Linear Equations in Two Variables

• Introduction

  • Definition of Linear Equation
  • Meaning of Two Variables
  • General Form: $ax + by + c = 0$
    • $a, b, c$ are real numbers
    • $a \neq 0$ and $b \neq 0$
  • Examples: $2x + 3y = 5$, $x - 2y = 4$

• Solutions of a Linear Equation

  • Meaning of a Solution
    • Pair of values $(x, y)$ that satisfies the equation
  • Number of Solutions
    • Every linear equation in two variables has infinitely many solutions
  • Finding Solutions
    • Method of substitution (assigning value to one variable to find the other)

• Graph of a Linear Equation in Two Variables

  • Steps to Draw Graph
    • Find at least two solutions (points)
    • Plot points on a Cartesian plane
    • Join points with a straight line
  • Properties of the Graph
    • Every point on the line is a solution
    • Every solution is a point on the line
  • Equation of $x$-axis: $y = 0$
  • Equation of $y$-axis: $x = 0$

• Lines Parallel to $x$-axis and $y$-axis

  • Parallel to $x$-axis
    • Form: $y = k$ (where $k$ is a constant)
  • Parallel to $y$-axis
    • Form: $x = k$ (where $k$ is a constant)
  • Role of Constants ($k$) in determining position