The document covers Grade 8 algebra topics including collecting like terms, expanding brackets, and constructing and solving equations. It provides rules for simplifying expressions, along with various examples and exercises to practice these concepts. Additionally, it explains how to set up equations for problem-solving using geometric diagrams.

1. Simplifying expressions
and solving equations
Grade 8

2. Topic(s)
• Collecting like terms
• Expanding brackets
• Constructing and solving equations
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3. Collecting like terms
• There are some rules you can follow when you write an
expression in algebra :
- Write products without the multiplication sign, so write
2 × 𝑛 as 2𝑛
- Write the number before the letter, so write 2𝑥 not 𝑥2
- Generally, write terms with letters before number
terms, so write 3𝑦 + 4 rather than 4 + 3𝑦
- Generally, write terms in alphabetical order, so write 4𝑎 + 5𝑏 not 5𝑏 + 4𝑎
- When a term has more than one letter, write them in alphabetical order, so write
6𝑐𝑑 rather than 6𝑑𝑐
- Write negative terms after positive terms, so write 5 − 4𝑧 not −4𝑧 + 5
(unless all the terms are negative, in which case follow the order rules, so write
− 3𝑥 − 8𝑦, not −8𝑦 − 3𝑥)
Algebraic expression
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4. • Like terms are terms that contain the same letter variables
which are raised to the exact same powers.
• You can also simplify expressions by collecting like terms
3𝑎 + 𝑏 + 2𝑎 + 6𝑏
2𝑥𝑦2
+ 𝑥 − 𝑥𝑦2
+ 𝑥2
𝑐2
+ 𝑑3
− 2𝑐2
example
Simplify these expressions
a. 4 + 2𝑥 + 5𝑥 b. 2𝑎𝑏 + 𝑎𝑏 − 5𝑏𝑎 c. 2𝑦 + 6𝑦2 − 3𝑦2 − 10𝑦
a. 2𝑥 + 5𝑥 + 4 = 7𝑥 + 4 2𝑥 and 5𝑥 are like terms
b. 2𝑎𝑏 + 𝑎𝑏 − 5𝑎𝑏 = −2𝑎𝑏 5𝑏𝑎 is the same as 5𝑎𝑏
c. 6𝑦2 − 3𝑦2 + 2𝑦 − 10𝑦 = 3𝑦2 − 4𝑎𝑏 6𝑦2 and −3𝑦2 like terms, also 2𝑦 and −10𝑦
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5. exercise
1. Use the guidelines opposite to rewrite these expressions
a. 8 × 𝑛 b. 4 + 3 × 𝑣 c. 6 × 𝑚 × 𝑛 d. 𝑝 × 2 + 𝑞7 e. −3𝑦𝑥 − 8𝑏𝑎
2. Simplify each expressions
a. 6𝑥 + 5𝑥 + 9𝑥 b. 8𝑐 − 4𝑑 − 2𝑐 + 𝑑
c. 4𝑥2 + 5𝑥2 + 8𝑥 − 5𝑥 d. 2𝑎𝑏 + 7𝑎𝑏 − 7𝑏𝑎
e. 11𝑦2
− 3𝑦 − 5𝑦2
f. 𝑎2
+ 2𝑎 + 2 − 𝑎2
+ 5𝑎
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6. Expanding brackets
• To expand brackets, multiply each term inside the brackets by the term outside
the brackets
example
a. Expand 3 𝑏 + 6
b. Expand and simplify 4 2𝑥 + 3𝑥2 − 𝑥 2 + 𝑥
a. 3 𝑏 + 6 = 3𝑏 + 18
b. 4 2𝑥 + 3𝑥2
− 𝑥 2 + 𝑥 = 8𝑥 + 12𝑥2
− 2𝑥 + 𝑥2
= 8𝑥 + 12𝑥2
− 2𝑥 − 𝑥2
= 12𝑥2
− 𝑥2
+ 8𝑥 − 2𝑥
= 11𝑥2 + 6𝑥
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7. exercise
1. Expand each expression
a. 3(𝑥 + 7) b. 12(3𝑎 − 4) c. 2(𝑎𝑏 + 4𝑐)
d. 𝑥(3𝑦 + 4) e. 𝑔(3ℎ + 6𝑔) f. 2𝑓(2𝑓 + 𝑔 − 2)
2. Expand and simplify each expression
a. 2 𝑥 + 3 + 2(𝑥 + 2) b. 5 5 + 4𝑣 − 4(3𝑣 + 7)
c. 8 𝑧 + 3 + 5(4 + 3𝑧) d. 𝑥 𝑥 + 2 + 𝑥(𝑥 + 4)
e. 𝑢 2𝑢 − 6 − 𝑢 𝑢 + 3 f. 𝑦 𝑦 + 2𝑥 + 4(𝑥 − 2)
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8. Constructing and solving equations
• When you are given a problem to solve, you may need to construct, or write, an
equation to help you solve the problem
example
The diagram shows a rectangle.
Work out the values of 𝑥 and 𝑦
3(𝑥 + 3) cm
3𝑦 + 8 cm5𝑦 − 4 cm
24
3 𝑥 + 3 = 24
3𝑥 + 9 = 24
3𝑥 + 9 − 9 = 24 − 9
3𝑥 = 15
𝒙 = 𝟓
5𝑦 − 4 = 3𝑦 + 8
5𝑦 − 3𝑦 = 8 + 4
2𝑦 = 12
𝒚 = 𝟔
The two lengths must be equal to find 𝑥
The two widths must be equal to find 𝑦
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9. exercise
1. Work out the value 𝑥 and 𝑦 in each of these diagrams. All measurements are
centimetres
3𝑥 + 1
2𝑦 + 154𝑦 + 5
2(𝑥 + 5)
5𝑥 − 3
8𝑦 − 43𝑦 + 16
3𝑥 + 11
3𝑥
165𝑦 + 1
18
6(𝑥 + 1)
202(𝑦 + 3)
72
a. b.
c. d.
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10. 2. Work out the value of 𝑥 or 𝑦 in each of these shapes. All measurements are
centimetres
27 3(𝑥 + 5)
8𝑦 − 5
3(𝑦 + 5)
a. b.
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11. Source : Cambridge Checkpoint Mathematics 8
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