antegrated mathematics sample lgebra 1 for...integrated mathematics algebra for secondary 1 ©...

INTEGRATED MATHEMATICS

ALGEBRAForSECONDARY

1ALEX LIMMaster Trainer atMavis Tutorial Centre

02_Integrated Maths Algebra S1_TP.pdf 1 9/11/2016 10:38:01 AM

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k As a seasoned secondary level Mathematics teacher who has taught for more than

10 years, I have been exposed to, and therefore am very familiar with, the changes

in the secondary mathematics curriculum over the years. It is evident that our

curriculum focuses on 4 fundamental and important themes: Algebra, Numbers,

Geometry and Statistics.

While schools differ in their pace and the sequence in which the topics are being

taught, their objective remains largely similar: to ensure their students are ready to take on the demands of the upper secondary mathematics syllabus and be ready for national examinations.

It is this same desire to help my students that this series of books is conceived.

I have written six books to cover all the topics in the entire lower secondary

mathematics course: Algebra, Numbers, Graphs & Statistics and Geometry,

Mensuration & Trigonometry.

The examples and questions are paced to help students understand concepts, recognize common steps, see through typical trick questions and apply mathematical knowledge taught and learned.

While the progressive practices ensure familiarity and exposure to a wide variety of question types, there are opportunities to exercise higher order thinking through the challenging questions provided in every topic.

Detailed solutions are included to guide and help students through critical thinking processes.

While my students will certainly benefit more from my personal teaching of these

questions, I am sure many others will also benefit from working through the same

questions systematically and studying the solutions diligently.

AlexLim

01 Pre&Cont_ALGEBRA Sec 1.indd 2 9/22/2017 10:30:55 AM

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AcknowledgmentsFirstly, I would like to thank Mr Anthony Ng, Mr Kelvin Ng and Mr Melvin Ng for their utmost support in the production of this book. Mr Kelvin Ng and Mr Melvin Ng are my mentors and teachers, and have advised me throughout the process.

The contributions of the Editorial and Production teams in Singapore Asia Publishers Pte Ltd and their help in getting this book published are gratefully appreciated.

I would also like to thank my colleagues at Mavis Tutorial Centre, especially the teachers of Secondary Mathematics Division, namely Cheryl Teo, Cecilia Chee, Jason Ong and Edmund Tan for their guidance and feedback in the development of the manuscript.

I want to especially thank Imran for his guidance and for selflessly sharing his experiences as a writer with me.

To my students, thank you for the invaluable feedback and suggestions as I was writing this book.

I want to thank my family members, especially my late mother, who had been my greatest pillar of support.

Finally, I would like to thank God for providing me the inspiration and courage to embark on this project.

About the author

Alex Lim received his Bachelor

of Science (Applied Mathematics)

degree from the National University of

Singapore in 2005. He is currently the

Mathematics Curriculum Specialist

for Mavis Tutorial Centre. He

spearheaded the development of the

Secondary Mathematics program in

Mavis Tutorial Centre and continues

to train and conduct workshops for

the teachers in the team. Thousands

of students have also benefitted from

his classes and he remains an active

advisor and instructor in related

mathematics seminars and camps.

Alex Lim currently teaches secondary

level mathematics in Mavis Tutorial

Centre at the following branches:

Parkway, Bedok and Tampines.


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Algebra

1 Introduction to Algebra ..............................................................................1

2 Expansion ................................................................................................15

3 Operations on Algebraic Expressions ......................................................25

4 Algebraic Fractions:

Addition and Subtraction of Algebraic Fractions ....................................35

5 Factorisation: Extracting Common Factors .............................................55

6 Solving Simple Linear Equations ............................................................69

7 Solving Fractional Equations ..................................................................87

8 Linear Inequalities .................................................................................105

9 Problem Solving ....................................................................................121

10 Number Patterns ....................................................................................143

solutions ................................................................................................ S1 - S36


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�Integrated Mathematics ALGEBRA for Secondary 1© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre

Introduction to Algebra1

Topic 1

Simplify the following. (a) q × 6 (c)(c) 10x ÷ 10y(b) r × r × 3 (d) 2p – 3x ÷ 7y

Solutions:(a) q × 6 = 6q

(b) r × r × 3 = 3r2

(c) 10x ÷ 10y = 10x ____ 10y

= x __ y

(d) 2p – 3x ÷ 7y = 2p – 3x ___ 7y

In algebra, we use letters to represent numbers.

A variable is a letter to represent a number.

An expression is a mathematical term or a sum or difference of mathematical terms that may use numbers, variables, or both.

Example 1

Algebraic notation

Notation Meaningpq p × q 4p 4 × p4(6) 4 × 6

p(q + r) p × (q + r)

p2 p × pp3q2 p × p × p × q × q

p __ q p ÷ q

p __ q + r _ s (p ÷ q) + (r ÷ s)

pq

___ r (p × q) ÷ r

Numbers are always written before variables.

It is a good practice to express it as a fraction.

02 Topic 1_ALGEBRA Sec 1.indd 1 9/22/2017 10:36:56 AM

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�Integrated Mathematics ALGEBRA for Secondary 1© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre Topic 1

� Simplify the following.

a 4x + 4 + 12 b – 6 – 5x – 10x

c 6p – (– 6p) + (– 4x) – 4x d 8 ___ 3p × 6p2

___ 4

e 4 ___ 6x ÷ 15 ___ 20 f (5p × q × 4) ÷ 8

g (– 4pq × p) ÷ 8 h (– 3p) × 2 1 __ 6 ÷ ( – 1 ___ 13 p )

i x3y2 × x2y ÷ ( 1 __ 3 × 3xy ) j 16x2y2 – 6x3y4 ÷ 2xy2

Let’s Practise Algebraic expressions


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Write an algebraic expression for each of the following.

(a) 14 more than 3x2.

(b) Subtract 29 from 6x.

(c) Product of 8 and 3x.

(d) Divide 3x by x2.

Solutions:

(a) 14 more than 314 more than 3x2 3x2 + 14

(b) Subtract 29 from 6x 6x – 29

(c) Product of 8 and 3x 8(38(3x) = 24x

(d) Divide 3x by x2 3x ____ x2 = 3 __

x

Here is a partial table of some common key words.

Translation of word sentences to algebraic expressions

Word sentence Algebraic expression

sum of ofof a and 3and 3 33b a + 3b

difference of 5 and 7 of 5 and 7of 5 and 7 and 7and 7 77a either (5 – 7a) or (7a – 5)

product of 6p andand q (6p)(q)

quotient of 4 of 4of 4 44x divided by 3divided by 3 33 4x ___ 3

square of ofof a a2

cube of ofof b b3

square root of x √__

x

cube root of y 3 √__

y

Example 2


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� Write, in its simplest form, an algebraic expression for each of the following.

a quotient of q divided by t2 b product of m and n

c difference of 3 and q times 8 d 4q plus 6q

e 9k minus 5h f subtract 4p from the sum of 6q and r

g add 6y to the quotient of 3x divided h multiply the cube of m to the square by y of n

i divide the sum of a and the cube j square the quotient of h divided root of b by c by j

k sum of x cubed, y squared and l cube of m divided by 3n the product of p and q

m average of 6c and 2d n three more than half of z

Let’s Practise Translation of word sentences to algebraic expressions


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Evalulating algebraic expressions

(a) When a = 8 and b = 6, evaluate the following.

(i) 5ab – (7a)2 (ii) 2a – 8b ___ a (iii) (a – b)3

_______ ab

(b) Given that x = −1, y = 4 and z = 6, find the value of each of the following.

(i) 5x – y + 2z (ii) 1 __ 2 (2y + 3z) – x

(iii) (xz)3 + 2y2

(c) If p = −1, q = 9, r = −6 and s = 1 __ 3 , evaluate the following.

(i) q(3r + 6s) (ii) pr

__qs (iii) r – 3s _______4p + 5q

Solutions:(a) (i) 5ab – (7a)2 = 5(8)(6) – (7(8))2

= 240 – (56)2

= 240 – 3136 = – 2896

(ii) 2a – 8b ___ a = 2(8) – 8(6)

____ 8 = 16 – 6 = 10

(iii) (a – b)3

_______ ab = (8 – 6)3

_______ (8)(6)

= (2)3

____ 48

= 8 ___ 48

= 1 __ 6

To evaluate an expression means we replace each of the variable in an expression with a number, and leave the answer in its simplest form.

Example 3


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(b) (i) 5x – y + 2z = 5(–1) – 4 + 2(6) = –5 – 4 + 12 = 3

(ii) 1 __ 2 (2y + 3z) – x = 1 __ 2 (2(4) + 3(6)) – (–1)

= 1 __ 2 (8 + 18) + 1

= 1 __ 2 (26) + 1

= 13 + 1 = 14

(iii) (xz)3 + 2y2 = ((–1)(6))3 + 2(4)2

= (– 6)3 + 2(16) = – 216 + 32 = – 184

(c) (i) q(3r + 6s) = 9 ( 3(–6) + 6 ( 1 __ 3 ) ) = 9(–18 + 2) = (9)(– 16) = – 144

(ii) pr

___ qs =

(–1)(–6)

_________

9 ( 1 __ 3 )

= 1 __ 3

(iii) r – 3s _______ 4p + 5q =

–6 – 3 ( 1 __ 3 )

_______________ 4(–1) + 5(9)

= –6 – 1 ___________ – 4 + 45

= – 7 ___ 41

Example 3


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� Given that w = 9, x = 5, y = 7 and z = 8, evaluate the following.

a z(y – x) b 4w – x + z

c 6x(z – 3y) d (3x + y)(x – y + 2)

e x3yz f z – y

____ w2

g xy z – 5 h y __ 2 – 3 __ z

_____ w – x

i x __ z – w __ y j (z – x)2

______ y

Let’s Practise Evaluating algebraic expressions


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� Write down an algebraic expression for the following statement: five times of a divided by the sum of b and c.

� Express the following word statements algebraically: (a) subtract 4x from the quotient of 5y divided by z; (b) add the product of m and n to the square of a.

� Express the following word statements algebraically: (a) add 42 to the product of x and y; (b) subtract 9z from the square of the quotient of m divided by n.

� Write the following as algebraic expressions, giving your answers in their simplest form.

(a) Divide 9 by the product of w2 and s3.

(b) Multiply 1 __ 5 by the square of 5x.

Let’s Practise Mixed practice


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� During a book fair, a book costs $5 and a magazine costs $2. 4x books and (x + 30) magazines were sold on a particular day. Express, in terms of x,

(a) the amount collected from selling the books. (b) the total amount, in its simplest form, that was collected.

� Fiona bought x boxes of oranges. There were 10 oranges in each box. (a) Write down an expression, in terms of x, for the number of oranges she bought. (b) When the boxes were opened, she found that p oranges were spoilt. She decided

to distribute the good oranges into 20 containers. Write an expression, in terms of x and p, for the number of oranges in each container.

�0 ABC is an isosceles triangle with AB = (3x + 4) cm. Given that the perimeter and area of triangle ABC are 30 cm and 40 cm2 respectively, express, in terms of x, the length of

(a) BC, and P

(3x + 4) cm

B CM

(b) AM.

�� The average score of two classes A and B is (3x + 40) marks. The average score of class A of 40 students is (2x + 31) marks. Find the average score, in terms of x, of class B of 20 students.


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�0Integrated Mathematics ALGEBRA for Secondary 1© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre Topic 1

�� x pieces of sushi cost 95 cents. Find an expression for the number of sushi that can be bought for $y.

�� A carton of drinks costs x cents. Alex buys a number of cartons and they cost y dollars altogether. Express, in terms of x and y, for the number of cartons that Alex buys.

�� On valentine’s day, a shop sold x cents for a dozen of roses. Daniel bought y roses. Find an expression, in terms of x and y, for the amount of money, in dollars, that he spent on buying the roses.

�� Given that a = 4 and b = −3, evaluate (a) 1 __ a + 1 __ b , (b) √

_____________

a2 – b2 + 2 _____________ 4 .


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��Integrated Mathematics ALGEBRA for Secondary 1© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre Topic 1

�� Given that x = – b – √

___________

b2 – 4ac

_____________ 2a , find the value of x when a = −3, b = 5 and c = 4,

giving your answer correct to 3 significant figures.

�� Given that x = 1 __ 4 , y = 3 and z = 2, find the value of

(a) (x – z)2,

(b) xy ____ z – x .

�� Given that x = 3, y = 2.5 and z = −3, evaluate 4x4y – 3yz

_________z2 .


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�� Given that x = −4, y = 4 __ 5 and z = −3, evaluate (a) x3z2, (b) 5(x + z) – 5xy.

�0 Given that p = 3, q = −4 and r = −1, find the value of

(a) 4p – 3q + r, (b)

4r2 – 3q2

________pq + r , (c)

r __ q + q __ p –

p __ r .

�� Find the value of −8a – 5b + 4c when a = −4, b = −2 and c = 6.


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�� Given that a = 2, b = −2 and c = 3, evaluate b2 + 3ac – 4b.

�� Given that a = 40, b = −6 and c = 8, evaluate a – 5b2 ________ (a + bc)3 .

�� Find the value of a – 6 _____a + 8 – 4a2 when a = −2.

�� Evaluate 4x3yz

______ 3y – 4z – yz

__ x when x = 3, y = −4 and z = −5.


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�� Evaluate 1 + 1 ____________ 1+ 1 _________

1 + 1 ______ 1+ √__x , when x = 19 ___ 16 .

�� Eric boughtEric bought x boxes of matches for $50. He decided to sell each box at a profit of y cents. Find an expression, without simplifying, the selling price of each box of matches.

�� A bus was travelling at x km/h for the first y hours. The bus stopped for half an hour before proceeding on with the second part of the journey. For the next z hours, its speed was (x – 20) km/h.

(a) Find its average speed, in terms of x, y and z. (b) Given that x = 85, y = 2 and z = 3, find its average speed.

�� Evaluate a __ b + c __ d

______ a __ b – c __ d

when a = 91 __ 2 , b = 11 __ 5 , c = 51 __ 4 and d = 17 __ 8 .

�0 Express the statement below algebraically: the square of the difference between z and w subtracted from the cube root of the sum

of x and y.

Try These!


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S�Integrated Mathematics ALGEBRA for Secondary 1© Singapore Asia Publishers Pte Ltd & Mavis Tutorial Centre Topic 1

Topic 1�. (a) 4x + 4 + 12 = 4x + 16 (b) –6 – 5x – 10x = – 6 – 15x (c) 6p – (–6p) + (–4x) – 4x = 12p – 8x

(d) 8 ___ 3p × 6p2

___ 4 = 8 × 6p2

______ 3p × 4

= 4p

2

11

2

(e) 4 ___ 6x ÷ 15 ___ 20 = 4 ___ 6x × 20 ___ 15

= 8 ___ 9x

2 4

3 3

(f) (5p × q × 4) ÷ 8 = 20pq

_____ 8

= 5pq

____ 2

(g) (–4pq × p) ÷ 8 = –4p2q

______ 8

= – p2q

___ 2

(h) (–3p) × 2 1 __ 6 ÷ ( – 1 ___ 13 ) p

= (–3p) × 13 ___ 6 ÷ ( – p ___ 13 )

= –3p × 13 ___ 6 × ( – 13 ___ p ) = 169 ____ 2

= 84 1 __ 2

(i) x3y2 × x2y ÷ ( 1 __ 3 × 3xy ) = x3 + 2y2 + 1 ÷ (xy)

= x5y3

____ xy

= x4y2

(j) 16x2y2 – 6x3y4 ÷ 2xy2

= 16x2y2 – 3x2y2

= 13x2y2

2. (a) q ÷ t2 = q __ t2

(b) mn (c) 3 – q × 8 = 3 – 8q or q × 8 – 3 = 8q – 3

(d) 4q + 6q = 10q

(e) 9k – 5h

(f) (6q + r) – 4p = 6q + r – 4p

(g) 6y + 3x ___ y

(h) m3 × n2 = m3n2

(i) (a + 3 √__

b ) ÷ c = a + 3 √__

b ______ c

(j) ( h __ j ) 2 = h2 __ j2

(k) x3 + y2 + (p × q) = x3 + y2 + pq

(l) m3 ÷ 3n = m3 ___ 3n

(m) 6c + 2d _______ 2 = 2(3c + d)

________ 2

= 3c + d

(n) 3 + 1 __ 2 × z = 3 + z __ 2

3. (a) z(y – x) = 8(7 – 5) = 8(2) = 16

(b) 4w – x + z = 4(9) – 5 + 8 = 36 – 5 + 8 = 39

(c) 6x(z – 3y) = 6(5)(8 – 3(7)) = 30(8 – 21) = 30(–13) = –390

(d) (3x + y)(x – y + 2) = (3(5) + 7)(5 – 7 + 2) = (15 + 7)(0) = 0

(e) x3yz = (5)3(7)(8) = 125(56) = 7000

(f) z – y

____ w2 = 8 – 7 _____ 92

= 1 ___ 81

(g) xyz – 5 = 5(7)8 – 5

= 5(7)3

= 5(343) = 1715

12 Solutions_ALGEBRA Sec 1.indd 1 9/22/2017 11:45:20 AM

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