BEAMS Module Unit 3: Algebraic Expressions, Quadratic ... · Unit 1: Negative Numbers UNIT 3...
Transcript of BEAMS Module Unit 3: Algebraic Expressions, Quadratic ... · Unit 1: Negative Numbers UNIT 3...
Unit 1:
Negative Numbers
UNIT 3
ALGEBRAIC EXPRESSIONS
AND
ALGEBRAIC FORMULAE
B a s i c E s s e n t i a l
A d d i t i o n a l M a t h e m a t i c s S k i l l s
Curriculum Development Division
Ministry of Education Malaysia
TABLE OF CONTENTS
Module Overview 1
Part A: Performing Operations on Algebraic Expressions 2
Part B: Expansion of Algebraic Expressions 10
Part C: Factorisation of Algebraic Expressions and Quadratic Expressions 15
Part D: Changing the Subject of a Formula 23
Activities
Crossword Puzzle 31
Riddles 33
Further Exploration 37
Answers 38
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
1 Curriculum Development Division
Ministry of Education Malaysia
MODULE OVERVIEW
1. The aim of this module is to reinforce pupils’ understanding of the concepts and skills
in Algebraic Expressions, Quadratic Expressions and Algebraic Formulae.
2. The concepts and skills in Algebraic Expressions, Quadratic Expressions and
Algebraic Formulae are required in almost every topic in Additional Mathematics,
especially when dealing with solving simultaneous equations, simplifying
expressions, factorising and changing the subject of a formula.
3. It is hoped that this module will provide a solid foundation for studies of Additional
Mathematics topics such as:
Functions
Quadratic Equations and Quadratic Functions
Simultaneous Equations
Indices and Logarithms
Progressions
Differentiation
Integration
4. This module consists of four parts and each part deals with specific skills. This format
provides the teacher with the freedom to choose any parts that is relevant to the skills
to be reinforced.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
2 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Pupils who face problem in performing operations on algebraic expressions might have
difficulties learning the following topics:
Simultaneous Equations - Pupils need to be skilful in simplifying the algebraic
expressions in order to solve two simultaneous equations.
Functions - Simplifying algebraic expressions is essential in finding composite
functions.
Coordinate Geometry - When finding the equation of locus which involves
distance formula, the techniques of simplifying algebraic expressions are required.
Differentiation - While performing differentiation of polynomial functions, skills
in simplifying algebraic expressions are needed.
Strategy:
1. Teacher reinforces the related terminologies such as: unknowns, algebraic terms,
like terms, unlike terms, algebraic expressions, etc.
2. Teacher explains and shows examples of algebraic expressions such as:
8k, 3p + 2, 4x – (2y + 3xy)
3. Referring to the “Lesson Notes” and “Examples” given, teacher explains how to
perform addition, subtraction, multiplication and division on algebraic expressions.
4. Teacher emphasises on the rules of simplifying algebraic expressions.
PART A:
PERFORMING OPERATIONS ON
ALGEBRAIC EXPRESSIONS
LEARNING OBJECTIVES
Upon completion of Part A, pupils will be able to perform operations on algebraic
expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
3 Curriculum Development Division
Ministry of Education Malaysia
PART A:
PERFORMING BASIC ARITHMETIC OPERATIONS ON ALGEBRAIC EXPRESSIONS
1. An algebraic expression is a mathematical term or a sum or difference of mathematical
terms that may use numbers, unknowns, or both.
Examples of algebraic expressions: 2r, 3x + 2y, 6x2 +7x + 10, 8c + 3a – n
2,
g
3
2. An unknown is a symbol that represents a number. We normally use letters such as n, t, or
x for unknowns.
3. The basic unit of an algebraic expression is a term. In general, a term is either a number
or a product of a number and one or more unknowns. The numerical part of the term, is
known as the coefficient.
Examples: Algebraic expression with one term: 2r, g
3
Algebraic expression with two terms: 3x + 2y, 6s – 7t
Algebraic expression with three terms: 6x2 +7x + 10, 8c + 3a – n
2
4. Like terms are terms with the same unknowns and the same powers.
Examples: 3ab, –5ab are like terms.
3x2,
2
5
2x are like terms.
5. Unlike terms are terms with different unknowns or different powers.
Examples: 1.5m, 9k, 3xy, 2x2y are all unlike terms.
LESSON NOTES
6 xy Coefficient Unknowns
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
4 Curriculum Development Division
Ministry of Education Malaysia
6. An algebraic expression with like terms can be simplified by adding or subtracting the
coefficients of the unknown in algebraic terms.
7. To simplify an algebraic expression with like terms and unlike terms, group the like terms
first, and then simplify them.
8. An algebraic expression with unlike terms cannot be simplified.
9. Algebraic fractions are fractions involving algebraic terms or expressions.
Examples: .2
,2
4,
6
2,
15
322
22
2
2
yxyx
yx
grg
gr
h
m
10. To simplify an algebraic fraction, identify the common factor of both the numerator and the
denominator. Then, simplify it by elimination.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
5 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following algebraic expressions and algebraic fractions:
(a) 5x – (3x – 4x) 64
)e(ts
(b) –3r –9s + 6r + 7s z
yx
2
3
6
5)f(
(c) 2
2
2
4
grg
gr
g
f
e2)g(
qp
43)d(
(h) x
x
3
2
13
Solutions:
(a) 5x – (3x – 4x)
= 5x – (– x)
= 5x + x
= 6x
(b) –3r –9s + 6r + 7s
= –3r + 6r –9s + 7s
= 3r – 2s
2
2
2
4)c(
grg
gr
gr
r
grg
gr
2
4
)2(
4
2
2
Perform the operation in the bracket.
Arrange the algebraic terms according to the like terms.
.
Unlike terms cannot be simplified.
Leave the answer in the simplest form as shown.
Algebraic expression with like terms can be simplified by
adding or subtracting the coefficients of the unknown.
Simplify by canceling out the common factor and the
same unknowns in both the numerator and the
denominator.
1
1
EXAMPLES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
6 Curriculum Development Division
Ministry of Education Malaysia
pq
pq
pq
p
pq
q
qp
43
43
43)d(
12
23
26
2
34
3
64)e(
ts
ts
ts
z
xy
z
yx
z
yx
4
5
22
5
2
3
6
5)f(
fg
e
gf
eg
f
e
2
2
12)g(
x
x
x
x
x
x
x
x
x
x
6
16
3
1
2
16
3
2
16
3
2
1
2
)2(3
3
2
13
)h(
The LCM of p and q is pq.
The LCM of 4 and 6 is 12.
Simplify by canceling out the common
factor, then multiply the numerators
together and followed by the
denominators.
Change division to multiplication of the
reciprocal of 2g.
Equate the denominator.
2
1
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
7 Curriculum Development Division
Ministry of Education Malaysia
ALTERNATIVE METHOD
Simplify the following algebraic fractions:
(a) x
x
3
2
13
= x
x
3
2
13
2
2
= )2(3
)2(2
1)2(3
x
x
= x
x
6
16
(b) 5
23
x = 5
23
x
x
x
x
x
x
xxx
5
23
)(5
)(2)(3
x
x
x
xx
x
x
xxx
4
316
)2(2
)2(2
3)2(8
2
2
2
2
38
2
2
38
)c(
The denominator of x2
3 is 2x. Therefore,
multiply the algebraic fraction byx
x
2
2.
Each of the terms in the numerator and
denominator is multiplied by 2x.
.
The denominator of 2is2
1. Therefore,
multiply the algebraic fraction by2
2.
Each of the terms in the numerator and
denominator of the algebraic fraction is
multiplied by 2.
The denominator of x
3 is x. Therefore,
multiply the algebraic fraction byx
x.
Each of the terms in the numerator and
denominator is multiplied by x.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
8 Curriculum Development Division
Ministry of Education Malaysia
x
x
x
xx
36
21
288
21
)7(4)7(7
8
)7(3
7
7
47
8
3
47
8
3)d(
The denominator of 7
8 x is 7.
Therefore, multiply the algebraic
fraction by7
7.
Each of the terms in the numerator
and denominator is multiplied by 7.
Simplify the denominator.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
9 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following algebraic expressions:
1. 2a –3b + 7a – 2b
2. − 4m + 5n + 2m – 9n
3. 8k – ( 4k – 2k )
4. 6p – ( 8p – 4p )
xy 5
13.5
5
2
3
4.6
kh
c
ba
2
3
7
4.7
dc
dc
3
8
2
4.8
yzz
xy.9
w
uv
vw
u
2.10
65
2.11
x
54
24
.12
x
x
TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
10 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Pupils who face problem in expanding algebraic expressions might have
difficulties in learning of the following topics:
Simultaneous Equations – pupils need to be skilful in expanding the
algebraic expressions in order to solve two simultaneous equations.
Functions – Expanding algebraic expressions is essential when finding
composite function.
Coordinate Geometry – when finding the equation of locus which
involves distance formula, the techniques of expansion are applied.
Strategy:
Pupils must revise the basic skills involving expanding algebraic expressions.
PART B:
EXPANSION OF ALGEBRAIC
EXPRESSIONS
LEARNING OBJECTIVE
Upon completion of Part B, pupils will be able to expand algebraic
expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
11 Curriculum Development Division
Ministry of Education Malaysia
PART B:
EXPANSION OF ALGEBRAIC EXPRESSIONS
1. Expansion is the result of multiplying an algebraic expression by a term or another
algebraic expression.
2. An algebraic expression in a single bracket is expanded by multiplying each term in the
bracket with another term outside the bracket.
3(2b – 6c – 3) = 6b – 18c – 9
3. Algebraic expressions involving two brackets can be expanded by multiplying each term of
algebraic expression in the first bracket with every term in the second bracket.
(2a + 3b)(6a – 5b) = 12a2 – 10ab + 18ab – 15b
2
= 12a
2 + 8ab – 15b
2
4. Useful expansion tips:
(i) (a + b)2 = a
2 + 2ab + b
2
(ii) (a – b)2 = a
2 – 2ab + b
2
(iii) (a – b)(a + b) = (a + b)(a – b)
= a2 – b
2
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
12 Curriculum Development Division
Ministry of Education Malaysia
Expand each of the following algebraic expressions:
(a) 2(x + 3y)
(b) – 3a (6b + 5 – 4c)
Solutions:
(a) 2 (x + 3y)
= 2x + 6y
(b) –3a (6b + 5 – 4c)
= –18ab – 15a + 12ac
1293
2)c( y
= 123
29
3
2 y
= 6y + 8
= (a + 3) (a + 3)
= a2 + 3a + 3a + 9
= a2 + 6a + 9
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
1293
2)c( y
2523)e( k
2)3()d( a
)5)(2()f( pp
2)3()d( a
When expanding a bracket, each term
within the bracket is multiplied by the term
outside the bracket.
When expanding a bracket, each term
within the bracket is multiplied by the term
outside the bracket.
1
3
1
4
EXAMPLES
Simplify by canceling out the common
factor, then multiply the numerators
together and followed by the denominators.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
13 Curriculum Development Division
Ministry of Education Malaysia
(c) (4x – 3y)(6x – 5y)
– 18 xy
– 20 xy
– 38 xy
= 24x2 – 38 xy + 15y
2
2523)e( k
= –3(2k + 5) (2k + 5)
= –3(4k2 + 20k + 25)
= –12k2 – 60k – 75
)5( )2( )f( qp
= pq – 5p + 2q – 10
ALTERNATIVE METHOD
Expanding two brackets
(a) (a + 3) (a + 3)
= a2 + 3a + 3a + 9
= a2 + 6a + 9
(b) (2p + 3q) (6p – 5q)
= 12p2 – 10 pq + 18 pq – 15q
2
= 12p2 + 8 pq – 15q
2
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
When expanding two
brackets, write down the
product of expansion and
then, simplify the like
terms.
When expanding two brackets, each term
within the first bracket is multiplied by
every term within the second bracket.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
14 Curriculum Development Division
Ministry of Education Malaysia
Simplify the following expressions and give your answers in the simplest form.
4
324.1 n
162
1.2 q
yxx 326.3
)(22.4 baba
)6()3(2.5 pp
3
26
3
1.6
yxyx
121.72
ee
nmmnm 2.82
gfggfgf 2.9
ihiihih 32.10
TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
15 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
Some pupils may face problem in factorising the algebraic expressions. For
example, in the Differentiation topic which involves differentiation using the
combination of Product Rule and Chain Rule or the combination of Quotient
Rule and Chain Rule, pupils need to simplify the answers using factorisation.
Examples:
2
2
2
32
3
32
2433
43
)27(
)154()3(
)27(
)2()3(])3(3)[27(
27
)3(.2
)1549()57(2
)6()57(])57(28[2
)57(2.1
x
xx
x
xxx
dx
dy
x
xy
xxx
xxxxdx
dy
xxy
Strategy
1. Pupils revise the techniques of factorisation.
PART C:
FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND
QUADRATIC EXPRESSIONS
LEARNING OBJECTIVE
Upon completion of Part C, pupils will be able to factorise algebraic expressions
and quadratic expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
16 Curriculum Development Division
Ministry of Education Malaysia
PART C:
FACTORISATION OF
ALGEBRAIC EXPRESSIONS AND QUADRATIC EXPRESSIONS
1. Factorisation is the process of finding the factors of the terms in an algebraic expression. It
is the reverse process of expansion.
2. Here are the methods used to factorise algebraic expressions:
(i) Express an algebraic expression as a product of the Highest Common Factor (HCF) of
its terms and another algebraic expression.
ab – bc = b(a – c)
(ii) Express an algebraic expression with three algebraic terms as a complete square of two
algebraic terms.
a2 + 2ab + b
2 = (a + b)
2
a2 – 2ab + b
2 = (a – b)
2
(iii) Express an algebraic expression with four algebraic terms as a product of two algebraic
expressions.
ab + ac + bd + cd = a(b + c) + d(b + c)
= (a + d)(b + c)
(iv) Express an algebraic expression in the form of difference of two squares as a product of
two algebraic expressions.
a2 – b
2 = (a + b)(a – b)
3. Quadratic expressions are expressions which fulfill the following characteristics:
(i) have only one unknown; and
(ii) the highest power of the unknown is 2.
4. Quadratic expressions can be factorised using the methods in 2(i) and 2(ii).
5. The Cross Method can be used to factorise algebraic expression in the general form of
ax2 + bx + c, where a, b, c are constants and a ≠ 0, b ≠ 0, c ≠ 0.
LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
17 Curriculum Development Division
Ministry of Education Malaysia
(a) Factorising the Common Factors
i) mn + m = m (n +1)
ii) 3mp + pq = p (3m + q)
iii) 2mn – 6n = 2n (m – 3)
(b) Factorising Algebraic Expressions with Four Terms
i) vy + wy + vz + wz
= y (v + w) + z (v + w)
= (v + w)(y + z)
ii) 21bm – 7bs + 6cm – 2cs
= 7b(3m – s) + 2c(3m – s)
= (3m – s)(7b + 2c)
Factorise the first and the second terms
with the common factor y, then factorise
the third and fourth terms with the
common factor z.
.
(v + w) is the common factor.
Factorise the first and the second terms with
common factor 7b, then factorise the third
and fourth terms with common factor 2c.
(3m – s) is the common factor.
EXAMPLES
Factorise the common factor m.
.
Factorise the common factor p.
.
Factorise the common factor 2n.
.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
18 Curriculum Development Division
Ministry of Education Malaysia
(c) Factorising the Algebraic Expressions by Using Difference of Two Squares
i) x2 – 16 = x
2 – 4
2
= (x + 4)(x – 4)
ii) 4x2
– 25 = (2x)2 – 5
2
= (2x + 5)(2x – 5)
(d) Factorising the Expressions by Using the Cross Method
i) x2
– 5x + 6
xxx
x
x
523
2
3
x2
– 5x + 6 = (x – 3) (x – 2)
ii) 3x2
+ 4x – 4
xxx
x
x
462
2
23
3x2 + 4x – 4 = (3x – 2) (x + 2)
The summation of the cross
multiplication products should
equal to the middle term of the
quadratic expression in the
general form.
The summation of the cross
multiplication products should
equal to the middle term of the
quadratic expression in the
general form.
a2 – b
2 = (a + b)(a – b)
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
19 Curriculum Development Division
Ministry of Education Malaysia
ALTERNATIVE METHOD
Factorise the following quadratic expressions:
i) x 2 – 5x + 6
ac b
+ 6 – 5
–2 –3
(x – 2) (x – 3)
)3)(2(65 2 xxxx
ii) x 2 – 5x – 6
ac b
– 6 – 5
+1 – 6
(x + 1) (x– 6)
)6)(1(65 2 xxxx
+1 (–6) = –6
+1 (–6) = –6
+1 – 6 = –5
a=+1 b= –5 c = –6
REMEMBER!!!
An algebraic expression can
be represented in the general
form of ax2 + bx + c, where
a, b, c are constants and
a ≠ 0, b ≠ 0, c ≠ 0.
+1 (+ 6) = + 6 –2 (–3) = +6
–2 + (–3) = –5
a=+1 b= –5 c =+6
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
20 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF C
(iii) 2x2 – 11x + 5
ac b
+ 10 –11
–1 – 10
2
10
2
1
52
1
(2x – 1) (x – 5)
)5)(12(5112 2 xxxx
(iv) 3x2 + 4x – 4
ac b
– 12 + 4
– 2 +6
23
2
3
6
3
2
The coefficient of x2 is 2,
divide each number by 2.
(+2) (+5) = +10
–1 (–10) = +10
–1 + (–10) = –11
–2 + 6 = 4
The coefficient of x2 is 3, divide each
number by 3.
3 (– 4) = –12
a=+2 b = –11 c =+5
a =+ 3 b=+ 4 c = –4
(3x – 2) (x + 2)
The coefficient of x2 is 2,
multiply by 2:
5)(12
52
5
21
21
xx
xx
xx
The coefficient of x2 is 3, multiply by 3:
2)(23
23
2
32
32
xx
xx
xx
)2)(23(443 2 xxxx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
21 Curriculum Development Division
Ministry of Education Malaysia
Factorise the following quadratic expressions completely.
1. 3p 2 – 15
2. 2x 2 – 6
3. x 2 – 4x
4. 5m 2 + 12m
5. pq – 2p
6. 7m + 14mn
7. k2 –144
8. 4p 2 – 1
9. 2x 2 – 18
10. 9m2 – 169
TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
22 Curriculum Development Division
Ministry of Education Malaysia
11. 2x 2 + x – 10
12. 3x 2 + 2x – 8
13. 3p 2 – 5p – 12
14. 4p2 – 3p – 1
15. 2x2
– 3x – 5
16. 4x 2 – 12x + 5
17. 5p 2 + p – 6
18. 2x2
– 11x + 12
19. 3p + k + 9pr + 3kr
20. 4c2 – 2ct – 6cw + 3tw
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
23 Curriculum Development Division
Ministry of Education Malaysia
TEACHING AND LEARNING STRATEGIES
If pupils have difficulties in changing the subject of a formula, they probably
face problems in the following topics:
Functions – Changing the subject of the formula is essential in finding
the inverse function.
Circular Measure – Changing the subject of the formula is needed to
find the r or from the formulae s = r or 2
2
1rA .
Simultaneous Equations – Changing the subject of the formula is the
first step of solving simultaneous equations.
Strategy:
1. Teacher gives examples of formulae and asks pupils to indicate the subject
of each of the formula.
Examples: y = x – 2
hrV
bhA
2
2
1
y, A and V are the
subjects of the
formulae.
PART D:
CHANGING THE SUBJECT
OF A FORMULA
LEARNING OBJECTIVE
Upon completion of this module, pupils will be able to change the subject of
a formula.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
24 Curriculum Development Division
Ministry of Education Malaysia
PART D:
CHANGING THE SUBJECT OF A FORMULA
1. An algebraic formula is an equation which connects a few unknowns with an equal
sign.
Examples:
hrV
bhA
2
2
1
2. The subject of a formula is a single unknown with a power of one and a coefficient
of one, expressed in terms of other unknowns.
Examples: bhA2
1
a2 = b
2 + c
2
hTrT 2
2
1
3. A formula can be rearranged to change the subject of the formula. Here are the
suggested steps that can be used to change the subject of the formula:
(i) Fraction : Get rid of fraction by multiplying each term in the formula with
the denominator of the fraction.
(ii) Brackets : Expand the terms in the bracket.
(iii) Group : Group all the like terms on the left or right side of the formula.
(iv) Factorise : Factorise the terms with common factor.
(v) Solve : Make the coefficient and the power of the subject equal to one.
LESSON NOTES
A is the subject of the formula because it is
expressed in terms of other unknowns.
a
2 is not the subject of the formula
because the power ≠ 1
T is not the subject of the formula
because it is found on both sides of the
equation.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
25 Curriculum Development Division
Ministry of Education Malaysia
1. Given that 2x + y = 2, express x in terms of y.
Solution:
2x + y = 2
2x = 2 – y
x = 2
2 y
2. Given that yyx
52
3
, express x in terms of y.
Solution:
yyx
52
3
3x + y = 10y
3x = 10y – y
3x = 9y
x = 3
9y
x = 3y
No fraction and brackets.
Group:
Retain the x term on the left hand side of the
equation by grouping all the y term to the
right hand side of the equation.
Fraction:
Multiply both sides of the equation by 2.
Group:
Retain the x term on the left hand side of the
equation by grouping all the y term to the
right hand side of the equation.
Solve:
Divide both sides of the equation by 2 to
make the coefficient of x equal to 1.
Solve:
Divide both sides of the equation by 3 to
make the coefficient of x equal to 1.
EXAMPLES
Steps to Change the Subject of a Formula
(i) Fraction
(ii) Brackets
(iii) Group
(iv) Factorise
(v) Solve
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
26 Curriculum Development Division
Ministry of Education Malaysia
3. Given that yx 2 , express x in terms of y.
Solution:
yx 2
x = (2y)2
x = 4y2
4. Given that px
3, express x in terms of p.
Solution:
px
3
2
2
9
)3(
3
px
px
px
5. Given that yxx 23 , express x in terms of y.
Solution:
2
2
2
2
2
22
23
23
yx
yx
yx
yxx
yxx
Solve:
Square both sides of the equation to make the
power of x equal to 1.
Fraction:
Multiply both sides of the equation by 3.
Solve:
Square both sides of the equation to make
the power of x equal to1.
Group:
Group the like terms
Solve:
Divide both sides of the equation by 2 to
make the coefficient of x equal to 1.
Solve:
Square both sides of equation to make the
power of x equal to 1.
Simplify the terms.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
27 Curriculum Development Division
Ministry of Education Malaysia
6. Given that 4
11x – 2(1 – y) = xp2 , express x in terms of y and p.
Solution:
4
11x – 2 (1 – y) = xp2
11x – 8(1 – y) = xp8
11x – 8 + 8y = 8xp
11x – 8xp = 8 – 8y
x(11 – 8p) = 8 – 8y
x = p
y
811
88
7. Given that n
xp
5
32 = 1 – p , express p in terms of x and n.
Solution:
n
xp
5
32 = 1 – p
2p – 3x = 5n – 5pn
2p + 5pn = 5n + 3x
p(2 + 5n) = 5n + 3x
p = n
xn
52
35
Fraction:
Multiply both sides of the equation
by 4.
Bracket:
Expand the bracket.
Group:
Group the like terms.
Factorise:
Factorise the x term.
Solve:
Divide both sides by (11 – 8p) to
make the coefficient of x equal to 1.
Fraction:
Multiply both sides of the equation by
5n.
Solve:
Divide both sides of the equation by
(2 + 5n) to make the coefficient of p
equal to 1.
Group:
Group the like p terms.
Factorise:
Factorise the p terms.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
28 Curriculum Development Division
Ministry of Education Malaysia
1. Express x in terms of y.
a) 02 yx
b) 032 yx
c) 12 xy
d) 22
1 yx
e) 53 yx
f) 43 xy
TEST YOURSELF D
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
29 Curriculum Development Division
Ministry of Education Malaysia
2. Express x in terms of y.
a) xy
b) xy 2
c) 3
2x
y
d) xy 31
e) 13 xyx
f) yx 1
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
30 Curriculum Development Division
Ministry of Education Malaysia
3. Change the subject of the following formulae:
a) Given that 2
ax
ax, express x in terms
of a .
b) Given that x
xy
1
1, express x in terms
of y .
c) Given that vuf
111 , express u in
terms of v and f .
d) Given that 4
3
2
2
qp
qp, express p in
terms of .q
e) Given that mnmp 23 , express m in
terms of n and p .
f) Given that
C
CBA
1, express C in
terms of A and B .
g) Given that yx
xy2
2
, express y in
terms of x.
h) Given that g
lT 2 , express g in
terms of T and l.
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
31 Curriculum Development Division
Ministry of Education Malaysia
CROSSWORD PUZZLE
HORIZONTAL
1) – 4p, 10q and 7r are called algebraic .
3) An algebraic term is the of unknowns and numbers.
4) 4m and 8m are called terms.
5) hrV 2 , then V is the of the formula.
7) An can be represented by a letter.
10) 21232 xxxx .
ACTIVITIES
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
32 Curriculum Development Division
Ministry of Education Malaysia
VERTICAL
2) An algebraic consists of two or more algebraic terms combined by
addition or subtraction or both.
6) 252212 2 xxxx .
8) terms are terms with different unknowns.
9) The number attached in front of an unknown is called .
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
33 Curriculum Development Division
Ministry of Education Malaysia
RIDDLES
RIDDLE 1
1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box
below.
3. Rearrange the alphabets to form a word.
4. What is the word?
1
2 3 4 5 6 7 8 9
1. Calculate
.3
5
12
D) 5
1 O) 1
W) 3
11 N)
15
11
2. Simplify yxyx 7693 .
F) yx 23 W) yx 169
E) yx 23 X) yx 29
3. Simplify 23
qp .
L) 6
32 qp A)
6
32 qp
N) 6
23 pq R)
6
23 qp
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
34 Curriculum Development Division
Ministry of Education Malaysia
4. Expand )7()4(2 xx .
A) 1x D) 15x
U) 13 x C) 153 x
5. Expand )52(3 cba .
S ) acab 156 C) acab 156
T) acab 156 R) acab 156
6. Factorise 252 x .
E) )5)(5( xx T) )5)(5( xx
I) )5)(5( xx C) )25)(25( xx
7. Factorise qpq 4 .
D) )41( qpq E) )4( pq
T) )4( qp S) )4( pq
8. Factorise 1282 xx .
I ) )6)(2( xx W) )6)(2( xx
F) )3)(4( xx C) )3)(4( xx
9. Given that 42
3
x
yx, express x in terms of y.
L) 5
yx C)
5
yx
T) 11
yx N)
3
8 yx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
35 Curriculum Development Division
Ministry of Education Malaysia
RIDDLE 2
1. You are given 9 multiple-choice questions.
2. For each of the questions, choose the correct answer and fill the alphabet in the box
below.
3. Rearrange the alphabets to form a word.
4. What is the word?
1
2 3 4 5 6 7 8 9
1. Calculate
.3
15
x
A) 3
5 x O)
x
x
3
5
I ) 5
3
x
x N)
5
3
x
2. Simplify r
qp
54
3 .
F) q
pr
4
15 R)
pr
q
15
4
W) r
pq
20
3 B)
r
pq
5
3
3. Simplifyz
xy
yz
x
2 .
N)2
2
y D)
2
2
2z
x
L) 22z
x I)
2
2
z
x
4. Solve ).3(2
yxxyx
E) xyyx 222 D) xyyx 222
I ) xyxyx 222 3 N) xyyx 222
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
36 Curriculum Development Division
Ministry of Education Malaysia
5. Expand 25p .
I) 252 p N) 252 p
D) 25102 pp L) 25102 pp
6. Factorise 1572 2 yy .
F) )5)(32( yy D) )5)(32( yy
W) )5)(32( yy L) )52)(3( yy
7. Factorise 5112 2 pp .
R) )5)(12( pp B) )5)(12( pp
F) )5)(1( pp W) )52)(1( pp
8. Given that ACC
B )1( , express C in terms of A and B.
L) AB
BC
R)
ABC
1
C) AB
ABC
N)
AB
ABC
9. Given that 25 xyx , express x in terms of y.
O) 16
42
yx B)
24
42
yx
I )
2
2
1
yx U)
2
4
2
yx
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
37 Curriculum Development Division
Ministry of Education Malaysia
SUGGESTED WEBSITES:
1. http://www.themathpage.com/alg/algebraic-expressions.htm
2. http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut11_si
mp.htm
3. http://www.helpalgebra.com/onlinebook/simplifyingalgebraicexpressions.htm
4. http://www.tutor.com.my/tutor/daily/eharian_06.asp?h=60104&e=PMR&S=MAT&ft=F
TN
FURTHER
EXPLORATION
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
38 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF A:
1. 9a – 5b
2. – 2m – 4n
3. 6k
4. 2p
5. xy
yx
5
15
6.
15
620 kh
7. c
ab
7
6
8. dc
dc
3
)4(4
9. 2z
x
10. 2
2
v
11. x
x
65
2
12. x
x
54
24
TEST YOURSELF B:
1. – 8n + 3 6. x + y
2. 3q + 2
1
7. 2e
3. – 12x2 + 18xy 8. mnmn 22
4. – 3b 9. fgf 22
5. p 10. 22 52 iihh
ANSWERS
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
39 Curriculum Development Division
Ministry of Education Malaysia
TEST YOURSELF C:
1. 3(p 2 – 5)
2. 2(x 2 – 3)
3. x(x – 4)
4. m(5m + 12)
5. p(q – 2)
6. 7m (1 + 2n)
7. (k + 12)(k – 12)
8. (2p – 1)(2p + 1)
9. 2(x – 3)(x + 3)
10. (3m + 13)(3m – 13)
11. (2x + 5)(x – 2)
12. (3x – 4)(x + 2)
13. (3p + 4)(p – 3)
14. (4p + 1)(p – 1)
15. (2x – 5)(x +1)
16. (2x – 5)(2x – 1)
17. (5p + 6)(p – 1)
18. (2x – 3)(x – 4)
19. (1 + 3r)(3p + k) 20. (2c – t)(2c – 3w)
TEST YOURSELF D:
1. (a) x = 2 – y (b)
2
3 yx
(c) x = 2y – 1
(d) x = 4 – y (e) 3
5 yx
(f) x = 3y – 4
2. (a) x = y2
(b) 24yx
(c) 236 yx
(d)
2
3
1
yx
2
2
1)e(
yx (f) 12 yx
3. (a) ax 3
(b) 1
1
y
yx
(c) fv
fvu
(d) 2
7qp
(e) 32
n
pm
(f) AB
BC
(g)
)1(2
x
xy (h)
2
24
T
lg
Basic Essential Additional Mathematics Skills (BEAM) Module
Unit 3: Algebraic Expressions and Algebraic Formulae
40 Curriculum Development Division
Ministry of Education Malaysia
ACTIVITIES
CROSSWORD PUZZLE
RIDDLES
RIDDLE 1
2 F
3
A
1
N
5
T
4
A
7
S
6
T
8
I
9
C
RIDDLE 2
2
W 1
O
3
N
5
D
4
E
7
R
6
F
9
U
8
L