Linear Equation in One Variable - Class 8 - Chapter 2

Lecture Overview

• Presented by: Ritik Mishra (Mathematics Mentor)
• Objectives: Understanding Chapter, Solution of Equation, Word Problems
• Importance: Clear base for higher classes
• Lecture style: Fun and interactive

Lecture Goals

1. Understand the chapter concept
2. Learn what a solution of an equation means
3. Get comfortable with word problems

Key Concepts Explained

Constants and Variables

• Variable: Represents a quantity that changes continuously (denoted by letters like x, y, z, etc.)
• Constant: A fixed number (e.g., 2, -100, 5/2)
• Examples: Positions, speeds changing over time

Terms and Algebraic Expressions

• Term: Can be a variable, constant, or product of constants and variables (e.g., 2x, y, 100)
• Algebraic Expression: Collection of terms (e.g., x+2xy+3)
• Equation: Two expressions separated by an equals sign (e.g., x + 2 = 5)
• Linear Equation in One Variable: Highest power of the variable is 1 (e.g., x+2=3)

Solving an Equation

• Solution: The value of the variable that satisfies the equation (LHS = RHS)
• Methods:
  • Trial and Error: Plugging in values
  • Isolation: Moving terms to one side and solving
• Important Rules:
  • Adding, subtracting, multiplying, or dividing both sides by the same amount
  • Transposition (moving a term across the '=' sign changes its sign)
  • Cross multiplication for rational equations

Example Problems

1. Solve: 2x + 1 = 0:
   • Solution: x = -1/2
2. Solve: 2x-3 = 7:
   • Solution: x = 5
3. Solve: 5x + 9 = 5 + 3x:
   • Solution: x = -2
4. Solve through Cross Multiplication: (x+3)/4 = (2x-5)/5:
   • Solution: x = 1

Word Problems

• Example 1: Ratio Problem

  • Two numbers in the ratio 8:3, sum is 143
  • Let numbers be 8x and 3x
  • Equation: 8x + 3x = 143
  • Solution: x = 13; Numbers = 104 and 39

• Example 2: Age Problem

  • Aman’s age is three times his son's age; 10 years ago, Aman was 5 times his son's age
  • Let son's current age be x
  • Aman's age = 3x
  • 10 years ago: Aman’s age = 3x-10, Son’s age = x-10
  • Equation: 3x-10 = 5(x-10)
  • Solution: Son’s age = 10 years, Aman’s age = 30 years

Final Advice

• Focus on strengthening your basics
• Keep practicing different problems
• Understand the importance of each step and method

Summary

• Understand terms, variables, constants, algebraic expressions, and linear equations
• Practice solving equations systematically
• Apply the concept to real-world problems like age, distance, and ratio
• Enjoy learning and practicing math concepts