NCERT Solutions for Class 9 Political Science Chapter 2

August 8, 202239 Insightful Publications

Algebra is a branch of math dealing with several symbols. Algebra represents or helps in representing problems or situations as mathematical expressions. It comprises variables such as \(x,\;y,\;z\). Moreover, it also involves mathematical expressions like subtraction, addition, multiplication, and division in order to generate a significant mathematical expression. Moreover, Algebra has its application in all the branches of math like trigonometry, calculus, coordinate geometry, etc. In this blog let us understand the importance of algebra.

As mentioned above, Algebra is a branch of math that deals with variables, symbols, variables, numbers and the rules for manipulating it. It helps solve mathematical equations and assists in deriving unknown quantities such as the bank interest, proportions, percentages. Thus, before we dive deep into understanding Algebra, its applications, branches, etc., let us first understand the basic terms associated with Algebra.

- Variables
- Constant
- Coefficient
- Terms
- Expressions
- Equations

1. **Variable**: A variable is a symbol that we assign to an unknown value. The variables are often in the form of letters like \(x, y, a, b\).

2. **Constant**: It is a fixed quantity that does not vary. For instance, \(3, –6, π\).

3. **Coefficient**: The coefficient of a variable is the amount that is placed in front of a variable—for instance, \(3 × w=3w\), where \(3\) is the coefficient.

4. **Terms**: The term can be a constant, variable, coefficient. Examples involve \(3, 10, π, –3x, 11y,\) etc.

5. **Expressions**: An expression comprises one or more terms, e.g. \(4w+ 8xy + 5\).

6. **Equations**: An equation involves two expressions with an equal sign. The expression on one side of the equal symbol has the equivalent value as the expression on the opposite side. e.g. \(4 + 6 = 5 × 2\).

Algebra is very beneficial in expressing a mathematical equation and relationship by employing letters or other symbols describing the entities. Algebra helps in finding the unknown value in the equation. Moreover, the Algebraic formulas help the students to find the distance, volume of containers, sales prices, etc. Additionally, the importance of algebra can be seen in various fields.

Moreover, Algebra is a mathematical concept and a skill that we employ in our daily life without even realising it. Knowing algebra as a concept is more valuable than solving equations and getting the right answer. As algebra can be used for understanding various other topics of maths that you will learn or have learnt before.

Now that we know the basic terms, it is important to note that Algebra is divided into various sub-branches like Elementary, Advanced, Abstract, Linear, and Commutative.

Elementary Algebra incorporates the conventional topics studied in a modern elementary algebra course. Arithmetic involves numbers besides mathematical operations such as \(+, -, x, ÷\). However, in Algebra, numbers are depicted via symbols and are titled variables like \(x,\;y,\;a,\;b\). It also enables the general formulation of the laws of arithmetic like “\(a + b = b + a\)”, and it is the first step that confers the systematic examination of all the properties of a system of real numbers. Some of the concepts covered in Elementary Algebra are as follows:

- Variables
- Evaluating expressions & equations
- Properties of equalities & inequalities
- Solving the linear equations and algebraic equations having one or two variables.
- Positive rational roots and exponents

Advanced Algebra is also known as intermediate level Algebra. However, its level of equations is a little high as compared to elementary algebra. Additionally, the topics covered under Advanced Algebra are as follows:

- Matrices
- Solving system of linear equations
- Equations with inequalities
- Conic Sections
- Polynomial Equation
- Graphing of linear equations and functions
- Polynomials and expressions with radicals
- Sequence and Series
- Rational Expressions
- Trigonometry
- Discrete Mathematics and Probability
- Quadratic functions with inequalities

This branch of Algebra discovers the truths associated with algebraic systems independent of the specific nature of some operations. Additionally, abstract algebra is the collection of advanced topics of algebra that dispense with abstract algebraic structures instead of the standard number systems. Moreover, it deals with algebraic structures such as groups, fields, modules, rings, lattices, vector spaces, etc. The Abstract Algebra incorporates the following topics:

- Sets
- Binary Operations
- Identity Element
- Inverse Elements
- Associativity

Linear Algebra is a branch of algebra that deals with both applied as well as pure mathematics. Moreover, it also deals with the linear mappings between the vector spaces. It also deals with the study of planes and lines. This branch of algebra also deals with the linear sets of equations with the transformation properties. It has its applications in all the areas of Maths. It involves the linear equations for the linear functions with their symbol in vector spaces and matrices. Some of the essential topics incorporated in linear algebra are as follows:

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

This branch of algebra deals with the studies of commutative rings and their ideals. The algebraic geometry and the algebraic number theory relies on commutative algebra. It involves polynomial rings, rings of algebraic integers and so on. Further, there are several other areas of maths that represents commutative algebra in distinct ways. It involves differential topology, invariant theory, order theory, and general topology. Commutative Algebra holds a remarkable role in modern pure mathematics.

Algebra is divided into various topics that help you solve equations and excel in other fields of mathematics. Here we have listed some of the significant topics of algebra like equations, sequence and series, exponents, logarithms, and sets. However, if you intend to make a career in Algebra, the understanding that you will need in the subject is a little more complicated. But now is the time to make sure you get the basics right.

An algebraic expression comprises integer constants, variables, and basic arithmetic operations of addition(+), subtraction (-), multiplication (×), and division (/). One of the instances of an algebraic expression is 6x+7. Here 6 and 7 are fixed numbers, and \(x\) is a variable. Additionally, the variables can be simple variables employing alphabets such as \(a,\;b,\;x,\;y\) or complex variables such as \(x^2,\;x^3,\;xy,\;x^2y,\), etc.

Algebraic expressions are also recognized as polynomials. A polynomial is an expression including variables (also called indeterminates), coefficients, and non-negative integer exponents of variables. For instance: \(4\times3+2\times6+4x+2=0\).

A set of numbers having a relationship with the numbers is called a sequence. It is a set of numbers having a typical mathematical relationship with the number, and a series is the total of the terms of a sequence. However, in maths, we can witness two broad number sequence and series that involves geometric progression and arithmetic progression. The series is further classified into finite and non-finite series.

The two series are also called geometric progression and arithmetic progression and can be depicted as follows.

**Arithmetic Progression:**An AP (Arithmetic progression) is a particular kind of progression in which the distinction between two consecutive terms is a constant. Furthermore, the arithmetic progression series are: \(a, a+x, a + 2x, a + 3x, a + 4x, a + 5x, …..\)**Geometric Progression:**GP or Geometric Progression is the progression in which the ratio of adjacent terms is fixed. Further, the form of representation of a geometric sequence is \(a,ar,a{r^2},a{r^3},….\)

Exponentiation is a mathematical process where it involves two numbers, the base a and the exponent or power n. Exponents are employed to simplify algebraic expressions. Further, exponents involve squares, square root, cubes, and cube root. The exponents can be depicted in the form \(a^n\;=\;a\;x\;a\;x\;a\;x\;…\;n\;times.\;\)

The logarithm is the inverse function to exponents. Moreover, logarithms are a useful way to simplify significant algebraic expressions. The exponential form \(ax\;=\;n\) can be transformed into logarithmic form as \(Logan = x\). Moreover, John Napier invented the notion of Logarithms in 1614. Logarithms have now converted into a prominent part of modern maths.

A set is a well-defined group of distinct objects and represents algebraic variables. Moreover, the goal of applying sets is to describe the collection of relevant objects in a group. For instance: \(Set\;N=\{2,4,6,8\}\)……….(A set of even numbers.), $SetM=\left\{a,e,i,o,u\right\}$……(A set of vowels).

Algebraic formulas indicate that the left-hand side of the equation is equal to the right-hand side for all values of the variables. However, these formulae include squares and cubes of algebraic expressions. It also helps the students to solve the algebraic expressions in a few fast steps. So, the frequently used algebraic formulae are as follows:

- \((a+b)^2=a^2+2ab+b^2\)
- \((a\;–\;b)^2\;=\;a^2\;–\;2ab\;+\;b^2\)
- \((a\;+\;b)(a\;–\;b)\;=\;a^2\;-\;b^2\)
- \((x\;+\;a)(x\;+\;b)\;=\;x^2\;+\;(a\;+\;b)x\;+\;ab\)
- \((x\;+\;a)(x\;–\;b)\;=\;x^2\;+\;(a\;–\;b)x\;–\;ab\)
- \((x\;–\;a)(x\;+\;b)\;=\;x^2\;+\;(b\;–\;a)x\;–\;ab\)
- \((x\;–\;a)(x\;–\;b)\;=\;x^2\;–\;(a\;+\;b)x\;+\;ab\)
- \((a\;+\;b)^3\;=\;a^3\;+\;b^3\;+\;3ab(a\;+\;b)\)
- \((a\;–\;b)^3\;=\;a^3\;–\;b^3\;–\;3ab(a\;–\;b)\)
- \((a\;+\;b)^4\;=\;a^4\;+\;4a^3b\;+\;6a^2b^2\;+\;4ab^3\;+\;b^4\)
- \((a\;–\;b)^4\;=\;a^4\;–\;4a^3b\;+\;6a^2b^2\;–\;4ab^3\;+\;b^4\)
- \((x\;+\;y\;+\;z)^2\;=\;x^2\;+\;y^2\;+\;z^2+\;2xy\;+2yz\;+\;2xz\)
- \((x\;+\;y\;–\;z)^2\;=\;x^2\;+\;y^2\;+\;z^2\;+\;2xy\;–\;2yz\;–\;2xz\)
- \((x\;–\;y\;+\;z)^2\;=\;x^2\;+\;y^2\;+\;z^2\;–\;2xy\;–\;2yz\;+\;2xz\)
- \((x\;–\;y\;–\;z)^2\;=\;x^2\;+\;y^2\;+\;z^2\;–\;2xy\;+\;2yz\;–\;2xz\)
- \(x^3\;+\;y^3\;+\;z^3\;–\;3xyz\;=\;(x\;+\;y\;+\;z)\;(x^2\;+\;y^2\;+\;z^2\;–\;xy\;–\;yz\;-xz)\)
- \(x^2\;+\;y^2\;=\left[(x\;+\;y)^2\;+\;(x\;–\;y)^2\right]\)
- \((x\;+\;a)\;(x\;+\;b)\;(x\;+\;c)\;=\;x^3\;+\;(a\;+\;b\;+\;c)x^2\;+\;(ab\;+\;bc\;+\;ca)x\;+\;abc\)
- \(x^3\;+\;y^3\;=\;(x\;+\;y)\;(x^2\;–\;xy\;+\;y^2)\)
- \(x^3\;–\;y^3\;=\;(x\;–\;y)\;(x^2\;+\;xy\;+\;y^2)\)
- \(\begin{array}{l}x^2\;+\;y^2\;+\;z^2\;–\;xy\;–\;yz\;–\;zx\;=\;\left[(x\;–\;y)^2\;+\;(y\;–\;z)^2\;+\;(z\;–\;x)^2\right]\\\end{array}\)

**Formulating a Theorem and Formula Sheets**

Math and, in particular, Algebra involves loads of formulas which can sometimes be confusing when you try to memorize them. Write down all the formulas for algebra in a sheet and make it a point to review it every day so that it becomes easier to understand and memorize the formulas without confusion. Just make sure while you study these formulas and theorems, you also practice their applications. This will make the subject easier.

2. **Grasp Concepts**

While math is majorly all about numbers, there are comprehensive concepts to help explain different math problems. Sometimes, students make the question harder than it is. So students should make sure to read the question properly and to ensure they grasp the problem before starting to work on the questions.

3. **Make your Basics Strong**

One can succeed in a particular field if the basic or the foundation is strong. The same thing is applicable to Algebra. Until and unless your basics of algebra are not in place, you won’t be able to master the subject. Thus, make sure you clear all your doubts and have sound knowledge about the basic concepts in algebra. If required you can also take extra help to help you in understanding the basics.

4. **Practice Consistently**

As with all subjects, the best way to get better at Algebra is to practice it. Students can find practise problems online or in workbooks. Thus, practising consistently will help you to succeed in the subject without any confusion gradually. One needs to practice problems regularly so that the concepts get imprinted in the mind.

Notwithstanding, the subject you are seeking to master, learning takes time, and every one has distinctive methods that work for them. But when we talk about algebra, the initial requirement is having a stronghold on the basics. Once you know that your basics are strong you can work on other tips to help you with the subject.

Moving Forward, Algebra has several real-life applications apart from the world of rules and properties. However, understanding the practical application of algebra is essential to ignite interest, and it also helps you see algebra beyond rules and technicalities.

Many Mathematicians utilize their time wondering about how they can apply the collections of abstract symbols. In real life, algebra integrates into all other areas as a tool. Moreover, the equations you learn and solve are most likely to come across in your daily routine.

Here are some of the real-life applications of Algebra:

**Algebra in Business and Finance**

Algebra is as important in business as a chef in a kitchen. A business person employs algebraic operations to calculate the profits or losses in a business. Further, a business person makes use of Algebra to determine if a piece of equipment does not lose its worthwhile it is available. Also, Algebra helps to determine the finances required to run the business. Furthermore, the business owner also applies algebra to calculate the lowest price at which an item has to be introduced in the market to cover the cost. Algebra also helps to understand the cost and time ratio.

Moreover, it is also useful in the finance area. The business owner makes use of algebraic expressions to compute the finance rates, interest rates, exchange rates, tax liability, etc. Thus, having sound knowledge in the subject helps officials to carry out finances correctly. Also, to understand the terms and conditions of a loan or an investment account, a thorough understanding of numbers, particularly algebra, is needed. Furthermore, the extension in the business market is frequently exponential, which also demands a great knowledge of fundamental algebraic operations.

2. **Algebra in Technology**

The luxury of flat LCD’s with various applications is all because of Algebra. Moreover, the computer games that we play, the smartphones that we use, and the cars we drive are all possible due to Algebra. However, we all know that, in algebra, particular numbers are replaced by symbols. Moreover, while playing a computer game, you recognize a character; that character is nothing but a series of symbols. Also, a computer programmer applies his knowledge of algebra to put forth the sequence of symbols. Additionally, the computer programmer uses a set of rules that makes the symbols interact in the correct way. That obviously has its applications in algebra.

3. **Algebra Enhances** **Logical Thinking**

Understanding algebra also helps in logical thinking and allows a person to break down a problem and then solve it. However, you might not witness the theoretical algebraic problems every day; exposure to algebraic problems and equations at some point in life will help you to think logically. Moreover, this capability of thinking logically will help you at home and in your office and allow you to take mindful decisions rather than impulsive ones.

4. **Algebra and Career Path**

Some well-known career options have algebra as a prerequisite. It involves computer programmers, pilots, electrical engineers, software engineers, actuaries, investment banking analysts and business analysts. Thus, these fields require the students to be meticulous with numbers. Moreover, employers in these fields look for people who have good knowledge of algebra and know how to apply it effectively. Thus, students wanting to establish a career in Algebra have to work hard and master the subject. Nevertheless, having sound knowledge of algebra makes it easier for the students to get hired in top companies, move forward with a career goal, and seek it with missionary zeal.

Conclusively, Algebra is an essential life skill worth learning well. It moves us beyond elementary math and prepares us for statistics and calculus. Further, algebra is the language that mathematics employs to explain the general characteristics of objects. Algebra discovers its use in math, engineering, statistics, science. Moreover, understanding algebra becomes easy when you have a good hold of basics and try to apply them to the problems. Further, practising the subject helps you to succeed in the subject. Also, consistency plays a huge role in Algebra as it helps you to master the subject. Thus, algebra is a prominent subject that helps you in all walks of life.

*Q.1. What are some real-life applications of Algebra?*** Ans:** Some of the real-life applications of Algebra are as follows:

1. Algebra in Business and Finance

2. Algebra in Technology

3. Algebra in Logical Thinking

4. Algebra and Career Path

*Q.2. What are the Branches of Algebra?*** Ans:** The branches of Algebra are divided into the following categories:

1. Elementary Algebra

2. Advanced Algebra

3. Abstract Algebra

4. Linear Algebra

5. Commutative Algebra

*Q.3. What is the significance of Algebra?*** Ans: **Algebra is very beneficial in expressing a mathematical equation and relationship by employing letters or other symbols describing the entities.

*Q.4. What are the tricks to master Algebra?*** Ans: **Some of the tricks to master Algebra are as follows:

1. Formulating formula sheets

2. Practicing consistently

3. Making the basics strong

4. Grasping concepts

*Q.5. What is Elementary Algebra?*** Ans:** Elementary Algebra incorporates the conventional topics studied in a modern elementary algebra course.

*We hope you find this detailed article on Algebra useful. If you have any queries feel free to reach out to us in the comment section below, and we will get back to you at the earliest.*