Algebra

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.

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.

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.

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:

$\begin{array}{l}��������:\frac{7�+5}{�-4}-\frac{6�-1}{�-3}-\frac{1}{{�}^2-7�+12}=1\end{array}$

Solution:

Consider, x2 – 7x + 12

= x2 – 3x – 4x + 12

= x(x – 3) – 4(x – 3)

= (x – 4)(x – 3)

Now, from the given,

$\begin{array}{l}\frac{7�+5}{�-4}-\frac{6�-1}{�-3}-\frac{1}{{�}^2-7�+12}=1\end{array}$

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:

$\begin{array}{l}�����:\sqrt{17�-\sqrt{{�}^2-5}}=7\end{array}$

Solution:

Given,

$\begin{array}{l}\sqrt{17�-\sqrt{{�}^2-5}}=7\end{array}$

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,

$\begin{array}{l}�=\frac{-(-1666)\pm \sqrt{{(-1666)}^2-4.288.2406}}{2.288}\\ �=\frac{-(-1666)\pm 62}{576}\\ �=\frac{-(-1666)+62}{576},�=\frac{-(-1666)-62}{576}\end{array}$

Therefore, x = 3, 401/144

Example 4:

Solve for x: 

$\begin{array}{l}{\log }_2({�}^2-6�)=3+{\log }_2(1-�)\end{array}$

Solution:

Given,

$\begin{array}{l}{\log }_2({�}^2-6�)=3+{\log }_2(1-�)\end{array}$

We know that, log2 base 2 = 1

so,

$\begin{array}{l}{\log }_2({�}^2-6�)=3{\log }_{2}2+{\log }_2(1-�)\end{array}$

 

$\begin{array}{l}\Rightarrow {\log }_2({�}^2-6�)={\log }_2{2}^3+{\log }_2(1-�)\end{array}$

$\begin{array}{l}\Rightarrow {\log }_2({�}^2-6�)={\log }_{2}8+{\log }_2(1-�)\end{array}$

$\begin{array}{l}\Rightarrow {\log }_2({�}^2-6�)={\log }_{2}8(1-�)\end{array}$

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