Transcript ALGEBRA 1 - Miss Hudsons Maths

ALGEBRA 1
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
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
Variable
Meaning
A letter or symbol used to
represent a number or
unknown value
Algebraic
A statement using
Expression numerals, variables and
operation signs
Example
A=πr 2
has A and r as
variables
3a + 2b - c
Equation
2x + 5 = 8
Inequation
An algebraic statement
containing an “ = “ sign
An algebraic statement
containing an inequality
sign, e.g <, ≤, >, ≥
3x - 8 < 2
This is all
precious
Terms
Like Terms
The items in an algebraic
expression separated by
+ and - signs
Two terms that have
EXACTLY the same
variables (unknowns)
Constant Term
A term that is only a
number
Coefficient
The number (including
the sign) in front of the
variable in a term
4x, 2y2
3xy, 7
4x and -7x
3x2 and x 2
NOT x and x2
-7, 54
4 is the
coefficient of
4x2, -7 is the
coefficient of
-7y
Substitution into formulae
Putting any number, x
into the machine, it
calculates 5x - 7.
Input,
2 x
i.e. it multiplies x by 5
then subtracts 7
2
e.g. when x = 2
3
3
3
e.g Calculate 5a - 7 when a = 6
5x
- 7 = 30 - 7= 23
e.g. Calculate y2 - y + 7 when y = 4
Writing 4 where y occurs in the equation gives
42 - 4 + 7 = 19
Substitution
If x = 4 and y = -2 and z = 3
eg 4:
eg 1:
x + y +z
= 4 + - 2 +3
=
=
= 4-2+3
= 5
=
eg 2: xy (z - x)
=
4 x -2x( 3 - 4)
4 x -2 x -1
= 8
eg 3:
2 x 42
2 x 16
32
eg 5: (2x)2
= (2 x 4)2
= 82
= 64
-1
=
2 x2
y2
= ( -2)2
= 4
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=4x5
Perimeter = 20 cm
Example 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.
Solution:
C = 10 + 20h
C = 10 + 20x 4
Charge = $90
Collecting Like Terms
• Adding like terms is like
adding hamburgers.
e.g.
Like terms
should sound
the same
+
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
A number
owns the sign
in front of it
5a + 3b - 2a - 6b = 3a - 3b
10abc - 3cab = 7abc (or 7bca
or 7cba etc)
-4x2 - 2 + 3x + 5 -1x + 7x2 = 3x2 + 2x + 3
Note: x means 1x
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
6a
fraction eg 6a ÷ y becomes
y
Simplifying Expressions
Multiplying algebraic terms.
Follow the rules and algebra is easy
examples:
f x 4 = 4f
4a x 2b = 8ab
-2a x 3b x 4c = -24 abc
2 x a + b x 3 = 2a + 3b
The terms do
not have to be
‘like’ to be
multiplied
Index Notation
24 means 2 x 2 x 2 x 2 = 16
base
24
index, power or exponent
a xa xa = a3
2xa xa xa xb xb = 2a3 b2
mxm-5xnxn= m2 -5n2
Laws of Indices
When multiplying terms with the same base we add the
powers
eg 1: y 3 x y 4 =
y 7
eg 2: 2
3
x25
=
28
eg 3: 3 m 2 x 2 m 3 = 6 m 5
When dividing terms with the same base we
subtract the powers.
eg 1:
eg 2:
8
p
p8÷p2= 2 
p
5
20 x 7
1 4x 3
210m 5
p6
 5x 4
2m
eg 3:

4
3
315m
or
2m
3
Expanding Brackets
Each term inside the bracket is multiplied by
the term outside the bracket.
example 1: 4 ( x + 2) =
example 2:
4x
x ( x -- 4) = x 2
example 3: 3y ( y 2 + y - 3) =
example 4: -2 (m - 4) =
8
4x
Remember:
means
the terms are
multiplied
3y 3 + 3y 2 - 9y
-2m + 8
NB: -2 x -4 = +8
example 5:
x (x - 5) + 2 (x + 3)
= x 2 - 5x + 2x + 6
=x
- 3x + 6
4( 2x - 3) - 5( x + 2)
example 6:
example 7:
2
=
8x - 12 - 5x - 10
=
3x - 22
4( 2x - 3) - 5( x - 2)
= 8x - 12 - 5x + 10
= 3x - 2
Factorising
This means writing an expression with brackets
eg 1: 2x + 2y = 2 ( x + y )
eg 2: 3x + 12 = 3 ( x + 4 )
3 (2x - 5 )
eg 3: 6x - 15 =
eg 4: 4x2 + 8x =
xx
x
4 x (x +2)
eg 5: 10d 2 - 5d = 5d ( 2d - 1)
NB: Always take
out the highest
common factor.
eg 6: 12a 3b 4c 2 - 20a 2b 3c 3 + 8a 4b 4c
aaabbbbcc
aabbbccc
5
aaaabbbbccccc
= 4a 2b 3c 2 ( 3a b - 5 c + 2a 2 bc 3)
Patterns
Shape (s)
Number of cubes (c)
1
3
2
5
3
7
4
9
n
2
61
30
100
+2
+2
+2
+1
201
49
24
Formula:
c = 2s + 1
Patterns - example 2
Shape (s)
1
Matchsticks (m)
2
3
4
11
16
n
40
8
6
+5
+5
+5
21
5 n+ 1
201
41
m = 5s + 1
Patterns - example 3
Shape (s)
Number of dots (d)
1
2
2
6
3
10
4
14
n
4n -2
70
102
18
26
d = 4s - 2
+4
+4
+4