Chapter12 Algebra

Algebra

12Patterns and algebra

Scientists put a satellite into space; a business person orders materials for a factory; extra dinner guests mean that ingredients in a recipe must be increased; and engineers build a bridge. To understand these ideas, formulas and patterns, you must know how algebra works.

use concrete materials, such as cups and counters, to model variables and algebraic expressions

translate from words to algebraic symbols use algebraic abbreviations and recognise equivalent algebraic

expressions add and subtract like terms to simplify algebraic expressions recognise the role of grouping symbols, and expand expressions

involving them simplify algebraic expressions involving addition, subtraction,

multiplication and expanding substitute into algebraic expressions.

algebraic expression An expression that describes a quantity, using variables and numerals, for example 4m + 2.

like terms Terms with exactly the same variables, for example 6ab and 3ab. simplify (an algebraic expression) To write an algebraic expression in

the shortest way. expand To rewrite an algebraic expression without grouping symbols. substitute To replace a variable with a numeral. evaluate To find the value of an algebraic expression after substitution.

Can you think of a quick way of calculating 7 102? What about 8 99? What about 6 (2a + 5)?

In this chapter you will:

Wordbank

Think!

ALGEBRA 389

CHAPTER 12

390

NEW CENTURY MATHS 7

Algebraic expressions

In Chapter 6, we used letters of the alphabet to stand for numbers when describing number patterns. In this chapter, we will explore variables further.

1 If = 8, nd the value of:a + 3 b 4 + c 6d + e 3 f 7g 2 h 32 i 9 + 12j 2 + 5 k 11 + 1 l 8 3

2 If = 20, nd the value of:a 5 b 3 c 4d + 9 e f 2 g 60 h 25 + 6 i + 11 9j 6 + 2 k 100 5 l 10 2

3 Find the value of:a 3 (4) b 2 5 c 8 (2)d 7 + 5 e 4 10 f 3 (9)g 4 (5) h 1 (6) i 2 + (3)

Start up

Worksheet 12-01

Brainstarters 12

Example 1

An envelope contains an unknown number of paperclips. Find an expression to represent the number of paperclips in this diagram:

SolutionLet y stand for the number of paperclips in the envelope.So: 1 envelope of paperclips + 3 paperclips = y + 3 paperclips

Each cup holds the same unknown quantity of marbles. Let k stand for the number of marbles in each cup. Find an expression to represent the number of marbles in this diagram.

Solution3 cups of marbles + 5 marbles = 3 lots of k + 5

= 3 k + 5= 3k + 5

?

Example 2

k marbles k marblesk marbles

ALGEBRA 391 CHAPTER 12

The expression 3k + 5 is called an algebraic expression. Unlike an algebraic rule or formula, such as y = 2x 9, it does not contain an equals sign. An algebraic expression uses variables and numerals to describe a quantity. The algebraic expression 3k + 5 describes the total number of marbles in the diagram on the previous page.

If m stands for the number of coins in an envelope and p stands for the number of coins in a cup, write the algebraic expression for the number of coins in this diagram.

Solutionm + 2p

Example 3

m coins p coins p coins

1 If y stands for the number of paperclips in an envelope, draw what is represented by each of the following algebraic expressions.a y + 1 b 2 lots of y c 4 + yd 1 + y + 2 e y + 5 f y + yg y + 2 + y + 1 h 2y + 3 i 4 + 2yj y + 1 + 2y + 4 k y + 2 + y + 2 l 3 + 2y + 1

2 Write an algebraic expression for what is shown by each of the following diagrams. Let k stand for the number of marbles in a cup and stand for one marble.

a b

c d

e f

g h

i j

Exercise 12-01Example 1

Example 2

392 NEW CENTURY MATHS 7

3 Write an algebraic expression for the number of paperclips in each of the following diagrams, using y to stand for the number of paperclips in an envelope.

4 Write an algebraic expression for what is shown by each of the following diagrams, using m to stand for the number of coins in any envelope and p to stand for the number of coins in a cup.

5 a Which of the expressions you wrote for Question 4 mean the same amount?b Which expression in each group you listed do you think is the simplest? Why?

a b

c d

e f

g h

i j

a b

c d

e f

g h

i j

k l

m n

Example 3


Algebraic abbreviationsMathematicians prefer to write expressions as simply as they can.For instance:5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 9 5and:2 2 2 2 2 = 25 In algebra, we try to write expressions as simply as possible too. Already we know that 2m is a simpler way of writing 2 m, and is a simpler version of a 2.In more complicated expressions, such as 2m + 3w 4k, we call each part of the expression a term.

a

2---

algebraic expressions

three terms

2m 3w 4k+

Example 4

1 Simplify each of the following expressions:a 3 + 3 + 3 + 3 + 3b m + m + m + m + mc 13K + 5K 8KSolutiona 3 + 3 + 3 + 3 + 3 can be written as 5 3b m + m + m + m + m is the same as 1m + 1m + 1m + 1m + 1m and can be rewritten

as 5m.c 13K + 5K 8K can be rewritten as 10K (because 13 + 5 8 = 10)

2 Simplify each of the following expressions:a 3 m b m wc 5 B A d 8 a n b me 3 m 2 w f k kg m m m h 2 y ySolutionSimplifying an algebraic expression means writing the expression in the shortest way.a 3 m = 3mb m w = mw (or wm since multiplication can be done in any order)c 5 B A = 5AB (or 5BA)Note that we always write the number at the start of the term, usually followed by the variables in alphabetical order.d 8 a n b m = 8abmne 3 m 2 w = 6mwNote that we multiply the numbers rst, then write the variables in alphabetical order.f k k = k2g m m m = m3 h 2 y y = 2y2


Example 5

Write each of these expressions in expanded form:a 3AB b 47abcdSolutiona 3AB = 3 A B b 47abcd = 47 a b c dNote that expanding is the opposite of simplifying.

1 How many terms are there in each of the following expressions?a 2k + 1 b m + 3p + mc 4x d 2w + 7y + we 4 + 7n f 4m 2k 7g 9 h 3p + 2qi x + y2 + z j 4pq + 6pq + 7pk 6k 3k + 2m m l 6ef + 4em 7m2 + 3n 2m2 + 6n n 12cde 5defo j 2 p 12f 4g + 3f + 7g + 10f 9g

2 Simplify each of the following expressions:a 5 + 5 + 5 + 5 b 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2c a + a + a + a + a + a d e + e + e + e e w + w f 5m + 2m + 6mg 8h + 3h + 2h + h h 5m 2mi 23k + 17k j 12w 4w 5wk 7p + 5p 2p l 6n + 3n 10nm a + a + a + b + b n 3d 2d + 5do m + 2m 3m p 12q 4q + q

3 Simplify:a 4 w b 8 a bc 3 c a d d a c b 5e 2 w 3 h f g 2 f 3g 2 w c 3 m h 2 m s 4 h bi the product of m and w j 2 y 3 r q 5 p dk a a l d d dm 4 f f n g 2 go 4 h h h p l 4 l 2q 2 m 4 m r 6 r r 6 rs c c e e e t 5 j 7 j ku y z y z y v 4 n 3 p n 2 p n

4 Computer Algebra Software can be used to simplify expressions that you make up yourself. Use this link to go to an activity that will show you how to use TI InterActive! to simplify algebraic expressions.

5 What confusion would be caused by removing the multiplication sign from 5 3?6 What does the expression AB mean?

Exercise 12-02

CAS 12-01

Simplifying expressions

Example 4


From words to algebraic expressionsThis section will improve our ability to change information written in words into algebraic expressions. For example, an algebraic expression for the sum of A and B is A + B.

7 Write each of these expressions in expanded form:a AB b 2mns c 188ABC d 3mnabcdefe 4m f m g x2 h q3i 7d2 j 5s3 k x2y3 l 9m3n2

8 What does the word abbreviation mean? Use it in a sentence to show its meaning. Is this word used the same way in mathematics?

Example 5

Worksheet12-02

Whats the expression?Example 6

If A represents any number, write an expression for:a three times that number b three less than that numberc the next consecutive number d that number multiplied by itselfe the square root of that number f one-third of that numberSolutiona 3A b A 3c A + 1 d A2

e f A or or A 3

Write an expression for:a the sum of m and 5 b the double of mc m increased by 10 d the difference between N and Qe the product of m and 2 f the quotient of m and 2Solutiona m + 5 b 2 m or 2mc m + 10 d N Qe 2m f m 2 or or m

A 13---A3---

Example 7

m

2----12---

Skillsheet 12-01

Algebraic expressions

1 Write an expression for each of the following. Use N to represent any number.a double the number b half the numberc triple the number d one-quarter the numbere one-tenth the number f the next consecutive numberg 5 times the number h the sum of the number and 21i the difference between the number and 10j 2 more than the number k the number increased by 3l the number times itself m the square root of the number



2 Imagine that you must repeat Question 1 using the pronumeral A, instead of N. What difference would this make to your answers? Does it matter which letter of the alphabet you choose to use?

3 If A, B and C represent any three numbers, write an expression for:a the sum of A and Bb the sum of all three numbers A, B and Cc the difference between B and C, where B is greater than Cd the product of A and Ce the product of all three numbers A, B and Cf the quotient of A and Bg the sum of A and B, divided by Ch the quotient of C and B.

4 Write an expression for:a the number of students in a class if there are B boys and G girlsb the number of pies needed at a party if there are N children and each child can eat two

piesc the number of children remaining in class if X leave for the library out of a total group

of T d the amount of money earned by selling N cakes at the school fete, where each cake is

priced at $2e the cost of each lm ticket where the total cost is $M and there are three people going

to the lmf the total cost of buying A cans of lemonade and B ice-creams, where each can costs

$1 and each ice-cream costs $2.

5 Write an expression for:a the sum of 3 and A b 3 less than Bc 5 added to C d 8 increased by De 3 taken away from E f X decreased by Fg the sum of A, B and W h m increased by mi R increased by 2 j A decreased by B.

Example 7


Like termsEarlier we did some concrete exercises involving algebra. In Example 3 and Exercise 12-01, we used p to represent the number of coins in each cup and m to represent the number of coins in each envelope .

We found that these algebraic expressions were the same:a

p + m + p + m = 2p + 2m

b

m + p + m + p + m = 3m + 2pThese are examples of collecting like terms. We can add together things that are the same. For subtraction:

3m + 2p 2m = m + 2pThis is also collecting like terms. We can subtract things that are the same.These algebraic expressions represent the total number of coins in the envelopes and cups.

and

and

=

1 Match each expression in the left column with the correct expression in the right column.

m + p + m

a

p + m + m + p

b

p + m + p + m + p

c

p + 2 + p + 1

d

2m + 2 + m + 1

e

2 + p + 2 + 2p

f

2m + 2p

A

2p + 3

B

3m + 3

C

p + 2m

D

3p + 4

E

3p + 2m

F

Exercise 12-04


2 Match each expression in the left column with the correct expression in the right column.

3 If p represents the number of coins in a cup and m represents the number of coins in an envelope, write the simplest algebraic expression you can for each of these:

m + p m

a

m + 4 2

b

2m + p p

c

2p + 6 4

d

2m + 2p 2m

e

2p + 3m m

f

2m

A

2p + 2

B

2p

C

p

D

2p + 2m

E

m + 2

F

p + m + p

a

m + 2p p

b

3m + p m + 2p

c

2p + 2 + p + 3

d

2m + 4 m 2

e

p + m + 2 + p + m 1

f

+

+


Simplifying algebraic expressions with like termsThe following pairs of terms are called like terms because each set of terms has the same pronumeral(s).

3x and 5x 3mw and 2mw 12m and 32m3m and m 5ab and 2ab xyz and 2yzx

The following are not like terms because each set of terms has different variables or pronumerals.

3x and 5m 8wm and 2wq 5ab and 2abc 2p and 3p2

4 Simplify these expressions:a 3m + 2m b 6p pc m + m m d 2m + 3p + 2pe 3m + 4p + m + 2p f 2m + p + m + 3pg 6m + 4p 2m h 7p + 5m 3mi 12m + 6p 3m 2p j 6p 4p + 3m m

m + 2p + 3 2p

g

3p + 2m m p

h

p + m + p + m + 4 2m 3

i

3m + 4p m 2p

j

Worksheet12-03

Collecting like terms

Skillsheet 12-02

Algebra using diagrams

We can only add or subtract like terms (terms which have exactly the same variables).

Example 8

Simplify these expressions:a 3m + 5m b 3ab + 2bac 5x + x d 42mw 17wmSolutionsa 3m + 5m b 3ab + 2ba

= 8m (3m and 5m are like terms) = 5ab (ab and ba are like terms)c 5x + x d 42mw 17wm

= 6x (x is really 1x) = 25mw (mw and wm are like terms)


Example 9

Simplify these expressions by adding or subtracting like terms:a 3m + 5w + 2m + 3w b 24 + 5ab 14 2baSolutiona 3m + 5w + 2m + 3w

b 24 + 5ab 14 2ba

3m + 2m+ 5w + 3w 5m 8w+=

like terms5w + 3w

like terms3m + 2m

24 14+ 5ab 2ba 10 3ab+=

like terms24 14

like terms5ab 2ba

1 Find the like terms in each of these sets:a 2my, 3x, 6am, 16x, 4mb b 2mw, 3km, 4w, 5mw, 6m, 7awc 8k, 3x, 2w, 12g, 23w d 2p, 5mq, 5p, 7q, 7me 2b, 2k, 2m, 2g, 3k f 2ab, 2a, 2b, 2m, 3bag x, 3x2, y, x2 h 4mn, 3m, 4, 2nm, mn, 2ni x2y, 2x, 3y, 4x2y j 7, 2a, 4b, 5p, 9, d, 2k p2, q2, 3p, 2q2, pq, 7q l c2d, cd2, 3c, 2d, 5cd2

2 Simplify each of these expressions by adding like terms:a 2m + 5m b 4k + 7kc 2ab + 5ab d 5mn + 2nme xy + xy f 3abc + 4abc + 2bacg 4ab + 3ab + 2ab h 12mn + 6mn + mn + nmi 6cde + 5dec + 3edc j 6mp + mp + 9mpk 3x2 + 2x2 + x2 l ef 2 + 4ef 2

3 Simplify each of these expressions by subtracting like terms:a 5m 3m b 8d 3dc 12mk 7mk d 45abc 12abce 45fg 13fg f 48mn 29mng 30cd 12cd h 12xy 7yx xyi 8de 12de j 6k2 3k2k 4w2 w2 l 5xy2 2xy2 xy2

Exercise 12-05

SkillBuilder 8-06

Like terms

Example 8


4 Simplify each of these by adding or subtracting like terms:a 5m + 2m 3m b 12a 4a 5ac 3x + 5x + 7x d 12f + 15f 18fe 8s2 + 7s2 6s2 f 5mn + 6mn 4nmg 3pq 2pq + pq h 11abc + 3abc 5abci 8p 3p 5p j 3d2 8d2 + 2d2k 4x + 7x 3x l 8de + 12de 5de + de

5 Simplify:a 3x + 4 + 5x + 6 b 4m 6 + 4m + 10c 2mn + 3f + 5mn + 7f d 10k 4 6k + 12e 23xy + 23ab + 17xy 17ab f 15r + 15 15r rg 2p + 3 3p + 8 h 5k 11 8k + 6i 13 + 2ab 6ab 1 j 5ab + 2p 7ab 2pk 3p + 12m 6p 4m l 10de + 3d + 5de 2dm 4y2 + 2y + y2 + 5y n 7m2 + 2n2 4m2 + 3n2o 9g2 12 + 2g2 + 7 p x2 3x + 4x2 7x

6 Simplify:a 6m + 3m b k + 2 + 3kc 12mx + 4xm d 2d + 9 + 3de 2f + 3g + 4f + 7g f 4y + 3x + 2x + 8yg 11p p h 7 + 5q 3i 8abc 4abc abc j 4k + 5 + 17 + 11kk 3a + 2b + 6a + 7b l x2 + 3x2 + 9x2 m 2y + 1 + 2y + 2 + y + 3 n 7r + 6s 2r 4so 12 + 6x 4x 6 p 12a a 2a 6aq 2m2 + 4m2 6 r 12pq pq 7qps 20x + 3y2 7x 2x t 12ab + 3cd 4ab + 7cd

7 Computer Algebra Software can simplify expressions by adding and subtracting like terms. The accompanying activity will teach you how to use TI InterActive! to do this.

Example 9

CAS 12-02

Collecting like terms

BrahmaguptaMuch of our ancient mathematics was passed down to us through the Arabs over one thousand years ago. The Arabs first met these ideas studying Greek and Hindu writings. One of the most famous Indian astronomers and mathematicians was Brahmagupta (AD 598665).Brahmagupta wrote all his discoveries about mathematics and astronomy in a long poem, called Brahma-sphuta-siddhanta (The Opening of the Universe) in AD 628. It was a very long poem, with 25 chapters. Two chapters were devoted to mathematics (algebra, geometrical proofs, areas of triangles, and volumes of solids). The other 23 chapters dealt with astronomy.

Find out what type of numbers Brahmagupta introduced into mathematics.

Just for the record


Multiplying algebraic terms

When multiplying algebraic terms, multiply the numbers rst, then multiply the variables. Variables are usually written in alphabetical order.

Example 10

Simplify each of the following:a 2 3a b 4x (2) c 7k 3k 2Solutiona 2 3a = 2 3 a

= 6ab 4x (2) = 4 x (2)

= 4 (2) x= 8x

c 7k 3k 2 = 7 3 2 k k= 42k2

1 Simplify each of the following:a 3 2y b 7 8qc 4t 3 d 3k 9e 4 2 d f 4a 3bg 12c 4d h 6p 9qi 3m 2m j 5k 11kk 12f 9f l 6g 3 4hm 10 2m 6n n 4a 3 7ao 4x 7x 3 p 5m 2 3m

2 Simplify:a 2 4a 9b b 2mn 4c x xy y d 2j 3k 7e 5rst 2rs f 3ab 6cdg p 9 h m n 2mi hjk hj jk j 4q2 6p2 k 2xy 6xy 9yx l 2m 3n mn

3 Simplify each of the following:a 6 3x b 2 3kc 1 p d j 4e 6 (m) f 4 (3y)g 2 (2n) h k (5)i 1 (r) j 8x (7)k s (s) l 3d (d)



Expanding an expressionLet m represent the number of paperclips in an envelope and represent one paperclip.

The number of paperclips in each envelope is unknown but it is the same number for each envelope. So:a 2 lots of (m + 3)

= 2(m + 3) = 2m + 6

b 3 lots of (2m + 1)= 3(2m + 1) = 6m + 3

What is happening? What do the parentheses (brackets) mean?The number outside is multiplying each term inside the parentheses:

2(m + 3) = 2 m + 2 3= 2m + 6

3(2m + 1) = 3 2m + 3 1= 6m + 3

This is called expanding the expression, that is, removing the grouping symbols by multiplying each term separately, and simplifying.An example of expanding being used is when nding the perimeter of a rectangle of length Y and breadth X.It can be written in two ways.Method 1: The perimeter can be found by adding together

all the sides:P = X + X + Y + YP = 2X + 2Y

Method 2: The perimeter can be found by doubling the sum of the length and height:P = 2(X + Y)

Both answers must be equal, so:2(X + Y) = 2X + 2Y.

These are two ways of expressing the same thing.

Worksheet12-04

Expandominoes

Skillsheet 12-02

Algebra using diagrams

m m m

m=

=

mmm

m m m m

m

m

m

m

m

X X

Y

Y

2(X + Y) = 2X + 2Yexpanding


Example 11

Expand each of the following expressions:a 2(x + 3) b 3(x 4) c 5(2x m)Solutiona 2(x + 3)

2(x + 3) = 2 x + 2 3= 2x + 6

b 3(x 4)3(x 4) = 3 x + 3 (4)

= 3x 12c 5(2x m)

5(2x m) = 5 2x + 5 (m)= 10x 5m

Expand:a 2(x 3) b (5 4x)Solutiona 2(x 3)

2(x 3) = 2 x 2 (3)= 2x + 6

b (5 4x)1(5 4x) = 1 5 1 (4x) (5 4x) is the same as 1(5 4x)

= 5 + 4x

Expand:a m(c + 1) b f(2m 3r)Solutiona m(c + 1)

m(c + 1) = m c + m 1= cm + m

b f(2m 3r)f(2m 3r) = f 2m + f (3r)

= 2fm 3fr

Example 12

Example 13

Animation 12-01

Expanding brackets

1 Expand each of the following expressions:a 3(a + 2) b 2(h + 2) c 2(m + 3)d 4(x + 6) e 4(2m + 3) f 7(a + b)g 12(2p + 5) h 5(a + 2) i 6(3x + 4)j 12(2m + n) k 10(4p + 2q) l 3(2a + 4b)

Exercise 12-07

SkillBuilder 8-10

Using brackets

Example 11

ALGEBRA

405

CHAPTER 12

Expanding and simplifying

2 Expand:a 4(x 2) b 3(m 7) c 8(k 3) d 5(y 5)e 2(m p) f 7(3f 2g) g 4(3m 5) h 5(3m 3)i 6(3x 4) j 3(1 k) k 6(2 3p) l 10(2 2m)

3 Expand:a 3(x + 2) b 7(p + 1) c 2(m 3) d 5(k 4)e 4(6 y) f 9(a + 4) g (k + 3) h (m 2)i (6 2x) j 3(4m + 5) k 5(3y 6) l (4 7x)

4 Expand:a m(n + s) b x(y 2) c k(m + 12) d p(q 4)e y(2x 3) f f(4k 7) g b(2a 6) h n(2p + 7)i 2m(3k + 1) j 3k(5x 2) k 4m(1 m) l xy(x + 2)m 5w(7x 2m) n 2c(5b 3a) o 3p(6q + 2r) p 4j(5k + 7l)

5 Expand:a x(x + 4) b a(2a + 5) c n(n + 2) d 2y(y 4)e r(2r 1) f p(p + q) g d(d + e) h w(8 w)i 3v(v + 7) j pq(p + q) k (m n) l 3k(k 1)m mn(2m + 3n) n (2k + 4j) o ab(a + b + c) p 4m(m + 2n + 3p)

6 Computer Algebra Software can simplify expressions by expanding brackets. This link will take you to an activity that teaches you how to use TI InterActive! to do this. CAS 12-03

Expanding expressions

Example 12

Example 13

Example 14

Expand and simplify:a 2x + 3(x + y) b 2a (4 + a)Solutiona 2x + 3(x + y) b 2a (4 + a)

= 2x + 3 x + 3 y = 2a 1(4 + a)= 2x + 3x + 3y = 2a 1 4 1 a= 5x + 3y = 2a 4 a

= a 4

Expand and simplify:a 2(m + 1) + 3(m 4) b 6(p + 2) 3(p 4)Solutiona 2(m + 1) + 3(m 4) b 6(p + 2) 3(p 4)

= 2 m + 2 1 + 3 m + 3 (4) = 6 p + 6 2 3 p 3 (4)= 2m + 2 + 3m 12 = 6p + 12 3p + 12= 5m 10 = 3p + 24

Example 15

12_NCM7_SB_TXT.fm Page 405 Friday, September 26, 2003 3:42 PM


1 Expand each of the following and simplify by collecting like terms:a 3p + 2(p + q) b 7(a + 2) + 12c 3(k 5) + 7 d 12 + 2(n 2)e 6k + 11(k + 2) f 12(v + 4) + 4vg 16(t 2) 8t h 5a 2(a + 1)i 7x 3(x + 2) j 13x 3(x + 1)k 2(x + 1) 5 l a (2 + a)m b (b 4) n 3(x + 5) 2x + 7o 3(4 2x) 10 p 5(3 2x) + 5x 3

2 Expand and simplify:a 2(x + 2) + 4(x + 1) b 3(r + 3) + 2(r 4)c m(m + 1) + 3(m + 1) d 2(q 7) + 5(q + 9)e 4(d + e) + 3(d e) f 4(x + 1) 2(x 2)g 3(x 2) 3(x 4) h 12(x 3) 4(2x 4)i 7(y 2) 3(y + 4) j 2(2m + 1) + 4(2m 2)k 3(d + 2) (d + 8) l 5(2t + 3) + 8(3t + 1)m 6(m 4) + 12(2m + 3) n 5(p q) + 7(p + q)o 9(m + n) + 2(m 3n) p 4(k + 7) (k 3)

3 Correct this students homework by checking her working out for each problem. Which questions did she get wrong?

a 5x (2 3x) b 6x + (4 5x)= 5x 2 3x = 6x + 4 5x= 2x 2 = x + 4

c 2x (8 x) d 4m (3 + 2m)= 2x 8 + x = 4m 3 2m= 3x 8 = 2m 3

e 5x (4 2x) + 3x f 8 + (4x + 5) 2x= 5x 4 + 2x + 3x = 8 + 4x + 5 2x= 10x 4 = 2x + 13

g 2p 3 (3 4p) h 9x 1 + (3x 5)= 2p 3 3 4p = 9x 1 3x + 5= 2p 6 = 6x + 4

i 10 (3x 2) + (5x 3) j 8x (2x 5) (4x + 4)= 10 3x 2 + 5x 3 = 8x 2x + 5 4x 4= 5 + 2x = 2x + 1

Exercise 12-08

Example 15

Example 14


Algebraic substitutionSubstitution occurs in many sports, when one player replaces another during a game. Substitution in mathematics involves replacing a variable with a value.Substituting 5 for k in k + 7 gives 5 + 7.

Evaluate means to nd the value of.

Multiplying by 9, 11, 99 or 101We can use expanding when multiplying by a number near 10 or near 100.1 Examine these examples:

a 25 11 = 25 (10 + 1) b 14 9 = 14 (10 1)= 25 10 + 25 1 = 14 10 + 14 1= 250 + 25 = 140 14= 275 = 126

c 32 12 = 32 (10 + 2) d 7 99 = 7 (100 1)= 32 10 + 32 2 = 7 100 + 7 1= 320 + 64 = 700 7= 384 = 693

e 27 101 = 27 (100 + 1) f 18 8 = 18 (10 2)= 27 100 + 27 1 = 18 10 + 18 (2)= 2700 + 27 = 180 36= 2727 = 144

2 Now simplify these:a 16 11 b 33 11c 29 9 d 45 9e 62 11 f 7 101g 18 101 h 36 99i 19 8 j 45 12k 21 102 l 6 98

Skillbank 12 SkillTest 12-01

Multiplying by 9, 11, 99 or 101

Worksheet12-05

Substitution

Evaluating an expression means substituting a number into the expression and working out the answer.

Worksheet12-06

Formula 1game

Example 16

Evaluate k 9 when k = 15.Solutionk 9 = 15 9

= 6


Example 17

1 Evaluate 2k + 1 when k = 3.Solution2k + 1 = 2 3 + 1 (always do multiplication before addition)

= 6 + 1= 7

2 Evaluate 3x 2y if x = 2 and y = 1.Solution3x 2y = 3 2 2 (1)

= 6 (2)= 6 + 2= 8

3 Evaluate m(n 3) if m = 5 and n = 7.SolutionSubstitute 5 for m in the expression and substitute 7 for n.m(n 3) = 5 (7 3)

= 5 (4) (always do brackets rst)= 20

Animation 12-02

Substitution

1 Evaluate k + 3 when:a k = 2 b k = 18 c k =119 d k = 21

2 Evaluate 45 k when:a k = 5 b k = 13 c k = 28 d k = 45

3 Evaluate 4k when:a k = 2 b k = 11 c k = 4 d k = 8

4 Find the value of 5k + 1 when:a k = 3 b k = 10 c k = 21 d k = 38

5 Find the value of 14k 8 if:a k = 2 b k = 5 c k = 12 d k = 13

6 Find the value of if:a k = 15 b k = 39 c k = 9 d k = 57

7 Evaluate:a 5n 2, if n = 2 b 3k + 8, if k = 11c 23 + 5t, if t = 9 d 100 2a, if a = 23e 5(p + 6), if p = 5 f 4 (2m 3), if m = 8g n(3n + 1), if n = 0.5 h 2w + 19, if w = 0i (d + 1) 4, if d = 9 j 15 3m, if m = 4k 10(4 p), if p = 3 l (3 + 2m) m, if m = 5m (4d 1) 3, if d = 7 n y2 + y, if y = 6

k3---


Example 17


8 Evaluate these expressions if a = 0, b = 2, c = 5, d = 10, e = 16.a 2a + b b 3d + cc 4d e d c + d + ee 4b a c f 3e + 2c 3dg 4b 3d + 8c h 5a + 2b 3c + 4di 3e 2d + 6c 400a j b(2c + 3d)k 2d(19a + 3c d) l bc + de

9 Computer Algebra Software can simplify expressions by substituting numerical values for the variables. This link will take you to an activity that uses TI InterActive! to do this.

10 Copy and complete this table using the different substitution values given:

11 Use this table of values to evaluate the following expressions:

a x + y b K + P + bc 4d + 3c d P 5ae z + 3y f 2K + 3K + xg 5 (x + c) h 2 (a x)i ab + Kz j 7Pxyk zd P l db xym Ka + Pab n y2o z2 p a2 c2

Values 2x + y xy 8x 3x 2y 37 4y

a x = 3 and y = 4 10 12 24 1 21b x = 4 and y = 0c x = 2 and y = 1d x = 5 and y = 6

e x = and y =

f x = 8 and y = 9

g x = 21 and y = 5

h x = 0 and y = 8

i x = 2.5 and y = 1.5

j x = and y = k x = 3 and y = 4

l x = 0.7 and y = 0.2

a b c d x y z K P M7 3 2 8 7 0 20 1 100 0

12---

12---

34---

25---

CAS 12-04

Substituting into an expression


Communicating and reecting: Generalised arithmeticWe can use algebraic symbols to describe general laws about numbers and arithmetic.For example, if we add zero to any number, the answer is still that number. This can be written algebraically if we let N stand for any number:

N + 0 = N What does the rule below mean in words?

a + b = b + a It means that, if we add any two numbers, a and b, we will get the same answer as if we added b and a. For example, 4 + 7 is the same as 7 + 4. As a general property of numbers and arithmetic, two numbers can be added in any order.Form groups of two to four students, and complete these problems together.1 Describe what these rules about numbers mean in words.

a N 1 = N b N 0 = 0c a b b a d a + b + c = b + a + ce N 0 = N f ab = ba

2 Write each of these rules algebraically.a Any number divided by 1 equals itself.b Multiplying a number by 8 is the same as doubling it three times.c Any three numbers, a, b and c, can be multiplied together in any order.d Any number added to itself is the same as multiplying that number by 2.e Any number subtracted from itself equals 0.f Any number multiplied by its reciprocal equals 1.

3 Are these rules about numbers true or false?a a b = b a b N N = 1c 4a a = 4 d a is a factor of ae If N is even, then N + 3 is odd f N = N 2g a + (a) = 2a h 1 is a factor of ai 0 N = 0 j N 1 = N

4 Explain the meanings of 2a + 1 and 2(a + 1). How are they different?5 If k is an odd number, what is an expression for:

a the previous odd number?b the next even number?

6 N 0 has no answer (it is not equal to 0). Why?

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Working mathematically


1 Simplify:a 2a + b + 3a 4b b m 3n 5m 7nc 3k 2j 5 4k + 6j + 9 d a2 + 2b2 + 3b2 a2e 4x2 + 6x x2 12x f xy + yz + xz + 4xy + 3xz + yzg 5a + 4b + c 3a + 5c h 7p 8q + 3p 5q + 4qi d2 + 7 + 6e2 3d2 2e2 + 9 j 5f 2 + 6f + 2 + 8f 2 1

2 Expand and simplify:a 4(x 1) 3(x + 1) b 2(m + 4) (m 4)c x(x 3) + 5(x 3) d m(a + 4) + m(a 7)e 3(y 4) + 7(y 3) f 4(2a + b + 3c) 2(a + 3b c)g p(p 3q + 2r) + 2p(4r 3p 2q) h 7k2 + 3(k2 3k 1) (6k2 4k + 2)i 12 5(4f 6g 3) j 5(2x2 5x + 3) 4(3x2 8x 25)k (x + 3)(x + 5) l (p 1)(p + 10)

3 Write three algebraic expressions that can be simplied to get the following answers:a 2x + 6 b 6m + 12p c x2 xd 7 e x2 + y2 f 2k2 + 4k

4 Evaluate each of the following algebraic expressions, if p = 4, q = 6, r = 3 and t = 1:a p + q b pq c r2 + t2

d e q2 6p + rt f 6 2r + p2

g 2p + 3q + 4r 5t h i

j k + l q + pr qt5 Write, as an algebraic expression, the perimeter and the area of each of these shapes:

6 If x is any number, simplify these expressions:a x + 0 b 1 x c 0 xd x x e x x f x (x)g 0 x h x 0 i (x)2

pqrt------

p 4+2------------

q r+p t+-----------

r 7+5-----------

q t5----------

p 3+10------------

r 22-----------

l

2

x

a

b

yc

a

bd

x

y2

rq

p

m + 3

n

m

5

b c

d ef

Power plus


Topic overview Write in your own words the new things you have learnt about algebra. What parts of this topic did you like? Write any rules you have learnt. What parts of this topic did you nd difcult or did you not understand? Talk to a friend or

your teacher about them. Give examples of algebra in use. Copy this overview into your workbook and complete it. If necessary, refer to the Language

of maths section for keywords.

Language of mathsabbreviation algebraic expression algebraic term consecutiveevaluate expand expanded form expressiongrouping symbols like terms pronumeral simplifysubstitute substitution term variable

1 There are English expressions, numerical expressions and algebraic expressions. What does the word expression mean?

2 Explain in your own words the difference between an algebraic term and an algebraic expression.

3 What is a non-mathematical meaning for expand? Relate this to its algebraic denition.

4 What is another name for a pronumeral?5 Distinguish between simplifying an expression and simplifying a fraction.

Worksheet 12-07

Algebra find-a-word

Expanding

Substitution Adding/subtracting

ALGEBRA

Like terms

Expression


1 Simplify each of these expressions:a 5 a 2 b 8 c b 3 a c 2 m 4 w 10 w

2 Write in expanded form:a 5m b 4mnk c 6ab d abc

3 Write a mathematical expression for each of the following. Use N to represent any number.a 3 times the number b the difference between the number and 5c the next consecutive number d one-third of the number

4 If A, B and C represent any three numbers, write expressions for:a the sum of A and C b the product of all three numbersc the difference between B and C, where C is greater than B

5 Write expressions for:a the sum of M and 3 b 5 more than B c 2H decreased by k

6 Write the like terms in each of these sets of terms:a 8ab, 3x, 2a, 2g, 3a b 3mn, 7abc, 8nm, 3bc, 2ac 2xy, yx, y2, 4yx, y2x d 3a2b, 2ab, 4a2b, ba2, b2a

7 Simplify each of these expressions by adding or subtracting like terms:a 5a + 3a + 2a b 4mn + 2nm c abc + 3abc 2bcad 17a 3a e 9ab 3ba f 10fg 4fg + 3fgg 3k + 4k 2k h 12mn + 3mn 5mn i 5x + 12 + 9x 5j 8w 12 + 12w + 20 k 6a 3b + 5a 8b l 7a + 4b 9a 6bm 2x + 7y 5y 3x n 9a2 + 6a 5a2 3a o yz + xy yz + xy

8 Simplify:a 3b 5d b 2h 6n c a (10ab)

9 Expand:a 7(e + 4) b 5(k 8) c y(2 y)d m(m + 2) e 3(j + 2) f 2(t 4)g 9a(a 2) h 2k(5 k) i 3m(m 5)

10 Expand and simplify:a 5w + 2(w + 3) b 6(h 1) + 12 c 7(x + 2) + 3(x + 1)d 14 (a + 3) e 3(y + 4) 2(y + 5) f 10(d + 2) 5d + 5g 5(m 3) 3(m 8) h 24c 4(c 2) i 3(2p + 5) + 5(3p 2)

11 Find the value of these expressions if a = 2, b = 3, c = 5 and d = 6:a 5a 2 b 2a + 2b c c 5d + 3a 2cd 7a + 4b + d e d2 c2 f cd a

12 Evaluate each expression for the different substitution values given:

Values 2a + 3b 3a 4b 2ab 5a 2b

a a = 3 and b = 4

b a = 5 and b = 1

Topic testChapter 12

Chapter 12 Review

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Student textImprint pageTable of contentsPrefaceHow to use this bookHow to use the CD-ROMAcknowledgementsSyllabus reference grid1 The history of numbersDifferent number systemsThe HinduArabic number systemPlace valueExpanded notationThe four operationsArithmagonsDividing by a two-digit numberOrder of operationsThe symbols of mathematicsTopic overviewChapter review

2 AnglesNaming anglesComparing angle sizeThe protractorDrawing anglesAngle geometryNaming linesAngles and parallel linesFinding parallel linesTopic overviewChapter review

3 Exploring numbersSpecial number patternsTests for divisibilityFactorsPrime and composite numbersPrime factorsIndex notationSquares, cubes and rootsTopic overviewChapter review

Mixed revision 14 SolidsNaming solidsConvex and non-convex solidsPolyhedraPrisms and pyramidsCylinders, cones and spheresClassifying solidsEulers ruleEdges of a solidThe Platonic solidsDrawing and building solidsDifferent views of solidsTopic overviewChapter review

5 IntegersNumber linesNumbers above and below zeroDirected numbersOrdering directed numbersAdding and subtracting integersMultiplying integersDividing integersThe four operations with integersReading a map gridThe number planeThe number plane with negative numbersTopic overviewChapter review

6 Patterns and rulesNumber rules from geometric patternsUsing pattern rulesThe language of algebraTables of valuesFinding the ruleFinding harder rulesFinding rules for geometric patternsAlgebraic abbreviationsSubstitutionSubstitution with negative numbersTopic overviewChapter review

Mixed revision 27 DecimalsPlace valueUnderstanding the pointOrdering decimalsDecimals are special fractionsAdding and subtracting decimalsMultiplying and dividing by powers of 10Multiplying decimalsCalculating changeDividing decimalsDecimals at workConverting common fractions to decimalsRecurring decimalsRounding decimalsMore decimals at workTopic overviewChapter review

8 Length and areaThe history of measurementThe metric systemConverting units of lengthReading measurement scalesThe accuracy of measuring instrumentsEstimating and measuring lengthPerimeterAreaConverting units of areaArea of squares, rectangles and trianglesAreas of composite shapesMeasuring large areasTopic overviewChapter review

9 Geometric figuresPolygonsClassifying trianglesNaming geometric figuresConstructing trianglesClassifying quadrilateralsConstructing perpendicular and parallel linesConstructing quadrilateralsTopic overviewChapter review

Mixed revision 310 FractionsHighest common factor and lowest common multipleNaming fractionsEquivalent fractionsOrdering fractionsAdding and subtracting fractionsAdding and subtracting mixed numeralsFractions of quantitiesMultiplying fractionsDividing fractionsTopic overviewChapter review

11 Volume, mass and timeVolumeVolume of a rectangular prismCapacity and liquid measureMassTimelinesConverting units of timeTime calculationsWorld standard timesTimetablesTopic overviewChapter review

12 AlgebraAlgebraic expressionsAlgebraic abbreviationsFrom words to algebraic expressionsLike termsMultiplying algebraic termsExpanding an expressionExpanding and simplifyingAlgebraic substitutionTopic overviewChapter review

13 Interpreting graphs and tablesPicture graphsColumn graphs and divided bar graphsSector graphsLine graphsTravel graphs and conversion graphsStep graphsReading tablesTopic overviewChapter review

Mixed revision 4General revisionAnswersIndex

GlossaryABCDEFG HI JK LMNOPQRSTU VW X Y Z

All files menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

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Learning technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 11Chapter 12

Practice menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Revision menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 12Chapter 13

Using technology menuChapter 1Chapter 2Chapter 3Chapter 4Chapter 5Chapter 6Chapter 7Chapter 8Chapter 9Chapter 10Chapter 11Chapter 13

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