BENGKEL MATHS TKC TINGKATAN 2 , 2022

Example 2

Solve the following equations :
a) 2x 13  7

5

b) 5 ( x - 4) = x + 16 x  22
Solution 2
a) 2x + 13 = 7 x 5
 11
= 35
2x = 35 -13

= 22

Example 2

Solve the following equations :
a) 2x 13  7

5

b) 5 ( x - 4) = x + 16
Solution
a) 5 x - 20 = x + 16

5 x - x = 16 + 20
4x = 36
x  36  9
4

6.2 Linear equations in two variables

• 6p + 4q = 9 x 2y  7
3

• 3h - 1 = 5k

• These equations have two variables and the power of each
variable is 1 .

• These equations are known as linear equations in two
variables

6.3 Simultaneous Linear equations in two
variables

• x + y = 7 and 2x + y = 12 are simulatenous linear
equations in two variables because both the linear
equations have two similar variables

The simultaneous linear equations in two variables can be
solved by using

a) graphical method
b) substitution method
c) elimination method

Example 3 Solution:

Solve the following
simultaneous linear
equations by using graphical
method .
x + y = 6 and 2x + y = 8
Solution :
From the graph drawn , the
point of intersection is (2,4)
Thus , the solution is x = 2
and y = 4

Example 4

Solve the following simultaneous linear equations by using
substitution method .
x - 3y = 7 and 5x + 2y = 1
Solution:
x - 3y = 7 .......(1)
5x + 2y = 1 ......(2)
x = 7 + 3y ........(3)

Example 4

Solve the following simultaneous linear equations by using
substitution method .

x - 3y = 7 and 5x + 2y = 1

Solution:

Substitute (3) in ( 2)

5(7 + 3y) + 2y = 1

35 + 15y + 2y =1

35 + 17y = 1 y   34  2
17y = 1 -35 2

= - 34

Example 4

Solve the following simultaneous linear equations by using
substitution method .
x - 3y = 7 and 5x + 2y = 1
Solution:
Substitute y = -2 into (3)
x = 7 + 3 (-2)

=7-6=1
Thus , x = 1 and y = -2

Example 5

Solve the following simultaneous linear equations by
using elimination method .
x + 2y = 9 and 3x - 2y = 15
Solution:
x + 2y = 9....(1) Identify the variable with the same
coefficient
3x - 2y = 15 ..(2)
(1) + (2) 4x + 0 = 24 Eliminate the variable y by
adding (1) and (2)

x=6

Example 5

Solve the following simultaneous linear equations by
using elimination method .

x + 2y = 9 and 3x - 2y = 15

Solution:

Substitute x = 6 into (1)

x + 2y = 9

6 + 2y = 9

2y = 9 - 6 = 3

y  3
2

Chapter 7

LINEAR
INEQUALITIES

Pn Sazazila Abdullah

7.2 Linear inequalities in one
variable

How do you solve problems involving linear inequalities ?

Linear inequality in one variable is an unequal
relationship between a number and a variable with power
of one .
For example , the algebraic inequalities such as

( The power of the variable x is 1)

3x  7

( The power of the variable y is 1)

y4 52y

are known as linear inequalities in one variable

7.2 Linear inequalities in one
variable

How do you solve problems involving linear inequalities ?

Solving a linear inequality in x is to find the values of x
that satisfy the inequality . The process of solving linear
inequalities is similar to process of solving linear
equations . However , we need to consider the direction
of the inequality symbol when solving linear inequalities .

7.2 Linear inequalities in one
variable

How do you solve problems involving linear inequalities ?

Example 4
Solve the following inequalities

a) x  2  6

Solution x  2  6
a) x  6  2
x8

7.2 Linear inequalities in one
variable

How do you solve problems involving linear inequalities ?

Example 4
Solve the following inequalities

b) 7x  28
Solution 7x  28
b) x  28
7
x4

7.2 Linear inequalities in one
variable

How do you solve problems involving linear inequalities ?

Example 4

Solve the following inequalities

c)  x  9 Solution  x  9
3 c) 3

x  9 (3) Reverse the
inequality
symbol

x  27

7.2 Linear inequalities in one

variable

How do you solve problems involving linear inequalities ?

Example 4

Solve the following inequalities

d) 7  4x  15 7  4x  15

Solution  4x  15  7 Reverse the
d)  4x  8 inequality
symbol
x 8
4

x  2

7.2 Linear inequalities in one
variable

How do you solve simultaneous linear inequalities ?

Based on the World Health Report , the daily
consumption of sugar is between 25 g and 37.5 g .
If m gram represents the quantity of daily sugar
consumption , then we can write

m  25 and m  37.5

7.2 Linear inequalities in one
variable

How do you solve simultaneous linear inequalities ?

m  25 m  37.5

The two inequalities are simultaneous linear
inequalities in one variable . Therefore the amount of
sugar , in grams an individual consumes , can be any
values between 25 and 37.5 , such as 27 , 32 , and 34.8.

These values are common values of the
simultaneous linear inequalities .

7.2 Linear inequalities in one
variable

How do you solve simultaneous linear inequalities ?

Example 5
Solve the following simultaneous linear inequalities .

a) 8x  5  5x 13 and 3x  4  9x  20

Solution 8x  5  5x 13 3x  4  9x  20
a) 8x  5x  13  8 and 3x  9x  20  4

3x  18  6x  24

x  6 x  4(reverse the symbol)

7.2 Linear inequalities in one
variable

How do you solve simultaneous linear inequalities ?

Example 5
Solve the following simultaneous linear inequalities .

a) 8x  5  5x 13 and 3x  4  9x  20

Solution The solution is  6  x  4
a)



Chapter 1

PATTERN AND
SEQUENCES

Pn Sazazila Abdullah

1. A pattern is a sequence or certain shapes in a list
numbers or objects .
2. A pattern of a sequence of numbers is a pattern that
follows a certain order .

3. The following are the patterns of various sets of

numbers .

a) Even numbers - can be divided equally with 2 .

i) 8,16,24,32,... ii) 44,54,64,74, ...

b) Odd numbers - cannot be divided equally with 2.

i) 1,3,5,7,9,... ii) 7,13,179,25,...

3. The following are the patterns of various sets of numbers .
https://youtu.be/nt2OlMAJj6

Example 1: State the pattern of the following sets of numbers . Hence ,
complete the following patterns .
a)4,20,100,500, .............. , .................
Answer :
Multiply 5 to the previous number .
2 500 , 12 500

Example 2
Complete the following Fibonacci Numbers .
a) 0 , 1, ........ , ......... , ........ , 5 , 8 , ........... , ...........
Answer
1 , 2 ,3 , 13 , 21

Sequence is a set of numbers or objects arranged
according to certain pattern .
Example 3
Determine whether the following numbers is a sequence .
16,13,10,7,4,...
Answer:

Chapter 2

FACTORISE AND
ALGEBRAIC FRACTIONS

Pn Sazazila Abdullah

CONTENTS

2.1 Expansion
2.2 Factorisation
2.2 Algebraic expressions and laws of
basic arithmetic operations

Word link

Expansion - kembangan
Algebraic expressions - ungkapan algebra
Factor - faktor
Highest common factor(HCF) - faktor sepunya terbesar
(FSTB)
Algebraic fraction - pecahan algebra
Perfect squares - kuasa dua sempurna
Cross multiplication - pendaraban silang
Numerator - pengangka

Word link

Denominator - penyebut
Lowest term - sebutan terendah
Lowest common multiple (LCM) -gandaan sepunya
terkecil (GSTK)

Why do you learn this chapter ?

- Algebra is mostly used in price comparison , buying
and selling process,measurement , etc

- Algebra is also used in certain fields of study like
Chemistry , Physics and Forensics .

Expansion

Expand the following expressions
a) 6 ( 3 + 4w )
b) 3r (r- 2s )
a
Solution

a) 18 + 24w

b) 3r2 - 6rs

Expansion

Expand the following expressions
a) ( y + 1 ) ( y - 3)
b) ( 4 + 3r ) ( 2 + r )
c) (3p + 2 )2

Solution
a) y2 - 3y + y - 3

y2 - 2y - 3
b) 8 + 4r + 6 r + 3r2

8 + 10r + 3r2

Expansion

Expand the following expressions
a) ( y + 1 ) ( y - 3)
b) ( 4 + 3r ) ( 2 + r )
c) (3p + 2 )2

Solution
c) (3p +2)(3p+2)

= 9p2 + 6p + 6 p + 4
= 9p2 + 12p + 4

FACTORISATION OF ALGEBRAIC EXPRESSIONS
1) Using HCF(Highest common factor).
2) Using difference of squares of two terms.
3) Using cross multiplication.
4) Using common factors involving 4 algebraic terms

FACTORISATION OF ALGEBRAIC EXPRESSIONS

1) Using HCF(Highest common factor)

8x 12x2 3x 15

 4x(2  3x)  3(x  5)

HCF HCF

FACTORISATION OF ALGEBRAIC EXPRESSIONS
1) Using HCF(Highest common factor)

7m  21m2
 7m(1 3m)

HCF

FACTORISATION OF ALGEBRAIC EXPRESSIONS
2) Using difference of squares of two terms

x 2  y 2  ( x  y )( x  y )

because ( x  y )( x  y )
 x 2  xy  xy  y 2
 x2  y2

FACTORISATION OF ALGEBRAIC EXPRESSIONS

2) Using difference of squares of two terms

b2 1  x2  4
 b 2  12  x2  22
 (b  1)( b  1)  ( x  2 )( x  2 )

FACTORISATION OF ALGEBRAIC EXPRESSIONS

2) Using difference of squares of two terms

9 m 2  100 3 y 2  147
 (3 m ) 2  10 2  3 ( y 2  49 )
 (3 m  10 )( 3 m  10 )  3( y  7 )( y  7 )

FACTORISATION OF ALGEBRAIC EXPRESSIONS

3) Using cross multiplication

x2  6x  9

answer  ( x  3)( x  3)

x - 3 -3x

x -3 -3x
x2 +9 -6x

FACTORISATION OF ALGEBRAIC EXPRESSIONS
3) Using cross multiplication

m 2  2m  8 answer  ( m  4 )( m  2 )

m -4 - 4m

m 2 + 2m
m 2 - 8 - 2m

FACTORISATION OF ALGEBRAIC EXPRESSIONS
4) Using common factors involving 4 algebraic terms

pq  qr  ps  rs
 q( p  r)  s( p  r)

same

answer  ( q  s )( p  r )

FACTORISATION OF ALGEBRAIC EXPRESSIONS
4) Using common factors involving 4 algebraic terms

2 px  6 qy  4 py  3 qx
 2 px  4 py  6 qy  3 qx
 2 p(x  2 y)  3q(x  2 y)

same

answer  ( 2 p  3 q )( x  2 y )

ALGEBRAIC EXPRESSIONS AND BASIC ARITHMETIC
OPERATIONS
Simplify each of the following

a) 4 a  3 a
55

Solution 4a  3a
55
 7a

5

ALGEBRAIC EXPRESSIONS AND BASIC ARITHMETIC

OPERATIONS
Simplify each of the following

b) x  2  x  5 x  2  (x  5)
5w 5w 5w 5w

Solution  x  2  x  5
5w

7
5w