Pair of Linear Equations in Two Variables

Jul 3, 2024

Lecture Notes: Pair of Linear Equations in Two Variables

Key Concepts

Definitions

  1. Pair of Linear Equations in Two Variables: Equations of the form ax + by = c where a, b, c are constants, and x and y are variables. Multiple equations involving x and y represent lines.
  2. Types of Solutions:
    • Consistent System: At least one solution (either unique or infinitely many).
    • Inconsistent System: No solution.
    • Dependent System: Infinitely many solutions (coinciding lines).
    • Independent System: Exactly one solution (intersecting lines).

Geometric Interpretation

  1. Intersecting Lines: One solution (consistent, independent).
  2. Parallel Lines: No solution (inconsistent).
  3. Coinciding Lines: Infinitely many solutions (consistent, dependent).

Graphical Method

  • Plot both equations on a graph. The point of intersection (if any) is the solution.
  • Steps:
    1. Convert each equation to the form y = mx + c.
    2. Plot the lines on the graph.
    3. Identify the intersection point.

Algebraic Methods

  1. Substitution Method:

    • Solve one equation for one variable.
    • Substitute this expression into the other equation.
    • Solve for the second variable.
    • Back-substitute to find the first variable.

  2. Elimination Method:

    • Multiply equations to make coefficients of one variable equal.
    • Add or subtract equations to eliminate one variable.
    • Solve the resulting equation.
    • Substitute back to find the other variable.

Conditions for Consistency

  1. For equations of the form a1x + b1y = c1 and a2x + b2y = c2:</n

    • Intersecting Lines:

a1/a2 ≠ b1/b2 - One unique solution (consistent).

  • Coinciding Lines:
    • a1/a2 = b1/b2 = c1/c2
    • Infinitely many solutions (consistent).
  • Parallel Lines:
    • a1/a2 = b1/b2 ≠ c1/c2
    • No solution (inconsistent).

Word Problems

  1. Forming Equations:

    • Translate the problem into algebraic equations using x and y for unknowns.
    • Use given conditions to form a system of equations.

  2. Examples:

    • Hostel Charges Problem: x + 20y = 1000, x + 26y = 1180
    • Coin Problem: x + y = 50, x + 2y = 75

Miscellaneous Tips

  1. Convert to Standard Form: Ensure equations are in Ax + By = C form before solving.
  2. Checking Solutions: Verify by plugging back into original equations.
  3. Interchanged Coefficients Problem: Add and subtract equations to simplify complex systems.

Homework

  1. Solve the given system of linear equations by both substitution and elimination methods.

Review

  • Revise and practice the steps and concepts mentioned above with different problems to solidify understanding.
  • Make sure to revisit the graphical method, graphical interpretations, and conditions for consistency before moving on to more advanced problems.