1. ALGEBRA 1

2. Words and Symbols ‘Sum’ means add ‘Difference’ means subtract
‘Product’ means multiply
Putting words into symbols
Examples:
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 means
2(k + 4)
y minus 4 means
y - 4
y less than 4 means
4 - y

3. 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.

4. 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
This is all precious
Inequation
An algebraic statement containing an inequality sign, e.g <, ≤, >, ≥
3x - 8 < 2

5. 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, is the
coefficient of
-7y

6. Substitution into formulae
Input, x 2
Putting any number, x into the machine, it calculates 5x - 7.
i.e. it multiplies x by 5 then subtracts 7 2 3 3
e.g. when x = 2 3
e.g Calculate 5a - 7 when a = 6 6
5 x =
30 - 7= 23
e.g. Calculate y2 - y + 7 when y = 4
Writing 4 where y occurs in the equation gives = 19

7. Substitution If x = 4 and y = -2 and z = 3 eg 1: x + y + z eg 4: 2 x2
- 2 + 3 =
2 x 42 =
= 2 x 16
= 5
= 32
eg 2: xy (z - x)
eg 5: (2x)2 =
4 x -2
( 3 - 4) x =
(2 x 4)2 -1
= 82
= 4 x -2 x -1
= 64
= 8
eg 3: y2
( -2)2 =
= 4

8. Formulas & Substitution
Example 1: If the perimeter of a square is given by the formula P = 4x, find the perimeter if x = 5 cm
Solution: P = 4x,
P = 4 x 5
Perimeter = 20 cm
Example 2: If a gardener works out his fee by the formula C = h where h is the number of hours he works, work out how much he charges for a job that takes 4 hours.
Solution:
C = h
C = x 4
Charge = $90

9. Like terms should sound the same
Collecting Like Terms
Like terms should sound the same
Adding like terms is like
adding hamburgers.
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

10. A number owns the sign in front of it
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.
A number owns the sign in front of it
We can only add and subtract ‘like terms’
Examples:
5x + 7x = 12x
5a + 3b - 2a - 6b = 3a
- 3b
10abc - 3cab =
7abc (or 7bca
or 7cba etc)
-4x x x + 7x2 = 1
3x2
+ 2x
+ 3
Note: x means 1x

11. Rules
Curvy
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 x
Numbers are written in front of unknowns eg 5y not y5
Letters are written in alphabetical order
eg 6abc rather than 6bca
Instead
of using a division sign ÷, we write the term as a
fraction eg 6a ÷ y becomes

12. Simplifying Expressions
Multiplying algebraic terms.
Follow the rules and algebra is easy
The terms do not have to be ‘like’ to be multiplied
examples: f x 4 =
4a x 2b = 4f 8 ab
-2a x 3b x 4c =
-24
abc
2 x a + b x 3 =
2a + 3b

13. Index Notation 24 24 means 2 x 2 x 2 x 2 = 16 a x a x a = a 3
index, power or exponent 24
base
a x a x a =
a 3
2 x a x a x a x b x b =
2 a 3
b 2
m x m - 5 x n x n =
m 2
- 5 n 2

14. Laws of Indices When multiplying terms with the same base we add the
powers
eg 1: y 3 x y 4 =
y 7 28
eg 2: x =
eg 3: 3 m 2 x 2 m 3 = 6
m 5

15. When dividing terms with the same base we subtract the powers.
eg 1: p 8 ÷ p 2 =
p 6 5
eg 2: 5
x 4 1 2 m
eg 3: or 3

16. Expanding Brackets
Each term inside the bracket is multiplied by the term outside the bracket.
Remember:
means the terms are multiplied
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
NB: -2 x -4 = +8

17. 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

18. Factorising This means writing an expression with brackets eg 1:
2x + 2y = 2
( ) x + y
eg 2:
3x + 12 = 3 ( x + 4 )
eg 3:
6x - 15 = 3 ( 2x -
5 )
eg 4:
4x2 + 8x = 4 x ( x
+ 2 )
NB: Always take out the highest common factor.
x x x
eg 5:
10d 2 - 5d =
5d ( 2d - 1)
eg 6:
12a 3b 4c a 2b 3c 3 + 8a 4b 4c 5
aaabbbbcc
aabbbccc
aaaabbbbccccc
= 4a 2b 3c 2 (
3a b
- 5 c
+ 2a 2 bc 3)

19. Patterns (s) (c) Formula: c = 2s + 1 Shape Number of cubes 1 2 3 4 n30
100 49
(c) 3 +2 5 +2 7 +2 9 n 2
+ 1 61
201 24
Formula:
c =
2s + 1

20. Patterns - example 2 Shape Matchsticks 1 2 3 4 n 40 41 (s) (m) 6 + 511
+ 5 16
+ 5 21
5 n
+ 1
201 8
m =
5s + 1

21. Patterns - example 3 Shape Number of dots 1 2 6 3 10 4 14 n 18 102 (s)
+ 4
+ 4
+ 4
4 n
- 2 70 26
d =
4s - 2