SPC
Preliminary
General 2
AM1: Algebraic Manipulation
St Patrick’s College
Name:
SPC Preliminary General 2
Table of contents
Section 1: Syllabus Dot Points
page 2
Section 2: Terminology
page 2
Section 3: Operations with Algebraic Expressions page3
Section 4: Further multiplication and division
page 6
Section 5: Expanding and simplifying algebraic expressions
page 7
Section 6: Substitution
page 8
Section 7: Solving Linear Equations
page 10
Section 8: MIND MAP & REFLECTION
page 12
Section 9: Strategies, handy hints and Keywords
Year 11 GM – Algebraic Manipulation
Page 1
Section 1: Syllabus Dot Points
AM1
Date:__/__/__
Algebraic manipulation
The principal focus of this topic is to provide a foundation in basic algebraic skills, including the solution of
a variety of equations. The topic develops techniques that have applications in work-related and everyday
contexts.
Content
Students:

add, subtract, multiply and divide algebraic terms

simplify algebraic expressions involving multiplication and division,
eg
9y
4m m 4wn nb
 5y ,


,
4
5n 20n
b
2w

expand and simplify algebraic expressions

substitute numerical values into algebraic expressions,
3x
5p x  y
eg
, 5 2x  4  , 3a2  b ,
,
yx
5
4m

substitute given values for the other pronumerals in a mathematical formula from a vocational or other
context to find the value of the subject of the formula,
eg if A  P 1 r  , find A given P = 600, r = 0.05, n = 3
n

solve linear equations involving two steps, eg 5x 12  22 ,
4x
x 1
r
 3,
 6 ,  3  2.
10
3
5
For more information, refer to page 50, 51, 52 and 53 of the General Mathematics
Syllabus on the board of Studies Website.
Section 2: Terminology
Terminology
algebraic expression
common difference
constant
equation
evaluate
expand
formulae
linear
Year 11 GM – Algebraic Manipulation
simplify
solution
solve
substitute
Page 2
Section 3: Operations with Algebraic Expressions
Date:__/__/__
COEFFICIENT = NUMERAL = number
PRONUMERAL = letter
Watch before class: http://www.youtube.com/watch?v=RPLaKeSBN-Q
In algebra, each pro-numeral used stands in place of a number
Adding & Subtracting Like Terms
Algebraic expressions can be simplified by collecting like terms. Like terms are the only
terms that can be added and subtracted. The coefficient is the number in front of the pronumeral.
Examples:
1. Simplify the following by collecting like terms.
a) 7𝑦 + 𝑦 =
b) −3𝑝 − 18𝑝 + 21𝑝 =
c) 8𝑟 − 3 + 5𝑟 =
d) 5𝑎 − 𝑎 + 6𝑏 + 𝑏 =
e) 5𝑎 − 3𝑝 + 6𝑎 + 2𝑝 =
Year 11 GM – Algebraic Manipulation
Page 3
Index Laws
Example:
1. Write the following iin index form
a) 3 × 3 × 3 =
b) 𝑝 × 𝑝 × 𝑝 × 𝑝 × 𝑝 × 𝑝 =
c) 3 × 𝑎 × 𝑎 × 4 × 𝑏 × 𝑎 =
2. write the following in expanded form
a) 23
b) 𝑦 5
Year 11 GM – Algebraic Manipulation
Page 4
Multiplication & Division of Algebraic Terms
Algebraic terms can be muliplied and divided to form single expressions.
Examples:
Simplify each of the following:
1. 𝑎2 × 𝑎4 =
2. 5𝑑 4 × 4𝑑 5 =
3. 9𝑟 3 𝑠 2 𝑡 4 × 6𝑟𝑠𝑡 6 =
4. 𝑎5 ÷ 𝑎2 =
5. 𝑏 9 ÷ 𝑏 =
6. 56𝑦 7 𝑘 8 ÷ 8𝑦 2 𝑘 3
7.
8.
9.
72𝑒 6
8𝑒 3
=
8𝑝3 ×7𝑟 2 ×2𝑠
6𝑝×14𝑟
𝟏𝟖𝒂𝟓 𝒃𝟐
𝟐𝟕𝒂𝟑 𝒃𝟕
=
=
Homework
Pace yourself
MQ Ex 11A, P371, Q110, a, c, e…
Extend yourself MQ Ex 11A, P371, Q110, column 3, 1012
Year 11 GM – Algebraic Manipulation
Page 5
Section 4: Further multiplication and division
Date:__/__/__
Examples: Simplify each of the following:
1) 3 ×
5𝑎
2
3ℎ
2) 5ℎ ×
3)
4)
5)
6)
7)
8)
9)
3𝑥
5
𝑥
5
×
7𝑥
𝑥
10
=
÷2=
20𝑦
3
=
20
𝑦
=
21𝑧
×
5𝑦
=
5
÷ =
𝑥
2𝑥𝑦
5
2𝑥
3𝑦
×
4𝑥 3
8𝑦
÷
3𝑥𝑦
5
15𝑦
×
6𝑥
=
3𝑦 4
15
=
÷
𝑦4
3
Homework
Pace yourself
MQ Ex 11B, P373, Q14, a, c, e…
Extend yourself MQ Ex 11B, P373, Q14, column 3, Q5+6, a, c, e…
Year 11 GM – Algebraic Manipulation
Page 6
Section 5: Expanding and simplifying algebraic expression
Date:__/__/__
To expand an algebraic expression means to remove a set of brackets. This is done by
multiplying everything inside the brackets by the term directly outside the brackets.
Examples: Expand and simplify where possible:
1. 3(𝑥 − 2) =
2. −7(𝑚 + 3) =
3. −10𝑝(𝑞 − 9) =
4. 5(𝑥 − 5) + 11 =
5. 4𝑐(2𝑑 − 3𝑐) − 𝑐𝑑 − 5𝑐 =
6. 2(𝑥 + 2𝑦) + 3(2𝑥 − 𝑦) =
7. 2(𝑐 − 3𝑑) − 5(2𝑐 − 5𝑑) =
8. 4𝑚(3𝑚 − 5) − (8𝑚 + 4)
It’s important that you master questions 7 and 8 above as they are the ‘typical’ HSC
questions.
Homework
Pace yourself
MQ Ex 11C, P375, Q15, a, c, e…, 6
Extend yourself MQ Ex 11C, P375, Q15, column 3, 6, 7
Year 11 GM – Algebraic Manipulation
Page 7
Section 6: Substitution
Date:__/__/__
Watch before class: http://www.youtube.com/watch?v=nARfQA8Vz6c
When the numerical values of pronumerals are known, we can substitute (replace) them into
an algebraic expression and evaluate it. It can be useful to place any substituted values in
brackets (especially if it’s a negative number) when evaluating an expression.
Substitution is an integral part of this course. Its important you follow the following simple
steps when using this technique to evaluate or calculate an expression or a worded problem
accuratly.
STEP 1: Write down the expression
STEP 2: Substitude all pronumerals with the given numerals
STEP 3: Evaluate
All three steps MUST be shown at all times.
NO working out = NO marks in the HSC.
Regardless if the answer is correct.
Examples
1. If 𝑎 = 5 𝑎𝑛𝑑 𝑏 = −6, evaluate the following expressions.
a) 𝑎 + 𝑏
b) 𝑎𝑏(𝑏 − 𝑎)
c) 𝑎𝑏 2
d)
𝑎
𝑏
e) 6𝑎 + 2𝑏
f) 3.6𝑏 − 2.4𝑎
2. Given that 𝑃 = 2𝑙 + 2𝑤, find 𝑃 when 𝑙 = 16 𝑎𝑛𝑑 𝑤 = 22.
Year 11 GM – Algebraic Manipulation
Page 8
3. The cost of hiring a taxi is $3.75 plus $0.50 per kilometre.
a) Write a formula for the codt of a taxi journey, C, in terms of distance travelled , d.
b) Use the formula to calculate the cost of a taxi journey of 35 km.
1
4. The kinetic energy of an object is found using the formula 𝐸 = 𝑚𝑣 2 , where 𝑚 is the
2
mass and 𝑣 is the velocity of the object. Find 𝐸 when
𝑚 = 3.2 𝑎𝑛𝑑 𝑣 = 14.2
Homework
Pace yourself
MQ Ex 11D, P377, Q1, 37,
Extend yourself MQ Ex 11D, P377, Q1, 37, 917 odds, column 3, 1823
Year 11 GM – Algebraic Manipulation
Page 9
Section 7: Solving Linear Equations
Date:__/__/__
Watch before class:
http://www.youtube.com/watch?v=1HLGsKYoUU0
http://www.youtube.com/watch?v=HyUiYeLee3g
When we are given an equation, our task is to SOLVE
IT. That is, to find a value for the pro-numeral which
makes the sentence true.
Examples:
1. Solve the following linear equations.
a) 𝑧 + 36 = 50
b) 𝑡 − 16 = 33
𝑣
d) = −4
e) 5𝑎 + 13 = 28
g) 15 − 3𝑔 = 45
h)
8
j)
𝑦
3
−5=9
𝑝
11
= 33
𝑘
k) 14 − = 30
Year 11 GM – Algebraic Manipulation
2
c) 12𝑏 = −144
f) 9𝑣 − 10 = 22
i)
12𝑚
−3
= −9
𝑣
7
3
10
l) =
Page 10
2 Solve each of the following equations.
a) 3(𝑥 + 2) = 8
c) 5(4𝑓 − 2) = −125
b) 4(2𝑥 − 1) = 12
3
d) (6𝑑 + 10) = −2
2
3. In the formula 𝐴 = 𝑙𝑏, calculate 𝑙, 𝑤ℎ𝑒𝑛 𝐴 = 65 · 0136 𝑎𝑛𝑑 𝑏 = 8 · 24
4. An operated connected phone call cost $2·20 connection fee plus $1·75 per minute.
a) Write a formula connecting the cost of the call, C to the length of the call, m.
b) Calculate the cost of a call lasting 5 minutes.
c) Calculate the length of call for which the charge is $21·45
Homework
Pace yourself
MQ Ex 11A, P371, Q110, a, c, e…
Extend yourself MQ Ex 11A, P371, Q110, column 3, 1012
Also complete Chapter Review.
Year 11 GM – Algebraic Manipulation
Page 11
Section 8: MIND MAP & REFLECTION
Please rate the following outcomes:
1 – still have great difficulty, 2- still having some difficulty, 3 – ok
4 – can do most questions with ease, 5 – can do all questions with ease.
OUTCOME- Algebraic Manipulation
1
 Add and subtract algebraic terms
2
3
4
5
 multiply and divide algebraic terms
 simplify fractional algebraic expressions involving
multiplication and division
 expand and simplify algebraic expressions
 substitute numerical values into algebraic expressions
 substitute given values for the other pronumerals in a
mathematical formula from a vocational or other context to
find the value of the subject of the formula
 solve linear equations involving two steps
REFLECTION:
Year 11 GM – Algebraic Manipulation
Page 12
Section 9: Strategies, handy hints and Keywords
Year 11 GM – Algebraic Manipulation
Page 13