ALGEBRA FOR MIDDLE SCHOOL Topics : • Collecting like terms • Expanding brackets • Constructing and solving equations //&//

1. Simplifying expressions
and solving equations
Grade 8

2. Topic(s)
• Collecting like terms
• Expanding brackets
• Constructing and solving equations
nomadic-math.blogspot.com

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
nomadic-math.blogspot.com

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
nomadic-math.blogspot.com

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
nomadic-math.blogspot.com

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
nomadic-math.blogspot.com

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)
nomadic-math.blogspot.com

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
nomadic-math.blogspot.com

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.
nomadic-math.blogspot.com

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.
nomadic-math.blogspot.com

11. Source : Cambridge Checkpoint Mathematics 8
nomadic-math.blogspot.com