the difficulty here is the fact there is an 𝑥 term on
both sides. The first job is to make it so that there is only an 𝑥 on
one side.
Tip: Find the side with the lowest 𝑥 value and eliminate that one. For
this example, the lowest is the 2𝑥 term so subtract this from both
sides.
5𝑥 + 𝑦 = 2𝑥 − 𝑧
−𝟐𝒙
−𝟐𝒙
3𝑥 + 𝑦 = −𝑧
Now this has become a question that is similar to ones we have
looked at previously.
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Exercise A14
Make 𝑥 the subject of each of the following equations.
1.
3𝑥 + 𝑦 = 𝑥 + 𝑧
2.
8𝑥 − 3𝑦 = 2𝑥 − 8
3.
3𝑥 + 7𝑦 = 𝑥 + 12
4.
5𝑦 + 2𝑥 = 𝑥 − 25
5.
4𝑥 + 7𝑦 = 3𝑥 + 1
6.
4(𝑥 + 𝑦) = 2𝑥 + 𝑧
7.
8(2𝑥 + 5𝑎) = 9𝑥 + 2𝑎
8.
4(5𝑥 + 2𝑦) = 2(2𝑥 − 𝑦)
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Factorising to isolate the subject
This is the most difficult form of rearranging an equation. Consider
the following example,
Example 6
Make 𝑥 the subject of the formula
5𝑥 = 𝑥𝑦 + 𝑧
Firstly, you want to ensure all the 𝑥 terms are on one side of the
equation. To do this, subtract 𝑥𝑦 from both sides.
5𝑥 = 𝑥𝑦 + 𝑧
−𝒙𝒚
−𝒙𝒚
5𝑥 − 𝑥𝑦 = 𝑧
At this stage, you will see that you can't combine the terms with 𝑥 so
you will need to factorise out the 𝑥 from both terms.
Go back to the 'Factorising' book for more help on this if you are
unsure.
Factorising gives us:
𝑥(5 − 𝑦) = 𝑧
Remember:
This means
𝑥 × (5 − 𝑦)
÷ (𝟓 − 𝒚)
÷ (𝟓 − 𝒚)
𝑧
𝑥 =
(5 − 𝑦)
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Exercise A15
Make 𝑥 the subject of each of the following equations.
1.
4𝑥 = 𝑥𝑦 − 4
2.
3𝑥 = 𝑥𝑦 + 3
3.
4(𝑥 + 𝑦) = 𝑦(𝑥 + 𝑧)
4.
2(𝑥 + 𝑧) = 𝑦(𝑥 + 𝑤)
Exam Questions A5
1.
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4.
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