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© John Wiley & Sons Australia, Ltd 1
WorkSHEET 8.1 Algebra Name: ___________________________ 1 Substitution
Full setting out please.
2 If 𝑎 = 2, and 𝑏 = 3 determine the value of the expression 2𝑎 + 4𝑏.
2𝑎 + 4𝑏
= 2 × 2 + 4 × 3
= 4 + 12
= 16
3 If 𝑑 = 3, and 𝑔 = 5 determine the value of the expression 4𝑑 − 2𝑔
4𝑑 − 2𝑔
= 4 × 3 − 2 × 5
= 12 − 10
= 2
4 If 𝑔 = 2, and ℎ = 3 determine the value of the expression 5ℎ − 2𝑔
5ℎ − 2𝑔
= 5 × 3 − 2 × 2
= 15 − 4
= 11
5 If , then:
𝑑 + 3𝑒 =
𝑑 + 3𝑒
= 3 + 3 × 2
= 3 + 6
= 9
6 If , then:
4𝑑 − 2𝑒 =
4𝑑 − 2𝑒
= 4 × 3 − 2 × 2
= 12 − 4
= 8
3 and 2d e= =
3 and 2d e= =
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 2
7 If 𝑥 = 2, determine the value of the expression 4𝑥 + 3.
4𝑥 + 3
= 4 × 2 + 3
= 11
8 If 𝑥 = 3, determine the value of the expression 2𝑥 + 7.
2𝑥 + 7
= 2 × 3 + 7
= 13
9 If 𝑥 = 4, determine the value of the expression 3𝑥 − 10.
3𝑥 − 10
= 3 × 4 − 10
= 2
10 If 𝑥 = 5, determine the value of the expression 4𝑥 − 6.
4𝑥 − 6
= 4 × 5 − 6
= 14
11 If 𝑥 = 2, determine the value of the expression 𝑥! + 3𝑥 + 4.
𝑥! + 3𝑥 + 4
= 2! + 3 × 2 + 4
= 4 + 6 + 4
= 14
12 If 𝑥 = 3, determine the value of the expression 𝑥! + 5𝑥 − 6.
𝑥! + 5𝑥 − 6
= 3! + 5 × 3 − 6
= 9 + 15 − 6
= 18
13 If 𝑥 = 4, determine the value of the expression 𝑥! − 2𝑥 − 5.
𝑥! − 2𝑥 − 5
= 4! − 2 × 4 − 5
= 16 − 8 − 5
= 3
14 If 𝑥 = 2, determine the value of the expression 𝑥! − 5𝑥 − 7.
𝑥! − 5𝑥 − 7
= 2! − 5 × 2 − 7
= 4 − 10 − 7
= −13
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 3
15 Substitution – Problem Solving
Full setting out please.
16 The velocity of a car ( 𝑣 stands for Velocity which is speed) is given by the formula;
𝑣 = 𝑢 + 𝑎𝑡 Find the velocity of the vehicle where;
𝑢 = 50 𝑎 = 8
and 𝑡 = 6
Yes, this is a real equation you will use in Science. It doesn’t matter that you do not know what all the letters mean, but you can Substitute to get an answer … :
𝑣 = 𝑢 + 𝑎𝑡 By substitution,
𝑣 = 50 + 8 × 6
= 50 + 48
= 98 The car is travelling 98 km/h
17 The interest a Bank account receives (𝐼 stands for Interest) is given by the formula;
𝐼 = 𝑃𝑅𝑇 Find the Interest received when;
𝑃 = 5000
𝑅 =4100
and 𝑇 = 2
Yes, this is a real equation you will use in Financial Maths. It doesn’t matter that you do not know what all the letters mean, but you can Substitute to get an answer … :
𝐼 = 𝑃𝑅𝑇 By substitution,
𝐼 = 5000 ×4100 × 2
Use your calculator
= 640 The interest earned is $640
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 4
18 The sum of the angles in a polygon is given by , where is the number of sides. Find the sum of the angles of a triangle (a 3-sided polygon).
Given; 𝑆 = 180(𝑛 − 2)
Set 𝑛 = 3 𝑆 = 180(3 − 2)
= 180 × 1
= 180
Therefore, the sum of all angles in a triangle is 180 degrees. Yes, we already knew that, but this formula is how we know it! If your solution didn’t look like this, then do it again. You need to show your substitution into the equation and then get an answer. Setting out is IMPORTANT!
19 The sum of the angles in a polygon is given by , where is the number of sides. Find the sum of the angles of a quadrilateral (a 4-sided polygon).
Given; 𝑆 = 180(𝑛 − 2)
Set 𝑛 = 4 𝑆 = 180(4 − 2)
= 180 × 2
= 360
Therefore, the sum of all angles in a quadrilateral is 360 degrees.
20 The sum of the angles in a polygon is given by , where is the number of sides. Find the sum of the angles of a pentagon (a 5-sided polygon).
Given; 𝑆 = 180(𝑛 − 2)
Set 𝑛 = 5 𝑆 = 180(5 − 2)
= 180 × 3
= 540
Therefore, the sum of all angles in a pentagon is 540 degrees.
21 The sum of the angles in a polygon is given by , where is the number of sides. Find the sum of the angles of a hexagon (a 6-sided polygon).
Given; 𝑆 = 180(𝑛 − 2)
Set 𝑛 = 6 𝑆 = 180(6 − 2)
= 180 × 4
= 720
Therefore, the sum of all angles in a hexagon is 720 degrees.
)(S)2(180 -= nS n
)(S)2(180 -= nS n
)(S)2(180 -= nS n
)(S)2(180 -= nS n
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 5
22 Combining Like Terms
23 By combining like terms, simplify;
3𝑥 + 5𝑥
3𝑥 + 5𝑥
= 8𝑥
24 Simplify;
5𝑥 − 3𝑥
5𝑥 − 3𝑥
= 2𝑥
25 Simplify;
4𝑥 − 7𝑥
4𝑥 − 7𝑥
= −3𝑥
26 Simplify;
3𝑥 + 7 + 5𝑥
3𝑥 + 7 + 5𝑥
= 8𝑥 + 7
27 Simplify;
5𝑥 + 7 − 3𝑥
5𝑥 + 7 − 3𝑥
= 2𝑥 + 7
28 Simplify;
7𝑥 − 9 − 4𝑥
7𝑥 − 9 − 4𝑥
= 3𝑥 − 9
29 Simplify;
3𝑥 − 9 − 8𝑥
3𝑥 − 9 − 8𝑥
= −5𝑥 − 9
30 Simplify; 7𝑥 + 4 + 3𝑥 + 12
7𝑥 + 4 + 3𝑥 + 12
= 10𝑥 + 16
31 Simplify; 2𝑥 + 3 + 4𝑥 + 5
2𝑥 + 3 + 4𝑥 + 5
= 6𝑥 + 8
32 Simplify; 7𝑥 + 8 + 9𝑥 + 10
7𝑥 + 8 + 9𝑥 + 10
= 16𝑥 + 18
33 Simplify; 3𝑥 + 4 − 5𝑥 − 2
3𝑥 + 4 − 5𝑥 − 2
= −2𝑥 + 2
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 6
34 Simplify; 4𝑥 + 6 − 9𝑥 − 1
4𝑥 + 6 − 9𝑥 − 1
= −5𝑥 + 5
35 Simplify; 5𝑥 + 4 − 3𝑥 − 2
5𝑥 + 4 − 3𝑥 − 2
= 2𝑥 + 2
36 Simplify; 15𝑥 + 14 − 7𝑥 − 11
15𝑥 + 14 − 7𝑥 − 11
= 8𝑥 + 3
37 Simplify; 8𝑥 + 4 − 6𝑥 − 3
8𝑥 + 4 − 6𝑥 − 3
= 2𝑥 + 1
38 Simplify; 2𝑥 + 7 − 9𝑥 − 5
2𝑥 + 7 − 9𝑥 − 5
= −7𝑥 + 2
39 Simplify; 𝑥 + 1 − 2𝑥 − 2
𝑥 + 1 − 2𝑥 − 2
= −𝑥 − 1
40 Simplify; 25𝑥 + 30 − 10𝑥 − 20
25𝑥 + 30 − 10𝑥 − 20
= 15𝑥 + 10
41 Simplify; 3𝑥 − 2 − 5𝑥 − 2 − 2𝑥 + 5
3𝑥 − 2 − 5𝑥 − 2 − 2𝑥 + 5
= −4𝑥 + 1
42 Simplify; 4𝑥 − 3 − 6𝑥 − 2 − 𝑥 + 7
4𝑥 − 3 − 6𝑥 − 2 − 𝑥 + 7
= −3𝑥 + 2
43 Simplify; 9𝑥 − 2 − 5𝑥 − 7 − 2𝑥 + 5
9𝑥 − 2 − 5𝑥 − 7 − 2𝑥 + 5
= 2𝑥 − 4
44 Simplify; 10𝑥 + 3 − 5𝑥 − 2 − 2𝑥 + 5
10𝑥 + 3 − 5𝑥 − 2 − 2𝑥 + 5
= 3𝑥 + 6
45 Simplify; 3𝑥 − 2 − 5𝑥 − 2 − 2𝑥 + 5
3𝑥 − 2 − 5𝑥 − 2 − 2𝑥 + 5
= −4𝑥 + 1
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 7
46 Expanding Brackets
47 Expand; 5(𝑥 + 2)
5(𝑥 + 2)
= 5𝑥 + 10
48 Expand; 3(𝑥 + 3)
3(𝑥 + 3)
= 3𝑥 + 9
49 Expand; 3(𝑥 + 2)
3(𝑥 + 2)
= 3𝑥 + 6
50 Expand; 3(𝑥 + 4)
3(𝑥 + 4)
= 3𝑥 + 12
51 Expand; 7(3𝑥 − 2)
7(3𝑥 − 2)
= 21𝑥 − 14
52 Expand; 5(2𝑥 + 2)
5(2𝑥 + 2)
= 10𝑥 + 10
53 Expand; 3(4𝑥 + 7)
3(4𝑥 + 7)
= 12𝑥 + 21
54 Expand; 2(3𝑥 − 5)
2(3𝑥 − 5)
= 6𝑥 − 10
55 Expand; 8(2𝑥 − 1)
8(2𝑥 − 1)
= 16𝑥 − 8
56 Expand; 7(3𝑥 − 2)
7(3𝑥 − 2)
= 21𝑥 − 14
57 Expand; 25(2𝑥 − 3)
25(2𝑥 − 3)
= 50𝑥 − 75
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 8
58 Expanding Brackets and Combining Like terms
59 Simplify; 3𝑥 + 5(𝑥 + 2)
3𝑥 + 5(𝑥 + 2)
= 3𝑥 + 5𝑥 + 10
= 8𝑥 + 10
60 Simplify; 5𝑥 + 3(𝑥 + 3)
5𝑥 + 3(𝑥 + 3)
= 5𝑥 + 3𝑥 + 9
= 8𝑥 + 9
61 Simplify; 3(𝑥 + 2) + 5
3(𝑥 + 2) + 5
= 3𝑥 + 6 + 5
= 3𝑥 + 11
62 Simplify; 2𝑥 + 3(𝑥 + 3)
2𝑥 + 3(𝑥 + 3)
= 2𝑥 + 3𝑥 + 9
= 5𝑥 + 9
63 Simplify; 5𝑥 + 7(3𝑥 − 2) + 3
5𝑥 + 7(3𝑥 − 2) + 3
= 5𝑥 + 21𝑥 − 14 + 3
= 26𝑥 − 11
64 Simplify; 7(3𝑥 − 2) + 5𝑥 + 3
7(3𝑥 − 2) + 5𝑥 + 3
= 21𝑥 − 14 + 5𝑥 + 3
= 26𝑥 − 11
65 Simplify; 5𝑥 + 3 + 7(3𝑥 − 2)
5𝑥 + 3 + 7(3𝑥 − 2)
= 5𝑥 + 3 + 21𝑥 − 14
= 26𝑥 − 11
66 Simplify; 3(𝑥 + 4) + 5(𝑥 − 2)
3(𝑥 + 4) + 5(𝑥 − 2)
= 3𝑥 + 12 + 5𝑥 − 10
= 8𝑥 + 2
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 9
67 Simplify; 3(2𝑥 − 4) + 5(3𝑥 − 1)
3(2𝑥 − 4) + 5(3𝑥 − 1)
= 6𝑥 − 12 + 15𝑥 − 5
= 21𝑥 − 17
68 Simplify; 3(𝑥 + 4) + 5(𝑥 + 2)
3(𝑥 + 4) + 5(𝑥 + 2)
= 3𝑥 + 12 + 5𝑥 + 10
= 8𝑥 + 22
69 Simplify; 2(𝑥 + 3) + 3(𝑥 + 1)
2(𝑥 + 3) + 3(𝑥 + 1)
= 2𝑥 + 6 + 3𝑥 + 3
= 5𝑥 + 9
70 Simplify; 4(𝑥 + 3) + 2(𝑥 − 2)
4(𝑥 + 3) + 2(𝑥 − 2)
= 4𝑥 + 12 + 2𝑥 − 4
= 6𝑥 + 8
71 Simplify; 3(𝑥 − 4) + 5(𝑥 − 2)
3(𝑥 − 4) + 5(𝑥 − 2)
= 3𝑥 − 12 + 5𝑥 − 10
= 8𝑥 − 22
72 Simplify; 2(𝑥 − 4) + 3(𝑥 − 2)
2(𝑥 − 4) + 3(𝑥 − 2)
= 2𝑥 − 8 + 3𝑥 − 6
= 5𝑥 − 14
73 Simplify; 3(2𝑥 − 4) + 4(3𝑥 − 2)
3(2𝑥 − 4) + 4(3𝑥 − 2)
= 6𝑥 − 12 + 12𝑥 − 8
= 18𝑥 − 20
74 Simplify; 5(12𝑥 − 20) + 10(4𝑥 − 5)
5(12𝑥 − 20) + 10(4𝑥 − 5)
= 60𝑥 − 100 + 40𝑥 − 50
= 100𝑥 − 150
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 10
75 Simplifying – Different
76 Simplify; 2 × 3𝑥
2 × 3𝑥
= 2 × 3 × 𝑥
= 6𝑥
77 Simplify; 2𝑥 × 3
2𝑥 × 3
= 2 × 𝑥 × 3
= 2 × 3 × 𝑥
= 6𝑥
78 Simplify; 2 × 6𝑥
2 × 6𝑥
= 2 × 6 × 𝑥
= 12𝑥
79 Simplify; 2𝑥 × 3𝑦
2𝑥 × 3𝑦
= 2 × 𝑥 × 3 × 𝑦
= 2 × 3 × 𝑥 × 𝑦
= 6𝑥𝑦
80 Simplify; 2𝑦 × 3𝑥
2𝑦 × 3𝑥
= 2 × 𝑦 × 3 × 𝑥
= 2 × 3 × 𝑥 × 𝑦
= 6𝑥𝑦
81 Simplify; 2𝑥 × 3𝑦 × 4𝑧
2𝑥 × 3𝑦 × 4𝑧
= 24𝑥𝑦𝑧
82 Simplify; 3𝑦 × 4𝑧 × 5𝑥
3𝑦 × 4𝑧 × 5𝑥
= 60𝑥𝑦𝑧
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 11
83 Is this enough simplification practice? NO … I’d have to do these questions multiple times to have confidence to do them all in the test without error.
84 Since I can do these questions in class, it means that I can do the in the test.
NO … I’d have to do these questions multiple times to have confidence to do them all in the test without error.
85
86 Expand; 5(𝑥 + 2) Factorise; 5𝑥 + 10
5(𝑥 + 2) = 5𝑥 + 10
5𝑥 + 10 = 5(𝑥 + 2)
87
88 What is the connection between Expanding and Factorising that you can see?
Yes, they are the opposite process. Expanding and factorising are the exact opposite process to each other!
89
90 Is there anything in the year 8 achievement standard on this?
Yes, A year 8 student needs to be able to make a connection between expanding and factorising.
91
92 What is the connection between Expanding and Factorising that you can see?
Yes, they are the opposite process. Expanding and factorising are the exact opposite process to each other!
WorkSHEET 8.1 Algebra
© John Wiley & Sons Australia, Ltd 12
93 Words in Algbra
94 If tickets to a tennis match cost $23 for adults and $11 for children, write an expression for the cost of the following. (a) p adult tickets (b) q children tickets (c) x adult and y children tickets
Answers: (a) $ (b) $ (c) $( )
95 Bob is now d years old. (a) Write an expression for his age in 6 years
time. (b) How old is Bob 6 years ago?
(c) How old was Bob 3 years ago?
Answers: (a) (b) 𝑑 − 6
(c)
96 Kate is 𝑘 years old. Bob is double Kate’s age, plus 3 more years Write an expression showing how old Bob is.
ANSWER:
2𝑘 + 3
97 Kate is 𝑘 years old. Bob is double Kate’s age, plus 3 more years. By writing Bob’s age as an expression, use
substitution to find Bob’s age, if Kate is 9 years old.
ANSWER: Bob’s age as an expression
2𝑘 + 3 By substitution; Bobs age is
2 × 9 + 3 = 21
Bob is 21 years old
23p11q23 11x y+
6d +
3d -