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Chapter 3 – Algebra

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

Introduction

In the ICSE syllabus of class 9th mathematics, the topic ‘Algebra‘ is an essential chapter as it forms the basis of higher-level mathematics. It covers all the important topics related to algebra, numbers, and equations. It is also used in successive chapters of class 9th mathematics, so a clear understanding of algebra will help you learn other mathematical concepts and chapters better.

The chapter covers the following topics:

    • Introduction to algebra 
    • Expansions 
    • Factorisation
    • Simultaneous linear equations in two variables
    • Exponents

 

Algebra is an important branch of mathematics that involves numbers, equations, and mathematical analysis. There are different methods and formulas for solving algebraic problems. It is one of the most important chapters of mathematics. Without a proper understanding of algebra, solving other questions is challenging. 

For better understanding, algebra is divided into five branches:

  • Elementary algebra: It covers basic algebra topics like variables, algebraic problems, equations, linear equations, etc. 
  • Advanced algebra: Also known as intermediate algebra, this includes higher levels of equations, like inequality equations, linear graphing equations, trigonometry, quadratic equations, matrices, etc. 
  • Abstract algebra: This branch of algebra analyses algebraic equations independently according to the specific nature of algebraic operations. It covers topics like sets, inverse elements, identity elements, etc.
  • Linear algebra: This involves studying the lines, linear equations, and planes. It covers topics like linear equations, computations, relations, vector spaces, etc.
  • Commutative algebra: It studies the commutative rings of algebraic equations. It forms the basis for important algebraic theories, like algebraic number theory, invariant theory, order theory, etc.

Expansion is a mathematical function of expanding the problem by removing the brackets from the equation. It becomes crucial to remove brackets, mainly to solve equations that need to be simplified to get the solution. 

Following are a few formulas and identities used in the expansion of algebraic equations:

  • (x + y) (x + z) = x^2 + (y + z) x + yz
  • (x + y) (x – z) = x^2 + (y – z) x – yz 
  • (x – y) (x + z) = x^2 – (y – z) x – yz
  • (x – y) (x – z) = x^2 – (y + z) x + yz
  • (x + y)^2 = x^2 + 2xy + y^2
  • (x – y)^2 = x^2 – 2xy + y^2
  • (x + y) (x – y) = x^2 – y^2

Factorisation is the act of factoring a number or a polynomial into various numbers or factors, such that when these factors are multiplied, they give the original number or polynomial. Factorisation is the opposite of expansion. 

Following are the various methods of factorisation:

  • Grouping method: It means arranging the even-numbered polynomials in groups, such that each group has a common factor. 
  • Common factors method: It means taking out the common factors and then dividing the polynomial. 
  • Splitting the middle terms method: This method involves splitting the middle term of the trinomial polynomial arranged in descending order, such that their product is equal to the product of the first and last terms.
  • Difference of two squares method: It involves using the identity x^2 – y^2 = (x + y) (x – y) to solve the polynomial.
  • Difference or sum of two cubes method: It involves using the identity of cubes to solve the polynomial equations.

A group of linear equations in the same two variables is called simultaneous linear equations in two variables. The same pair of two variables solve these linear equations. Simultaneous linear equations can either have one solution, infinite solutions, or no solution. 

Following are the methods for solving simultaneous linear equations in two variables:

    • Substitution method: In this method, find the value of any one variable in the form of another variable for the first linear equation. Then substitute the value of the variable in the second equation to find the numerical value of the two variables. 
    • Elimination method: Multiply both the equations with a number such that one of the variables of both the equations becomes equal. Then subtract or add the equations to eliminate that variable and find the value of the two variables.
    • Cross product method: This method involves converting the linear equations into a general form to apply the cross product method to find the value of the two variables.

Exponent represents the power of the polynomial. For example, x^y represents that polynomial x is multiplied by itself y number of times, where y is the exponent and x is the polynomial or base.

There are 3 laws of exponents:

  • Product law: This law states that the solution to polynomial ×^y × x^z is x^(y+z)
  • Quotient law: This law states that the solution to polynomial x^y ÷ x^z is x^(y-z)
  • Power law: According to this law, the solution to polynomials (x^y)^z is x^yz.

Algebra is an important chapter for class 9th mathematics and an essential topic for advanced mathematics. Sound knowledge of this chapter will help you learn other chapters like geometry, equations, differentiation, etc.

  1. What do you mean by Algebra?
    Algebra is a branch of mathematics that involves studying numbers, equations, geometry, mathematical analysis, etc.
  2. What are the expansion identities for cubic equations?
    (x + y)^3 = x^3 + 3xy(x + y) + y^3
    (x – y)^3 = x^3 – 3xy(x – y) – y^3
  3. What are formulas used in the factorisation methods of difference or sum in two cubes?
    It involves using the identities of cubes to solve the polynomial equations, x^3 + 3xy(x + y) + y^3 = (x + y)^3 and x^3 – 3xy(x – y) – y^3 = (x – y)^3.
  4. Which method is the best to solve simultaneous linear equations?
    Substitution and elimination methods are the easiest and most used methods to solve simultaneous linear equations.

Algebra is an exciting chapter of mathematics. The visual representation of this chapter is essential. The MSVgo app is a video library platform that helps you learn through exploratory representations and practical models. Make sure to check out the explanatory videos at MSVgo for a better understanding of these concepts.

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
  • Electron Beams and Radioactivity
  • 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
  • Hydrogen
  • Ideal solutions
  • Introduction to Organic Chemistry
  • Ionic equilibria
  • Matter
  • Matter Around Us
  • Matter In Our Surroundings
  • Metallurgy
  • Metals And Non-Metals
  • Mole Concept and Stoichiometry
  • Natural Resources
  • Organic Chemistry – Basic Principles
  • Periodic Classification of Elements
  • Physical and Chemical Changes
  • Physical and Chemical Properties of Water
  • Polymers
  • 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
  • Adolescent Issues
  • 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
  • Algebra – Quadratic Equations
  • 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 – Quadrilaterals
  • 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
  • Chemistry in Your Life
  • 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
  • Light, Shadows And Reflections
  • 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

  • Addition
  • 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
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