# Algebra

This branch of mathematics, which uses letters or symbols to represent numbers, is very powerful as it allows statements to be made that are true for all cases, not just specific ones.

Vocabulary

Explanation

Examples

constant

This is a value, usually a number, which does not change

3      8           ∏

Variable

A variable is a symbol, usually a letter, standing for any value from a given range of numbers.

x    y

p  q w

coefficient

This is a constant in front of  a variable.

The variable is multiplied by the constant.

A coefficient of 1 is usually not written.

3x means
3 times x

27y2 means 27 times y times y

z means
1 times z.

expression

An expression is a list of constants and variables joined together by the operators  + -  x ÷

45a - 27b
15x + 9y

45x +32x2

(15/c) x 3a

terms

Each part of an expression is called a term

45x +32x2
has two terms

45x +32x2-9
has three terms

## Collect like terms

Like terms have the same variable.

When we collect like terms, we gather together all  the terms that are the same.

example

x+ 5  + 8  + Q + 9 +M + x + Q+ x+Q + Q + Q + x

re arranged becomes

x+x+x+x+ Q+ Q+ Q+ Q+ Q+M+5 + 8 + 9

=4x + 5Q + M + 22

### Simplify

All the like terms are collected together by addition or subtraction .

Usually, small letters are used.

The order is coefficients, variables, ordinary numbers

Examples

4x + 17x
= 21x

a + 6b +23a – 3
=24a +6b -3

3a +6b+6-2a +3b+4
= a +9b +10

## Evaluate

Substitute the given values into the expression and calculate a result.

Don’t forget the order of operations !

Example if a = 2 and b = 3

a + 2b
= 2 + 2 x 3
= 2 +6
= 8

and

a2 - 3b + 5
= 2x2 – 3x3 +5
= 4 - 9 + 5
= 0

## equation

An equation sets two expressions equal to each other, so that they have the same value.  One of the expressions  may be a constant.

2b +3 = 7

3(a+b) =3a + 3b

y = x + 2

y = mx + c

## inequation

An inequation occurs when there is an inequality between the expressions.

3x +4  < 7

12x ≥ 60

## Satisfy an equation

An equation is satisfied if a value is substituted for a variable and the statement  is still true.

The left hand side of the equation must equal the right hand side.

b = 2 satisfies
2b +3 = 7
Since
2x2 + 3
=4+3
=7

b = 3 does not satisfy
2b +3 = 7
Since
2x3 + 3
=6+3
=10

solution

The solution is the value, or set of values, which satisfies the equation

2b +3 = 7
has solution
b = 2

## Formulae

A formula is a statement, usually an equation, which allows quantities to be calculated for given values.

Formulae can be expressed in words :-

Example

The cost of tuition is £20 per hour, plus a travel fee of £5.

How much does 6 hours tuition cost ?
Cost of tuition =Number of hours x £20 +  travel fee
=6 x £20 +  £5
= £120 +  £5
= £125
6 hours of  tuition costs  £125

or in symbols :-

Let £C = cost of tuition and h = number of hours
Then
£C =Number of hours x £20 +  travel fee

£C =h x 20 +  5

£C =20h +  5
£C =20 x6 + 5
£C = 120 +  5
£C = 125

6 hours of  tuition costs  £125

## Evaluating Formulae

Example

The  formula

Ek = ½mv2

is used to calculate the kinetic energy (Ek) of an object with mass  (m) kilograms traveling at  speed (v) metres per second.

Use the formula to calculate the energy of an object of mass 5 kg traveling
at speed 12 m/s .

Ek = ½mv2

= ½x 5 x 12 x 12
= ½x 5 x 12 x 12
= ½x 720
= 360 Joules

Changing the subject of a formula © Alexander Forrest