# Chapter 3 – Algebra

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Do you know algebra is known as one of the oldest subjects in the history of mathematics? This study of various mathematical symbols and rules includes number theory, geometry, and analysis. Algebra can get used in solving numerous elementary equations and in-depth study of abstractions. It is one of the most pivotal chapters that need to get studied thoroughly to understand the concept better.

Algebra refers to mathematical symbols used in solving mathematical equations and establishes a mathematical relationship by using symbols, signs, or generalities. Algebra contains variables that can get used in rendering unknown quantities and derives meaningful equations. By using Algebra, one can solve complex equations and identify the unknown quantities swiftly. Do you know that we use algebra in our daily lives? Yes, algebraic expressions are used to find the distance, the volume of the bottle and estimate the sales prices.

#### Branches of Algebra

Truth be told that Algebra is all about equations and, outrightly, based on unknown values called variables. Thus, to perform arithmetic operations, there are various sets of rules that need to get followed. Algebra is a cardinal life skill that is worth understanding as it embraces the journey from basic mathematics to statistics and calculus. Algebra is divided into different branches: elementary algebra, advanced algebra, abstract algebra, linear algebra, and commutative algebra.

• Elementary algebra: Elementary algebra is one of the fundamental mathematics fields that talks about the properties of numbers and their relations. It helps in building good knowledge about the understanding of arithmetic. Various civilisations like Babylonian, Greek, Indian, Chinese, and Islamic have contributed to advancing elementary algebra. Elementary algebra’s basic knowledge helps in the apprehensions of other subjects like statistics, computer science, economics, and business.

• Advanced algebra: It is one of the cardinal branches of algebra that emphasised intermediate-level algebra expressions. It includes the topic related to Equations with inequalities, matrices, solving systems of linear equations, graphing of functions and linear equations, conic sections, polynomial equations, quadratic functions with inequalities, rational expressions, and trigonometry. It prepares students to solve high-level algebraic equations with ease.

• Abstract algebra: Abstract algebra encompasses abstract algebraic structures, preferably than the usual number systems. It is related to algebraic expressions independent of the distinct nature. Abstract algebra explains advanced algebra topics and ensures an in-depth understanding of commutative algebra, representation theory, and homological algebra that is a part of abstract algebra. This branch of mathematics is called abstract algebra.

• Linear algebra: It is the branch of mathematics dealing with linear equations. Today, linear algebra is a central locus apex of all facets of mathematics that works on the modern portrayal of geometry, lines, planes, and rotations. In a nutshell, it deals with the application of linear algebra to spaces of functions.

• Commutative algebra: It analyses commutative rings, their ideals, and modules of those rings. Commutative algebra is a part of algebraic geometry and algebraic number theory. With the help of commutative algebra, various other disciplines of mathematics can get derived swiftly. It is one of the pivotal components of modern pure mathematics.

#### Algebraic Expressions Properties of Inequalities

There are 5 properties of Inequalities.

1. Addition property: It states that if any inequality subsists when adding any same number on both sides will not change the discrepancy in that number.
2. Subtraction property- It states that if any inequality subsists when subtracting any same number on both sides, inequality will still exist.
3. Multiplication property: It affirms that if one side of the inequality can multiply by any number, then another side of the inequality can multiply by the same number.
4. Division property: It insists that if one side of the inequality can get divided by any number, then another side of the inequality can get divided by the same number.
5. Transitive property: It states that for any real numbers a, b, c. If a ≤ b and b ≤ c, then a ≤ c.

In the case of multiplication and division of algebraic expression, you need not consider whether the variable is the same or not; you need to multiply/divide every expression simultaneously to get the results.

#### Factorisation

Factorisation refers to any entity’s cleavage, such as a number, a matrix, or a polynomial. It converts that entity into a product of another object. This concept is mostly taught in secondary classes from 6 to 8. For example- -12, -6, -2, -1, 1, 2, 6, and 12 are called factors because their reminder is always 12. It is sometimes referred to as algebra factorisation.

#### Linear Equations in One Variable

The linear equations in one variable refer to an equation that gets manifested in ax+b = 0, where a and b are two integers, x is a variable with only one solution.

The first step of solving linear equations in one variable is to simplify and use add or subtract properties to move the variable term to one side and all other terms to the other side.

#### Conclusion

The true virtue of algebra is that it is the basic language that can get used to describe various real-world phenomena in no time. From measuring gravity to finding the value of an unknown object, Algebra can be used to solve any mathematical problem.

#### FAQs

1. What are the basics of algebra?
The basics of algebra include:

• Addition and subtraction of algebraic expressions
• Multiplications and division of algebraic expression
• Verbal Problems
• Literal equations
2. Which category of math can be termed algebra?
Algebra refers to the branch of mathematics that talks about various symbols, and dominions used manipulating symbols and signs. The symbols you see written in Latin and Greek letters are variables, such as “x” or “y.”

3. Is algebra easy to learn?
Algebra is never hard to learn, but all you need is a better understanding of the subject matter and in-depth apprehension of concepts. Many students study the mechanics of algebra than focussing on its face value. It helps them to get a better idea about the core approach of algebra before its implication.

4. What is the usage of algebra?
The primary usage of algebra is to build a mathematical relationship and its equation with the help of symbols, alphabet, and letters. With the help of algebra, you can supersede values and explain the equations for the unfamiliar quantities.

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### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
• Chemical Kinetics
• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
• Electrolysis
• Elements, Compounds and Mixtures
• Environmental Chemistry
• Equilibrium
• Ethers and Carbonyl compounds
• Haloalkanes and Haloarenes
• Hydrocarbons
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• Ideal solutions
• Introduction to Organic Chemistry
• Ionic equilibria
• Matter
• Matter Around Us
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• Mole Concept and Stoichiometry
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• Periodic Classification of Elements
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• Preparation, Properties and Uses of Compounds
• Principles and Processes of Isolation of Elements
• Redox Reactions
• Relative Molecular Mass and Mole
• States of Matter
• Structure Of The Atom
• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
• Surface Chemistry
• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
• Thermodynamics

### High School Biology

• Absorption and Movement of Water in Plants
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time