ALGEBRA 1ALGEBRA 1ALGEBRA 1ALGEBRA 1

Words and Symbols‘Sum’ means add‘Difference’ means subtract‘Product’ means multiply

Examples:

Putting words into symbols

the sum of p and q means p + q

a number 5 times larger than b means

5b

a number that exceeds r by w means

r + w

twice the sum of k and 4 means2(k + 4)

y less than 4 means

y minus 4 meansy - 4

4 - y

Algebraic Language• A Pronumeral is a letter that represents an unknown number. For example, X might represent the number of school days in a year.

• A Term usually contains products or divisions of pronumerals and numbers.

The term 3x2 means 3 x x x x or 3 lots of x2.

• The Coefficient is the number by which a pronumeral or product of pronumerals is multiplied.

3 is the coefficient of x2 in the term 3x2

• Like Terms have exactly the same letter make up, other than order.6x2 y and 14x2 y are like terms.

• A Constant Term is a number by itself without a pronumeral.• -7 is the constant term in the expression 4x2 – x – 7.

The Language of Algebra

Word Meaning Example

Variable A letter or symbol used to represent a number or unknown value

A = π r 2 has A and r as variables

Algebraic

Expression

A statement using numerals, variables and operation signs

3a + 2b - c

Equation An algebraic statement containing an “ = “ sign

2x + 5 = 8

Inequation An algebraic statement containing an inequality sign, e.g <, ≤, >, ≥

3x - 8 < 2

This is all precious

Terms The items in an algebraic expression separated by + and - signs

4x, 2y2

3xy, 7

Like Terms Two terms that have EXACTLY the same variables (unknowns)

4x and -7x

3x2 and x 2

NOT x and x2

Constant Term A term that is only a number

-7, 54

Coefficient The number (including the sign) in front of the variable in a term

4 is the coefficient of 4x2, -7 is thecoefficient of-7y

Substitution into formulae

Putting any number, x into the machine, it calculates 5x - 7.

i.e. it multiplies x by 5 then subtracts 7

e.g. when x = 2

2

2 3 3

3

Input, x

e.g Calculate 5a - 7 when a = 6

5 x - 7 =

30 - 7= 23

6

e.g. Calculate y2 - y + 7 when y = 4

Writing 4 where y occurs in the equation gives

42 - 4 + 7 = 19

SubstitutionIf x = 4 and y = -2 and z = 3eg 1: x + y + z

4 = 4 - 2 + 3

= 5

-

eg 2: xy (z - x)

4 x -2

-1

= 4 x -2 x -1

= 8

eg 4: 2 x2

2 x 42

= 2 x 16

= 32

eg 5: (2x)2

(2 x 4)2

= 82

= 64

eg 3: y2

( -2)2

= 4

+- 2 +3

( 3 - 4)x

=

=

=

=

=

Formulas & Substitution

Solution: P = 4x, P = 4 x 5

Perimeter = 20 cmExample 2: If a gardener works out his fee by the formula C = 10 + 20h where h is the number of hours he works, work out how much he charges for a job that takes 4 hours.

Example 1: If the perimeter of a square is given by the formula P = 4x, find the perimeter if x = 5 cm

Solution: C = 10 + 20hC = 10 + 20x 4Charge = $90

Collecting Like Terms

• Adding like terms is like

adding hamburgers.

Like terms should sound

the same

e.g. +

gives

You’ve started with hamburgers, added some more and you end up with a lot of hamburgers

2x + 3x = 5x You started with x, added more x and end up with a lot of x, NOT x2

Like Terms‘Like terms’ are terms which have the same letter or letters (and the same powers) in them. ie when you say them - they sound the same.

We can only add and subtract ‘like terms’

Examples: 5x + 7x = 12x

5a + 3b - 2a - 6b =3a - 3b

A number owns the

sign in front of it

10abc - 3cab =

7abc (or 7bca or 7cba etc)

-4x2 - 2 + 3x + 5 - x + 7x2 =3x2 + 2x+ 3

Note: x means 1x

1

Rules

of using a division sign ÷, we write the term as a

fraction eg 6a ÷ y becomes

Curvy

ya6

We usually don’t write a times sign eg 5y not 5 x y, 5(2a + 6)

The unknown x is best written as x rather than xNumbers are written in front of unknowns eg 5y not y5

Letters are written in alphabetical order eg 6abc rather than 6bca Instead

Simplifying ExpressionsMultiplying algebraic terms.

examples: f x 4 =

4a x 2b = 8ab

-2a x 3b x 4c = -24 abc

Follow the rules and algebra is easy

The terms do not have to

be ‘like’ to be multiplied

4f

2 x a + b x 3 = 2a + 3b

Index Notation24 means 2 x 2 x 2 x 2 = 16

24base

index, power or exponent

a x a x a =

2 x a x a x a x b x b =2 a 3

m x m - 5 x n x n =m 2

a 3

b 2

- 5 n 2

When multiplying terms with the same base we add the powers

eg 1: y 3 x y 4 = y 7

eg 2: 2 3 x 2 5 = 28

eg 3: 3 m 2 x 2 m 3 = 6 m 5

Laws of Indices

When dividing terms with the same base we subtract the powers.

eg 1: p 8 ÷ p 2 = 2p

p8

p 6

eg 2: 3

7

4

20

x

x1

5

5x 4

eg 3: 415

10

m

m52

3 32 m or

32m

Expanding BracketsEach term inside the bracket is multiplied by the term outside the bracket.

example 1:

4 ( x + 2) = 4x+ 8

example 2:

x ( x - 4) = x 2- 4x

example 3: 3y ( y 2 + y - 3) =3y 3 + 3y 2 - 9y

example 4:-2 (m - 4) = -2m + 8

Remember: means the terms are multiplied

NB: -2 x -4 = +8

example 5:

x (x - 5) + 2 (x + 3)

= x 2- 5x + 2x + 6

= x 2 - 3x + 6

example 6: 4( 2x - 3) - 5( x + 2)

8x= - 12 - 5x - 10

= 3x - 22

example 7: 4( 2x - 3) - 5( x - 2)

= 8x - 12 - 5x + 10

= 3x - 2

Factorising This means writing an expression with bracketseg 1: 2x + 2y =2( )x +y

eg 2: 3x + 12 =3 ( x +4)

eg 3: 6x - 15 = 3(2x - 5 )

eg 4: 4x2 + 8x =x x x

4 x ( x + 2) NB: Always take out the highest common factor.

eg 6: 12a 3b 4c 2 - 20a 2b 3c 3 + 8a 4b 4c 5 aaabbbbcc aabbbccc aaaabbbbccccc

= 4a 2b 3c 2 (3a b - 5 c + 2a 2 bc 3)

eg 5: 10d 2 - 5d =5d ( 2d - 1)

PatternsShape Number of cubes

1

2

3

4

n

30

100

49

3

5

7

9+2

+2

+2

2 + 161

201

24

n

(s) (c)

c = 2s + 1Formula:

Patterns - example 2

Shape Matchsticks

1

2

3

4

n

40

41

6

11

16

21+ 5+ 5

+ 5

5 n + 1201

8

(s) (m)

m =5s + 1

Patterns - example 3

Shape Number of dots

1 2

2 6

3 10

4 14

n

18

102

+ 4

+ 4

+ 4

4 n - 2

70

26

(s) (d)

d = 4s - 2