Lecture Notes on Pair of Linear Equations in Two Variables

Introduction

• Presenter: Sammit Sankanaver
• Topic: Revision of Pair of Linear Equations in Two Variables (SSLC Mathematics)

Basics

Understanding Variables

• Variables: Terms in equations whose values change
  • Commonly represented by x and y, but can be any letter
• Example: y = 2x + 1
  • When x = 1, y = 3
  • When x = 2, y = 5
  • When x = 3, y = 7
• Constants: Terms whose values don’t change
  • In the example: 2 and 1 are constants

Variables in Equations

• Equation Types:
  • One variable: Example 2x + 3 = 0
  • Two variables: Example 2x + 3y = 0
  • Three variables: Example 2x + 3y + z = 0

Linear Equations

• Definition: Equations that graph as straight lines
• Identification:
  • Algebraically, if the power of any variable is 1, it's a linear equation
  • Powers: 1 - Linear, 2 - Quadratic, 3 - Cubic

Plotting Linear Equations

• Example: 2x - y = 0
• Outcome: Points on a graph that form a straight line
• General Form: ax + by + c = 0
  • Each solution (x, y) satisfies the equation and points lie on the plotted line
• Important Point: Each solution of a linear equation in two variables represents a point on the line

Pair of Linear Equations in Two Variables

Need for Pair of Equations

• Single equation with two variables cannot be solved algebraically without additional info
• Number of equations required equals the number of variables

General Form

• First equation: a1x + b1y + c1 = 0
• Second equation: a2x + b2y + c2 = 0
• Coefficients: a, b, c are real numbers
• Important: a1 and b1 cannot be zero at the same time

Forming and Solving Pair of Equations

Example Problem

• Romela and Sonali bought pencils and erasers
• Initial Equations:
  • 2 pencils + 3 erasers = 9
  • 5 pencils + 6 erasers = 21
• Assign variables:
  • Cost of pencil = x
  • Cost of eraser = y
• Formation:
  • 2x + 3y = 9
  • 5x + 6y = 21

Methods of Solving Pair of Equations

Geometrical Method

• Plot both equations on the graph
• Intersection point gives the solution
• Example Result: Intersection at (3, 1) gives x = 3, y = 1

Algebraic Methods

1. Substitution Method

• Formulate one equation in terms of one variable
• Substitute it into the other equation
• Example:
  • Equation 1: x - 3y = 10
  • Equation 2: x - 7y = -30
  • Solve for x in one and substitute

2. Elimination Method

• Make coefficients of one variable equal
• Add or subtract equations to eliminate one variable
• Example:
  • Equation 1: x + y = 25
  • Equation 2: 50x + 100y = 2000
  • Multiply equation 1 and subtract

3. Cross Multiplication Method

• Use a formula involving coefficients to find x and y
• Pair of equations needs to be in standard form
• Example:
  • Substituted into formula for x and y

Special Cases and Reduction

Equations Reducible to Linear Form

• Substitution to convert non-linear equation to linear form
• Solving using any algebraic method after reduction
• Example: Variables in denominators

Example Problem

• Given equations with variables in denominators
• Substitution to convert and solve

Conclusion

• Overview of solving methods and concepts
• Importance of identifying the correct approach for different types of equations

End of Lecture