algebra 1. words and symbols ‘sum’ means add ‘difference’ means subtract ‘product’ means...
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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