What is Algebra?

algebra.jpgAlgebra is the part of mathematics that is used to help to solve problems. It provides a way for a problem and its solution to be written down.

In algebra, letters or symbols are used to represent numbers.

The letters are often called variables or pronumerals. In a problem the variable often stands for the unknown thing we are trying to find.

e.g. A problem might ask "What is the cost of 7 apples if each apple costs 50 cents?"
In this problem we could let c stand for the cost. So c would equal 7 x 50
This means we could write an equation: c = 7 x 50
Solving this equation would give c = 350 cents = $3.50

Most people would work out this problem in their head but algebra like this allows us to work out much more difficult problems.

 

Meanings

First of all, the meanings of some of the words used in algebra are given below;

  • Variable or pronumerals are usually represented by letters.
3x x is the varaible.
4 + p p is the variable.
mn = 5 m and n are the variables.
  • An algebraic term can be made up of coefficient, variables and exponents (sometimes called an power or index).
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  • Like terms have the same variables and exponents.

e.g. { 2a, 4a, -6a } are like terms.
a is common to all.

 

{ x, 3a2, 6 } are unlike terms. 
There is no common term.
  • An algebraic expression is a group of terms.

e.g. 4x + 3y is an expression.

 

Words into Symbols

Many mathematics problems are written in words and these words often have to be changed into algebraic expressions using letters and symbols.

We need to be able to write down mathematical operations using algebra:

 

Algebra Expression
Words

x + 3

3 is added to x

y − 5

5 is subtracted from y

4z

z is multiplied by 4
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 w is divided by 10
xy
 The product of x and y. 
It means x is multiplied by y
a − 5
 The difference between a and 5.
It means 5 is taken away from a.

7 + w

The sum of 7 and w.
It means 7 and w are added together.
pq + 5
 p is muliplied by q and 5 is added
6c = 30
 6 multipied by c is equal to 30

In this section on algebra, six topics will be used to introduce the following algebraic ideas:

Indices or Exponents

When 3 is multiplied by 3, this can be written as 32. In algebra if x is multiplied by x this can be written as x2. The two is called an index or exponent.
Working with indices or exponents is covered in detail in Topic 37 Exponents

Simplifying

Often in algebra, expressions can be simplified. For example, 2x + 3x can be simplified to 5x.
Simplifying expressions is covered in detail in Topic 38 Simplifying Expressions

 

Expanding and Factorising

If an expression such as 2x + 3 is multiplied by 5, this can be written 5(2x + 3). This is said to be in factorised form and if these factors are expanded this gives 10x + 15. The 5 is multiplied by each of the terms inside the brackets.
Expanding and factorising is covered in detail in Topic 39 Expanding and Factorising

Substitution

A variable in algebra can be given a value. For example if p is given the value of 4 then a term like 6p becomes 6 x 4 = 24. This is called substituting or evaluating the term.
Substituting values into expressions is covered in detail in Topic 40 Substitution

Equations and Formulae

When solving a problem using algebra an equation is often formed. For example if 6 bananas are bought and the total cost is $4.20 then an equation can be formed to find the cost of each banana. The equation would be 6x = 4.20
Working with equations is covered in detail in Topic 41 Equations

Patterns, Sequences and Graphs

Solving problems is often helped by looking for a pattern, finding a missing term in a sequenceor drawing a graph. This topic looks at how this can be done.
Working with patterns, sequences and graphs is covered in detail in Topic 42 Coordinates, Sequences and Graphs