MATH IMAN NAUFAL 101

THE LANGUAGE OF ALGEBRA

THE LANGUAGE OF ALGEBRA

THE LANGUAGE OF ALGEBRA

SUBSTITUTION INTO FORMULAE







SIMPLIFYING AN ALGEBRAIC EXPRESSION

By “simplifying” an algebraic expression, we mean writing it in the most compact or
efficient manner, without changing the value of the expression. This mainly involves
collecting like terms, which means that we add together anything that can be added
together.

Simplifying Expressions
1.Solution:
a) 14x + 5x = (14 + 5)x = 19x.
b) 5y – 13y = (5 –13)y = –8y.
c) p – 3p = (1 – 3)p = – 2p. ...

For example,
3x + 2y – 2x + 6. = 3x– 2x + 2y + 6. = (3 – 2)x + 2y + 6. ...

Example: Simplify 3x + 2a – 4x.
Solution: 3x + 2a – 4x. =3x– 4x + 2a. = (3 – 4)x + 2a.

SOLVING LINEAR EQUATIONS







SETTING UP EQUATIONS

SOLVING THE QUADRATIC EQUATIONS

SOLVING THE QUADRATIC EQUATIONS BY THE
QUADRATIC FORMULA

SOLVING THE QUADRATIC EQUATIONS BY
COMPLETING THE SQUARE













WHAT ARE INEQUALITIES ?

In our daily lives, we often make
comparisons between two quantities with
different values. We compare the quantities
in terms of number, price, temperature, size,
height, mass and so on.

The relationship between two quantities
that do not have the same value is known
as an inequality.

SOLVING LINEAR EQUATIONS

When you have an equation that
contains brackets, you first must multiply
out the brackets and then solve the
resulting equation.













RATIO

Ratio is used to compare two or more quantities of
the same kind that are measured in the same unit.
For example, the ratio of 5 000 g to 9 kg can be
represented as
5 000 g : 9 kg = 5 kg : 9 kg

= 5: 9

The ratio of a to b is written as a : b.

RATIOS AS FRACTIONS

A ratio in its simplest form can be expressed
as portions of a quantity by expressing the
whole numbers in the ratio as fractions with
the same denominator ( bottom number).

RATIO

Divide $28 in the ratio4 :3 Dividing Amounts In A Given Ratio
4+3=7 parts altogether To divide an amount in a given ratio,
So 7 parts = $28 first look at the ratio to see how many
Dividing by 7: parts there are altogether. Example
1 part = $4
4 parts = 4 x $4 = $16 and 3 parts = 3 x $4 = $12 Map Scales
So $28 divided in the ratio 4:3 = $16: $12 Map scales are often given as ratios in
the form 1:n. Example
A map of New Zealand has a scale of 1: 900 OOO.
The distance on the map from Auckland to Hamilton is 11.5 centimetres, Two business partner, Lubna and Adama, divided their total
What is the actual distance? profit n the rai : ratio 3:5,
1cm on the map = 900 OOO centimetres on the ground. Lubna received $2100. How much did Adama get? : sa
= 9000 metres (100 centimetres = 1 metre) Lubna $2100 was ofthe total oft (Check that you row why)
= Q kilometres (1000 metres = 1 kilometre). fof the total profit = $2100 = 3 = $700
The distance is 11.5 x 9 kilometres So Adama’ share hich was 8, amounted to $700 x 5 = $3600,
= 103.5 kilometres.

Calculating With
Ratios When Only

Part Of The
Information Is Known

INCREASES AND DECREASES USING RATIOS

Increasing or Decreasing a Quantity in a Given Ratio. If the ratio of a new
quantity to an old quantity can be expressed as an improper fraction, then the
new quantity is greater than the old quantity. Applying this ratio to the old

quantity is known as increasing the old quantity in a given ratio.

RATES

Rates are ratios, any ratio with different Summary:
units in the numerator and denominator is 1.A rate refers to the frequency by which a certain
called a rate. To solve most questions event happens while a ratio refers to the relationship
involving rates of all kinds, all we have to do
is to set up an equation of the form ratio = between the size, number, or degree of two or more
ratio. We just match the units on each side. things.
2.A rate is a comparison between two measurements
of the same units while a ratio is the proportion of
one thing to another.
3.A rate refers to the fixed quantity of two things
while a ratio refers to the relationship between
various things.
4.A ratio indicates the difference between things
while a rate indicates the changes in their
measurements or units.
5.A ratio is indicated by the quotient of one quantity
divided by the other while a rate is indicated by the
comparison between two things.

DIRECT PROPORTION

The term direct
proportion means that
two (or more) quantities
increase or decrease in the
same ratio.

If eight pens cost $2.64, what is the cost of five pens?
First, find the cost of one pen. This is $2.64 + & = $0.33
So, the cost of five pens is $0.33 x 5 = $1.65

INVERSE PROPORTION

The equation for inverse proportion is x y = k or x = k/ y. Therefore, for
finding the value of the constant k, you can use the known values and then
use this formula to calculate all the unknown values.