Algebra is one of the oldest branches in the history of mathematics that deals with

number theory, geometry, and analysis. The definition of algebra sometimes states
that the study of the mathematical symbols and the rules involves manipulating these
mathematical symbols. Algebra includes almost everything right from solving
elementary equations to the study of abstractions. Algebra equations are included in
many chapters of Maths, which students will learn in their academics. Also, there are
several formulas and identities present in algebra.

Table of Contents:

 Algebra Math
 Branches
 Elementary Algebra
 Advanced Algebra
 Abstract Algebra
 Linear Algebra
 Commutative Algebra
 Video lessons
 Parts
 Examples
 Related Articles
 FAQs

What is Algebra?
Algebra helps solve the mathematical equations and allows to derive unknown
quantities, like the bank interest, proportions, percentages. We can use the variables
in the algebra to represent the unknown quantities that are coupled in such a way as
to rewrite the equations.

The algebraic formulas are used in our daily lives to find the distance and volume of
containers and figure out the sales prices as and when needed. Algebra is
constructive in stating a mathematical equation and relationship by using letters or
other symbols representing the entities. The unknown quantities in the equation can
be solved through algebra.

Some of the main topics coming under algebra include Basics of algebra, exponents,
simplification of algebraic expressions, polynomials, quadratic equations, etc.
In BYJU’S, students will get the complete details of algebra, including its equations,
terms, formulas, etc. Also, solve examples based on algebra concepts and practice
worksheets to better understand the fundamentals of algebra. Algebra 1 and
algebra 2 are the Maths courses included for students in their early and later stages
of academics, respectively. Like, algebra 1 is the elementary algebra practised in
classes 7,8 or sometimes 9, where basics of algebra are taught. But, algebra 2 is
advanced algebra, which is practised at the high school level. The algebra problems
will involve expressions, polynomials, the system of equations, real numbers,
inequalities, etc. Learn more algebra symbols that are used in Maths.
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Branches of Algebra
As it is known that, algebra is the concept based on unknown values called
variables. The important concept of algebra is equations. It follows various rules to
perform arithmetic operations. The rules are used to make sense of sets of data that
involve two or more variables. It is used to analyse many things around us. You will
probably use the concept of algebra without realising it. Algebra is divided into
different sub-branches such as elementary algebra, advanced algebra, abstract
algebra, linear algebra, and commutative algebra.

Algebra 1 or Elementary Algebra

Elementary Algebra covers the traditional topics studied in a modern elementary
algebra course. Arithmetic includes numbers along with mathematical operations like
+, -, x, ÷. But in algebra, the numbers are often represented by the symbols and
are called variables such as x, a, n, y. It also allows the common formulation of the
laws of arithmetic such as, a + b = b + a and it is the first step that shows the
systematic exploration of all the properties of a system of real numbers.

The concepts coming under elementary algebra include variables, evaluating

expressions and equations, properties of equalities and inequalities, solving the
algebraic equations and linear equations having one or two variables, etc.
Algebra 2 or Advanced Algebra
This is the intermediate level of Algebra. This algebra has a high level of equations
to solve as compared to pre-algebra. Advanced algebra will help you to go through
the other parts of algebra such as:

 Equations with inequalities

 Matrices
 Solving system of linear equations
 Graphing of functions and linear equations
 Conic sections
 Polynomial Equation
 Quadratic Functions with inequalities
 Polynomials and expressions with radicals
 Sequences and series
 Rational expressions
 Trigonometry
 Discrete mathematics and probability

Abstract Algebra
Abstract algebra is one of the divisions in algebra which discovers the truths relating
to algebraic systems independent of the specific nature of some operations. These
operations, in specific cases, have certain properties. Thus we can conclude some
consequences of such properties. Hence this branch of mathematics called abstract
algebra.

Abstract algebra deals with algebraic structures like the fields, groups, modules,
rings, lattices, vector spaces, etc.

The concepts of the abstract algebra are below-

1. Sets – Sets is defined as the collection of the objects that are determined by some
specific property for a set. For example – A set of all the 2×2 matrices, the set of two-
dimensional vectors present in the plane and different forms of finite groups.
2. Binary Operations – When the concept of addition is conceptualized, it gives the binary
operations. The concept of all the binary operations will be meaningless without a set.
3. Identity Element – The numbers 0 and 1 are conceptualized to give the idea of an
identity element for a specific operation. Here, 0 is called the identity element for the
addition operation, whereas 1 is called the identity element for the multiplication
operation.
4. Inverse Elements – The idea of Inverse elements comes up with a negative number. For
addition, we write “-a” as the inverse of “a” and for the multiplication, the inverse form is
written as “a-1″.
 5. Associativity – When integers are added, there is a property known as associativity in
which the grouping up of numbers added does not affect the sum. Consider an example,
(3 + 2) + 4 = 3 + (2 + 4)

Linear Algebra
Linear algebra is a branch of algebra that applies to both applied as well as pure
mathematics. It deals with the linear mappings between the vector spaces. It also
deals with the study of planes and lines. It is the study of linear sets of equations with
transformation properties. It is almost used in all areas of Mathematics. It concerns
the linear equations for the linear functions with their representation in vector spaces
and matrices. The important topics covered in linear algebra are as follows:

 Linear equations
 Vector Spaces
 Relations
 Matrices and matrix decomposition
 Relations and Computations

Commutative algebra
Commutative algebra is one of the branches of algebra that studies the commutative
rings and their ideals. The algebraic number theory, as well as the algebraic
geometry, depends on commutative algebra. It includes rings of algebraic integers,
polynomial rings, and so on. Many other mathematics areas draw upon commutative
algebra in different ways, such as differential topology, invariant theory, order theory,
and general topology. It has occupied a remarkable role in modern pure
mathematics.

 Algebra Calculator
 Algebra For Class 6
 Algebra Formulas For Class 8
 Algebra Formulas For Class 9
 Algebra Formulas For Class 10
 Algebra Formulas For Class 11

Video Lessons
Watch the Below Videos to understand more about Algebraic
Expansion and Identities
Algebraic Expansion
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Algebraic Identities
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Parts of Algebra

Introduction to Algebra
 Algebra Basics
 Addition And Subtraction Of Algebraic Expressions
 Multiplication Of Algebraic Expressions
 BODMAS And Simplification Of Brackets
 Substitution Method
 Solving Inequalities

Exponents
 Introduction to Exponents
 Exponent
 Square Roots and Cube Roots
 Surds
 Simplifying Square Roots
 Laws of Exponents
  Exponents in Algebra

Simplifying
 Associative Property, Commutative Property, Distributive Laws
 Cross Multiply
 Fractions in Algebra

Polynomials
 What is a Polynomial?
 Adding And Subtracting Polynomials
 Multiplying Polynomials
 Rational Expressions
 Dividing Polynomials
 Polynomial Long Division
 Conjugate
 Rationalizing The Denominator

Quadratic Equations
 Solving Quadratic Equations
 Completing the Square

Solved Examples on Algebra

Example 1: Solve the equation 5x – 6 = 3x – 8.

Solution:

Given,

5x – 6 = 3x – 8

Adding 6 on both sides,

5x – 6 + 6 = 3x – 8 + 6

5x = 3x – 2

Subtract 3x from both sides,

5x – 3x = 3x – 2 – 3x

2x = -2
Dividing both sides of the equation by 2,

2x/2 = -2/2

x = -1

Example 2:

Simplify: 7x+5x−4−6x−1x−3−1x2−7x+12=1
Solution:

Consider, x2 – 7x + 12

= x2 – 3x – 4x + 12

= x(x – 3) – 4(x – 3)

= (x – 4)(x – 3)

Now, from the given,

7x+5x−4−6x−1x−3−1x2−7x+12=1
Here, LCM of denominators = (x – 4)(x – 3)

Thus,

[(7x + 5)(x – 3) – (6x – 1)(x – 4) – 1]/ (x – 4)(x – 3) = 1

7x2 – 21x + 5x – 15 – (6x2 – 24x – x + 4) – 1 = (x – 4)(x – 3)

x2 + 9x – 20 = x2 – 7x + 12

9x + 7x = 12 + 20

16x = 32

x=2

Example 3:

Solve: 17x−x2−5=7
Solution:

Given,

17x−x2−5=7
On removing the square roots of the LHS, we get;
x2 – 5 = 2401 – 1666x + 289x2

2401 – 1666x + 289x2 = x2 – 5

Adding 5 on both sides,

2401 – 1666x + 289x2 + 5 = x2 – 5 + 5

289x2 – 1666x + 2406 = x2

Subtracting x2 from sides,

289x2 – 1666x + 2406 – x2 = x2 – x2

288x2 – 1666x + 2406 = 0

Using quadratic formula,

x=−(−1666)±(−1666)2−4.288.24062.288x=−(−1666)±62576x=−(−1666)+62576,x=−(−1666)−625
76
Therefore, x = 3, 401/144

Example 4:

Solve for x:
log2⁡(x2−6x)=3+log2⁡(1−x)
Solution:

Given,
log2⁡(x2−6x)=3+log2⁡(1−x)
We know that, log2 base 2 = 1

so,
log2⁡(x2−6x)=3log2⁡2+log2⁡(1−x)

⇒log2⁡(x2−6x)=log2⁡23+log2⁡(1−x)
⇒log2⁡(x2−6x)=log2⁡8+log2⁡(1−x)
⇒log2⁡(x2−6x)=log2⁡8(1−x)
Now, by cancelling the log on both sides, we get;

(x2 – 6x) = 8(1 – x)

x2 – 6x = 8 – 8x

x2 – 6x + 8x – 8 = 0

x2 + 2x – 8 = 0
x2 + 4x – 2x – 8 = 0

x(x + 4) – 2(x + 4) = 0

(x – 2)(x + 4) = 0

Therefore, x = 2, -4

Example 5: Solve 2ex + 5 = 115

Solution:

Given,

2ex + 5 = 115

2ex = 115 – 5

2ex = 110

ex = 110/2

ex = 55

x = ln 55

Algebra Related Articles

Algebra Related Articles

Basics Of Algebra Determinants Substitution

Polynomials Relations and Functions Mean, Median and Mode

Polynomial Functions Inverse Functions Asymptotes

Factorizing of Polynomials Sequence and series Solving linear equations

Algebra Formulas Exponents Complex Numbers

Division Of Polynomial Matrices Rational Numbers

Algebraic Equations GCF’s and LCM’s Rational Function

Solving Inequalities Fractions Algebra of Matrices

Quadratics Percents Degree Of Polynomial

Algebra Related Articles

Boolean Algebra Algebraic Expressions Multiplication of Algebraic Expressions

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Frequently Asked Questions on Algebra

Q1

What is algebra?

Algebra is a branch of mathematics that deals with solving equations and finding the
values of variables. It can be used in different fields such as physics, chemistry, and
economics to solve problems. Algebra is not just solving equations but also
understanding the relationship between numbers, operations, and variables.
Q2

Why should students learn algebra?

Algebra is a powerful and useful tool for problem-solving, research, and everyday
life. It’s important for students to learn algebra to increase their problem-solving
skills, range of understanding, and success in both maths and other subjects.
Q3

Is algebra hard to learn?

Algebra is not that hard to learn, in fact, it can be simple and sometimes even fun.
Some people say that algebra is a hard subject to learn, while others confidently say
it is easy. If you think you are struggling with algebra, don’t be discouraged by what
other people have told you about it; work through the problems in your textbook until
you master the concepts without difficulty.
Q4

What are the basics of algebra?

The basics of algebra are:

Addition and subtraction of algebraic expressions
Multiplications and division of algebraic expression
Solving equations
Literal equations and formulas
Applied verbal problems
Q5

Mention the types of algebraic equations

The five main types of algebraic equations are:

Monomial or polynomial equations
Exponential equations
Trigonometric equations
Logarithmic equations
Rational equations
Q6

What are the branches of algebra?

The branches of algebra are:

Pre-algebra
Elementary algebra
Abstract algebra
Linear algebra
Universal algebra