The Main Idea

• System Structure
  • Each equation has form [latex]ax + by + cz = d[/latex]
  • Three equations with three unknowns ([latex]x[/latex], [latex]y[/latex], [latex]z[/latex])
  • Each equation represents a plane in 3D space
• Types of Solutions
  • One solution: [latex]{(x,y,z)}[/latex] (intersection point of three planes)
  • No solution (inconsistent): leads to contradiction like [latex]3=0[/latex]
  • Infinite solutions (dependent): leads to identity like [latex]0=0[/latex]

The Main Idea

• Step-by-Step Solution Process
  • Choose any two equations to work with first
  • Eliminate one variable to create a two-variable equation
  • Repeat with another pair to get a second two-variable equation
  • Solve the resulting two-by-two system
  • Use back-substitution to find the final variable
• Back-Substitution Strategy
  • Once you find one variable, plug it into simpler equations
  • Work from simplest to most complex equations
  • Keep track of positive/negative signs carefully
• Variable Selection
  • Choose the variable that’s easiest to eliminate first
  • Look for equations where variables are missing
  • Consider coefficients that make elimination easier
• Solution Verification
  • Always check solution in ALL original equations
  • A true solution works in every equation
  • One equation being false means the solution is incorrect

The Main Idea

• Preparation Steps
  • Write all equations in standard form: [latex]ax + by + cz = d[/latex]
  • Clear any fractions by multiplying through
  • Label equations (1), (2), (3) for tracking
  • Plan which variable to eliminate first
• Elimination Process
  • Choose same variable to eliminate from two pairs of equations
  • Create two new equations with two variables
  • Solve resulting two-by-two system
  • Back-substitute to find final variable
• Strategy
  • Look for equations where coefficients are already opposites
  • Choose variable that’s easiest to eliminate
  • Keep coefficients as simple as possible
  • Create upper triangular form when possible
• Systematic Approach
  • Always follow elimination steps in order
  • Track equations with numbers/labels
  • Show all work clearly
  • Verify solution in all original equations

The Main Idea

• Key Characteristics:
  • No solution exists that satisfies all equations
  • Results in a contradiction (e.g., [latex]0 = 5[/latex])
  • Represents planes that don’t intersect at a common point
• Geometric Interpretations:
  • Three parallel planes
  • Two parallel planes with one intersecting plane
  • Three planes intersecting in different locations
• Detection Method:
  • Elimination process leads to a contradiction
  • Often takes several steps to reveal the contradiction
  • Cannot be determined by looking at equations alone

The Main Idea

• Key Characteristics:
  • Infinite solutions exist
  • Results in an identity (e.g., [latex]0 = 0[/latex])
  • Can express solution using one variable
• Geometric Interpretations:
  • Three identical planes
  • Two identical planes intersecting a third
  • Three planes intersecting along a line