What Is Algebra?

Algebra is a branch of mathematics that uses letters and other symbols to represent numbers.

Dictionary Definition

The Oxford English Dictionary defines algebra as "the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations."
Here is an example of an algebraic equation. In this example, the letter x represents a number. Our job is find the value of x.

Equations ("Tetris" Game)

Here is an interactive game to help you learn about equations.

Tip: If the game is playing slowly, you can use the controls below to remove the background text and image to speed up the game.

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Where Does the Word "Algebra" Come From?

Algebra comes from the Arabic word al-jebr, which means "reunion of broken parts".

Albert Einstein and Algebra

The famous scientist Albert Einstein learned algebra from a young age. His Uncle Jakob gave him books on the subject and called algebra "a merry science". He compared algebra to hunting a little animal. You didn't know the name of the animal, so you called it "x". When you finally caught the animal you gave it the correct name.

The Curriculum

The lessons are grouped into mini-curriculum to help you organise your learning. A brief description is given for each mini-curriculum. Click the MORE button to learn more.

Algebra Definitions

There are many different words used in algebra: such as equation, variable, coefficient, term. Understanding algebra is easier when you understand what each of these words mean. In this mini-curriculum, you will learn all the definitions.

Equation
An equation has an equals sign (=) and tells us two values are equal. Here is an example of an equation:


Constant
A constant have a fixed value. In the equation below, the 2, 1, and 9 are constants.


Variable
A variable is a symbol that stands for a number. Its value is not fixed - it can take any value. In the equation below, the x is a variable.


Coefficient
A coefficient is a number that is placed in front of a letter. The coefficient is multiplying the variable. In the equation below, the 2 is a coefficient.


Exponent
An exponent tells you how many times a number (or symbol) is multiplied by itself. It is written to the right and above the number. In the equation below, the 2 is an exponent.


Operator
An operator defines an operation, such as +, , ×, and ÷. In the equation below, the + is an operator.


Term
A term is a collection of one or more numbers, letters and brackets written next to each other (they are multiplied together). Terms are separated by + or operators. In the equation below, the 2x2, 1 and 9 are terms.


Expression
An expression is one term or several terms that are added (+) or subtracted (−) together. In the equation below, the 2x + 1 and 5 are expressions. They are separated by the = sign.


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Read more about equations
Read more about constants
Read more about variables
Read more about coefficients
Read more about exponents
Read more about operators
Read more about terms in algebra
Read more about expressions in algebra

Arithmetic with Letters

Algebra uses letters and other symbols to represent numbers. Just like numbers, the letters and symbols can be added, subtracted, multiplied and divided. In this mini-curriculum, you will learn arithmetic with letters.

Adding Letters
Adding letters works when the letter is added to numbers, other letters and the same letter.


Subtracting Letters
Subtracting letters works when the letter is subtracted by numbers, other letters and the same letter.


Multiplying Letters
Multiplying letters works when the letter is multiplied by numbers, other letters and the same letter.


Dividing Letters
Dividing letters works when the letter is divided by numbers, other letters and the same letter.


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Read more about how to add letters
Read more about how to subtract letters
Read more about how to multiply letters
Read more about how to divide letters

Arithmetic with Terms

Terms are a collection of one or more numbers, letters and brackets written next to each other. Terms can be added, subtracted, multiplied and divided. In this mini-curriculum, you will learn arithmetic with terms.

Adding Terms
Terms can be added.


Subtracting Terms
Terms can be subtracted.


Multiplying Terms
Terms can be multiplied. The coefficients are multiplied, and each letter must appear in the result as many times as they appear in the multiplication.


Dividing Terms
Terms can be divided.


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Read more about how to add terms
Read more about how to subtract terms
Read more about how to multiply terms
Read more about how to divide terms

Laws of Exponents in Algebra

An exponent tells you how many times a number, letter or term is multiplied by itself. In this mini-curriculum, you will learn how to use the law of exponents in algebra.

Laws of Exponents
The laws of exponents in algebra let us understand how a letter can be multiplied by itself.


Exponent of −1
A letter with an exponent of −1 is equal to 1 divided by the letter.


Multiplying Powers
When the same letters with different exponents are multiplied together, the exponents can be added.


Dividing Powers
When the same letters with different exponents are divided, the exponents can be subtracted.


Powers of Powers
A power of a power is where a letter with an exponent is itself raised to an exponent. When a letter with an exponent is raised to another exponent, the exponents can be multiplied.


Powers of Products
A power of a product is where a term containing letters multiplied together is raised to an exponent. Each letter in the term is raised to the exponent.


Negative Exponent
A letter with a negative exponent is equal to 1 divided by the letter with the exponent made positive.


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Read more about the laws of exponents
Read more about an exponent of −1
Read more about how to multiply powers
Read more about how to divide powers
Read more about how to find a power of a power
Read more about how to find a power of a product
Read more about how to find a negative exponent

Like Terms

Like terms are terms which have the same combination of letters, each with the same exponent. The only difference allowed between terms that are like terms are the numbers and signs in front of them. In this mini-curriculum, you will learn how to find like terms and how to collect like terms to simplify expressions.

Like Terms
Like terms have the same letters with the same exponents. x, 2x and −x are like terms, they all contain x; only the number or sign in front of them is different. a2b, 5a2b and 12a2b are like terms, they all contain a2b; only the number or sign in front of them is different.


Identifying Like Terms
Identify like terms in an expression by finding terms with the same letters and exponents.


Adding Like Terms
Like terms can be added. Add the coefficients together.


Subtracting Like Terms
Like terms can be subtracted. Subtract the coefficients from each other.


Collecting Like Terms
Collect like terms by adding and subtracting the like terms you have identified. This simplifies the expression.


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Read more about like terms
Read more about identifying like terms
Read more about adding like terms
Read more about subtracting like terms
Read more about collecting like terms

Expanding Brackets

Expanding brackets means multiplying terms inside of a pair of brackets in order to remove the brackets from an expression. In this mini-curriculum, you will learn how to expand brackets.

Expanding Brackets
Expanding a single pair of brackets means multiplying the term outside of the brackets with each term inside the brackets. This removes the brackets.


Expanding Brackets with the Grid Method
Brackets can be expanded using the grid method. This is a useful method if you prefer using pictures to understand a method.


Expanding Double Brackets
Expanding double brackets means multiplying the term in two pairs of brackets that are written next to each other. This removes the brackets.


The FOIL Method
FOIL stands for Firsts, Outsides, Insides, Lasts. It helps you remember an order to multiply terms when expanding two pairs of brackets.


Multiplying Expressions
Multiplying expressions means writing each expression in a pair of brackets and expanding them.


Expanding Double Brackets with the Grid Method
Double brackets can be expanded using the grid method.


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Read more about expanding brackets
Read more about expanding single brackets using the grid method
Read more about expanding double brackets
Read more about the FOIL method
Read more about multiplying expressions in algebra
Read more about expanding double brackets using the grid method

Solving Equations with Brackets

Some equations in algebra contain brackets. In this mini-curriculum, you will learn about how to solve equations with brackets.

Solve Equations with Brackets
To solve equations with brackets, expand the brackets and rearrange the equation to find what x equals.


Solve Equations with Two Sets of Brackets
To solve equations with two sets of brackets, expand both brackets and rearrange the equation to find what x equals.


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Read more about how to solve an equation with brackets
Read more about how to solve an equation with two sets of brackets

Factoring

Factoring is the opposite of expanding brackets. It is a way of simplifying an expression. In this mini-curriculum, you will learn about factors in algebra, the greatest common factor and how to factor expressions.

Factors
A factor is one of the numbers, letters and brackets (or a product of them) that are multiplied together to make a term.


Finding Factors
You can find factors of terms by looking at the numbers, letters and brackets (with any exponents) that appear in the term. The factors of 2x are 1, 2, x and 2x.


Greatest Common Factor
The greatest common factor is the largest factor that is common to two or more terms. The greatest common factor of 2xy and 2xz is 2x.


Finding the Greatest Common Factor
Find the greatest common factor by looking at the numbers, letters and brackets that appear in two or more terms.


Factoring Expressions
Factoring simplifies an expression by writing it as a product of factors. It lets you write an expression where terms are added together to one where they are multiplied together to make a single term.


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Read more about factors in algebra
Read more about how to find factors in algebra
Read more about the greatest common factor in algebra
Read more about how to find the greatest common factor in algebra
Read more about factoring an expression

Algebraic Fractions

Algebraic fractions are fractions which contain letters and other symbols. In this mini-curriculum, you will learn about algebraic fractions, including arithmetic and simplifying.

Algebraic Fractions
An algebraic fraction is a fraction which contains letters and other symbols.


Adding Algebraic Fractions
Algebraic fractions can be added.


Subtracting Algebraic Fractions
Algebraic fractions can be subtracted.


Multiplying Algebraic Fractions
Algebraic fractions can be multiplied.

Dividing Algebraic Fractions
Algebraic fractions can be divided.


Simplifying Algebraic Fractions
Algebraic fractions can be simplified by cancelling out terms that appear on the top and bottom of the algebraic fraction.


Powers of an Algebraic Fraction
An algebraic fraction can be raised to an exponent. The top and bottom of the fraction are both raised to the same exponent.


Reciprocal of an Algebraic Fraction
The reciprocal of an algebraic fraction is found by turning the algebraic fraction upside down.


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Read more about algebraic fractions
Read more about adding algebraic fractions
Read more about subtracting algebraic fractions
Read more about multiplying algebraic fractions
Read more about dividing algebraic fractions
Read more about simplifying algebraic fractions
Read more about how to find a power of an algebraic fraction
Read more about how to find the reciprocal of an algebraic fraction