ALGEBRAIC MODELLING
DEFINITIONS AND FORMULAS
 PRONUMERAL is a symbol (letter) used in place of a number
 LIKE TERMS have exactly the same pronumeral part
(eg. 3a, -7a, a, 12a)
 UNLIKE TERMS have different pronumerals (eg. 8v, 5w, -9z)
Note: 3y and 7y2 are UNLIKE TERMS because the pronumerals are not
exactly the same
 You can only add and subtract LIKE TERMS. The operation sign in
front of a term belongs to that term.
 When multiplying or dividing algebraic expressions, multiply or divide
the number part and pronumeral separately
(eg. 4a x 3b = 12ab, 15pq  3q = 5p)
 When removing grouping symbols (EXPANDING) every term inside
the bracket must be multiplied by the number or term outside
(eg 4(a + b) means 4 x a + 4 x 8 = 4a + 32,
-(f + g) means –1 x f + -1 x g = -f – g)
 The opposite of EXPANDING is called FACTORISING (putting
brackets in)
(eg. 2h + 14 = 2(h +7), 24ab + 16a = 4a(6b + 4), 18ij – 14j = 2j(9i – 7)
 When we SUBSTITUTE we replace a pronumeral with a given
number. This allows us to convert a formula to a mathematical figure.
(eg. A = ½ b x h is the area formula for a triangle where b is the base
and h is the height, F = 9/5 C + 32 converts temperature from Celsius to
farenheit)
 The aim of solving equations is to find the value of the pronumeral.
 A FUNCTION – is an expression into which we can substitute different
values.
 Any function of type y = mx + b is called a LINEAR function and is a
straight line when graphed
 If the highest power of x in a function is squared (x2) it s a
QUADRATIC function
 If the highest power is cubed (x3) we have a CUBIC function
 Where y = a/x it is a HYPERBOLIC function
 Any function of type y = b(ax) is called an EXPONENTIAL function.
If a>1 y will increase as x increases (exponential growth). The opposite
is exponential ‘decay’.
Note 3(45) means 3 x 45
THE ZERO INDEX
Any base to the power of zero always equals 1.
5 0= 1
Eg.1
x 0= 1
(7y 2) 0 = 1
(3x) 0 = 1
Watch out for these!!!!!!!!!
3x 0 = 3 X x 0
=3X1
=3
Eg.2
5m0
= 5 x m0
=5x1
=5
2a 0 = 2 x a0
=2x1
=2
TERMINOLOGY
Cubic
Linear
Power
Quadratic
Proportional
Gradient
Substitution
Function
Expanding
Pronumeral
Exponential
Factorisation
Hyperbolic
Midpoint
INDEX LAWS
1. When multiplying like terms add the powers.
a m xa n= amn
a6 x a4 = a 6 + 4 = a 10
2. When dividing like terms, subtract the powers
a m = aman
n
a7 = a 7 – 3 = a 4
a3
3. When an expression in brackets is raised to a power, multiply
the indicies.
( a m) = a mxn
( a 3 ) 5 = a 3 x 5 = a 15
4. Any letter or number raised to the power of zero ( 0 ) is equal to
1.
a 0= 1
7 0= 1
5. If two terms in brackets are raised to a power, then simply raise
each term separately to that power.
( ab ) m = a m x b m
( 2m ) 4 = 2 4 x m 4 = 16 m 4
SIMPLIFYING EXPRESSIONS BY MULTIPLYING
Eg 1. 3a2 x 2a3
= 3 x 2 x a 2 x a3
= 6 a 2+3
= 6a5
Eg 2. 5m4 x 3m6 x 2m3
= 5 x 3 x 2 x m 4 x m6 x m3
= 30 m 4+6+3
= 30 m13
Eg 3. (2a3)5
= 2a3 x 2a3 x2a3 x2a3 x2a3
= 32a 3x5
= 32a15
SIMPLIFYING EXPRESSIONS BY DIVIDING
Eg 1. 12m8
________
3m6
m8
12
=
_____
x
3
_____
m6
Eg 2. 18a5  6a3
= 18 x a 5-3
6
= 3a2
Eg 3. 12x10y8
8x6y2
= 12
x10
y8
8 x x6 x y2
3
= 2 x x10-6 x y8-2
3
= 2 x x4 x y6
= 3x4y6
2
= 4 x m8-6 = 4m2
REMOVING GROUPING SYMBOLS OR EXPANDING BRACKETS
“Grouping symbols” means exactly the same as “brackets”. To remove
grouping symbols we multiply the outside term by the term inside the
brackets.
a ( b + c )= a x b + a x c
= ab + ac
a ( b – c )= a x b – a x c
= ab - ac
3 ( a – 4) = 3 x a – 3 x 12
= 3a – 12
Eg. 5 ( m + 2) = 5 x m + 5 x 2
= 5m + 10
- 2 ( x – 5)= - 2x x – 2 x ( - 5)
= - 2x + 10
- ( 2 + 3y) = -1 ( 2 + 3y)
= -1 x 2 + (-1) x 3y
= -2 – 3y
EXPANDING BINOMIAL PRODUCTS
Eg.
(a + b) (c + d)
= ac + ad + bc + bd
(2f + 3) (2f + 2)
= 2f x 2f + 2f x 2 + 3 x 2f + 6
= 4f 2 + 4f + 6f + 6
= 4f 2 + 10f + 6
FACTORISING
This process is the reverse process to “removing the brackets”. First we
find the highest common factor (just like in fractions!) and write it
outside the brackets.
Eg. 18m – 12 = 6 (3m – 2)
15m 3 – 12m 2 = 3m2 ( 5m – 4)
-8
a 2 + 12 a = - 4a ( 2a – 3)
(HCF = 6)
(HCF = 3 m 2)
(HCF = -4a)
CO-ORDINATE GEOMETRY
GRAPHING LINEAR EQUATIONS
There are 4 ways of expressing a relationship between 2 variables x
and y.
1. Algebraically – is an equation with an equal (=) sign.
Eg. y = 2x + 1 (so if x = 5, y=11)
2. Table Form – must consist of at least 3 ‘x’ values.
X
0
1
2
Y
1
3
5
3. Ordered Pairs – this is done simply by changing the pairs of
numbers in the table into ordered pair form using brackets.
(0,1) (1,3) (2,5)
4. Graph – you can now plot the ordered pairs on a number plane.
y
x
MIDPOINT
The midpoint of an interval (line) is halfway between the end points of
that interval.
x1 + x2 , y1 + y2
2
2
Eg. Find the midpoint between points (0,4)(6,10)
midpoint =
midpoint =
0+6 , 4+10
2
2
(3,7)
DISTANCE BETWEEN POINTS
d=  (x2 – x1)2 + (y2 – y1)2
let one point points have co-ordinates (x1, y1) and the other (x2, y2)
OR
You can use Pythagoras’s Theorem to find the distance between two
points. (c2=a2 + b2)
SLOPE/ GRADIENT
Talks about the degree of steepness of a line.
Slope/ Gradient
m
(m)
=
vertical rise
horizontal run
= y2 – y1
x2 – x1
= 10-4
6-0
= 6
6
= 1
POSITIVE AND NEGATIVE GRADIENTS
When a line is forward sloping ( / ) we say it has a positive gradient.
When a line is backward sloping ( \ ) we say it has a negative gradient