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Chapter 11 – Algebra

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

Introduction

The oldest branch of mathematics that deals with number theory, geometry, and analysis is known as algebra. The study of algebra involves manipulating mathematical symbols, and deals with the study of their rules and application. In Algebra, you must have come across the concepts of variables, expressions with variables, and equations. Additionally, you also need to learn about multiple formulae and identities.

Algebra is the branch of mathematics that helps in deriving unknown quantities using equations and variables. In your life, you may have heard of some terms like bank interests, or percentage of profit. Did you know that you can use algebra to derive the true value of these unknown quantities, known as variables, by using something called an equation?

There are more examples of variables that you can find in your daily life. For example, the distance between your home and school is also a variable, and so are the prices of the grocery items that your parents buy at the nearby store. You can use algebraic equations and formulas to solve for these unknown variables.

Have you seen a letter being used to denote a number while talking about an unknown quantity? That letter is a variable. The value of a variable varies, hence the name. When used in an algebraic expression, the variable may denote different numbers, and maybe denoted by any alphabet. Conventionally, alphabets x, y, z, or a,b,c are the most widely used variables. 

All algebraic expressions and terms must consist of at least one variable. The idea of a variable is the distinguishing factor between an algebraic and a numerical expression. You would be surprised to know that the presence of a variable in any algebraic expression expands the possibilities of determining the value of the expression to infinity.

A mathematical expression that is made up of variables, constants, and algebraic operations like addition, subtraction, etc. is called a mathematical expression or mathematical equation. Another important component of an algebraic expression is a term. A term is a representation of grouping together of more than one factor in an algebraic expression. 

As you grow up and study more complex mathematical concepts, you will come across different types of algebraic expressions and equations. 

  • Linear Equations

These are equations of the first order and have a single, homogeneous variable. They are used to represent a straight line on a coordinate system. It is represented by,

y= mx+b, where, 

m is the slope of the line

b is the y-intercept

x and y are the coordinates  of the x and y axes, respectively

  • Non-linear Equations

Polynomial equations of the second order in one variable. It is represented as a function of x as, 

f(x)= ax2 + bx + c

  • Solving Linear Equations

I. Solve for x

x= 24+2x

Or, x-2x=24

Or, -(2-1)x=24

Or, -x= 24

Or, x = -24

II. Solve for x and y

x+y=15

x-2y=8

From the first equation, 

x= 15-y

Substitute this in equation 2

(15-y)-2y= 8

Or, 15-y-2y= 8

Or, 15- (y+2y)=8

Or, 15-3y= 8

Or, 15-8= 3y

Or. 7=3y

Or 3y=7

Or, y= 7/3 

Substituting this in equation 1,

x= 15-7/3

x= [3(15)-7]/3

x=[45-7]/3

x=38/3

x= 122/3

Imagine that you are planning a holiday with your father and trying to understand important factors like time of travel, mode of transport, etc. to reach the destination. Here is some basic information that you have,

The bus that you are planning to take to Goa from Bombay travels at 60 kmph.

You also know that it takes 5 hours to reach your grandmother’s home from your place, and the former is 20km from Goa. 

Can you find out the distance between Bombay and Goa using an algebraic expression?

Solution

Speed of the bus= 60 kmph

So, the distance (d) traveled in 5 hours can be represented by the equation

d= 60×5

d= 300

Therefore, the distance (D) from Bombay to Goa can be represented as 

D= d+20

D= 300+20 (substituting the value of d)

D= 320 km

Did you know that you can get help from a simple everyday tool like a matchstick to learn algebra? Matchstick patterns are a great way to understand the idea of a variable, or the use of variables in common rules.

  • Examples of matchstick patterns

Using the following matchstick patterns, find out the rule to represent the number of matchsticks to be used in each pattern.

  1. Pattern of A
  2. Pattern of T

Solutions

To write the letter A, you need 3 matchsticks. Therefore, the rule to represent this is 3n, where n is the number of matchsticks.

To write the letter T, you need 2 matchsticks. Therefore, the rule to represent this is 2n, where n is the number of matchsticks.

You can find more matchstick patterns to learn how to come up with algebraic rules using variables and constants. 

  • What are the basics of algebra?

The basics of algebra consist of variables, constants, and mathematical operations like addition, subtraction, etc. 

  • How do you calculate algebra?

You can calculate algebra with the use of algebraic expressions and equations.

  • Who invented algebra?

Al-Khwarizmi is credited with the invention of Algebra. 

  • Why is algebra so hard?

The use of symbols and equations may make algebraic expressions look complicated. 

  • What does XY mean in algebra?

XY is used to denote a term or the product of two variables X and Y.

  • What are the four rules of algebra?

The four basic rules of algebra include

  1. Symmetry rule
  2. Commutative rule
  3. Inverse of adding
  4. Two rules of equation

Algebra can be fun too, if you want it. With the MSVGo app, you can now have access to an interactive learning platform and vast video library, where you can learn about the most important algebraic modules from experts and peers. You can simply download the app to start learning today!

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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