Elimination Method for Systems of Equations

Nov 11, 2024

Lecture Notes: Solving Systems of Equations by Elimination Method

Key Concepts

  • Elimination or Addition Method: A method used to solve systems of equations by adding the equations together to eliminate one variable.
  • The elimination method is faster than graphing and estimating the point of intersection for the solution.

Steps to Solve Using Elimination Method

  1. Multiply Equations: Adjust one or both equations by multiplying them by appropriate numbers to ensure one variable has the same coefficient but opposite signs.
  2. Add Equations: Add the two equations together so that one variable is eliminated.
  3. Solve Equation: Solve the resulting equation with one variable.
  4. Substitute Solution: Substitute the solution back into one of the original equations to find the remaining variable.

Example Problems

Example 1: Solve for x and y

  • Given Equations:
    • 4x + 3y = 8
    • x - 3y = 7
  • Add equations:
    • Result: 5x = 15, where y terms cancel out.
    • Solve: x = 3
  • Substitute x in the first equation:
    • 4(3) + 3y = 8
    • Solve: y = -4/3
  • Solution: x = 3, y = -4/3
  • Classification: The system is consistent and independent.

Example 2: General Steps

  • Choose which variable to eliminate by ensuring opposite coefficients.
  • Multiply and add equations:
    • Example transformation:
      • 2x and 3x to 6x and -6x.
  • Eliminate variable, solve for remaining:
    • 19y = 38, y = 2
    • Substitute y: x = -2
  • Solution: x = -2, y = 2
  • System is consistent and independent.

Example 3: No Solution

  • Adjust y terms to opposites and solve:
    • End result: 0 = 3 (False statement)
  • Conclusion: No solution, system is inconsistent and independent.

Application: Comparing Phone Plans

  • Problem Statement: Sunset Phone offers two plans:
    • Plan A: $5/month + $0.08/minute
    • Plan B: $0.12/minute, no monthly fee
  • Objective: Find when the two plans cost the same.
  • Variable: x = number of minutes
  • Set Equations:
    • Plan A total cost: 0.08x + 5
    • Plan B total cost: 0.12x
  • Equality: 0.08x + 5 = 0.12x
  • Solve:
    • Simplify to: 8x + 500 = 12x
    • Result: x = 125 minutes
  • Conclusion: Plans cost the same at 125 minutes usage.

Summary

  • The elimination method provides a systematic approach to solving systems of equations efficiently.
  • Real-world applications, such as phone plan comparisons, can be solved using this method.
  • Helps in determining consistency and independence of systems.