Systems of Linear Equations: Substitution and Elimination

Definitions

• Consistent: At least one solution.
• Inconsistent: No solutions.
• Independent: Exactly one solution (applies to consistent systems).
• Dependent: Infinite number of solutions (applies to consistent systems).
• Classification Summary:
  • One solution: Consistent and independent.
  • No solutions: Inconsistent.
  • Infinite solutions: Consistent and dependent.

Solving Systems: Methods

1. Substitution
2. Elimination (Addition Method)

Example A: Elimination Method

• Equations:
  • ( x + 3y = 5 )
  • ( 2x - 3y = -8 )
• Steps:
  • Add equations to eliminate ( y ):
    • ( 3x = -3 ) → ( x = -1 ).
  • Substitute ( x = -1 ) back into one equation:
    • ( -1 + 3y = 5 ) → ( 3y = 6 ) → ( y = 2 ).
• Solution: ( (-1, 2) ).
• Classification: Consistent and independent.

Example B: Substitution Method

• Equations:
  • ( x = 5y + 2 )
  • Substitute ( x ) into another equation:
    • ( 3(5y + 2) - 15y = 6 ).
    • Simplify to find ( 6 = 6 ) (True statement).
• Solution: Infinite solutions.
• Classification: Consistent and dependent.
• Solution Set: All ( (x, y) ) pairs that satisfy ( x - 5y = 2 ).

Application Example

• Problem: Passengers on a flight with fares and totals given, find number getting off at Chicago.
• Variables:
  • ( x = ) passengers to Chicago.
  • ( y = ) passengers to New York.
• Equations:
  • ( x + y = 185 )
  • ( 45x + 60y = 10500 )
• Steps:
  • Solve by substitution to find: ( x = 40, y = 145 ).
• Conclusion: 40 passengers got off at Chicago.

Solving Three Variables Systems

1. Write equations in standard form.
2. Select a variable to eliminate.
3. Use pairs of equations to eliminate chosen variable.
4. Solve new system of two equations.
5. Substitute back to find third variable.
6. Write solution as an ordered triple.

Example: Three Equations

• Equations:
  • ( x + y + z = 1 )
  • ( 2x + y + z = -1 )
  • ( 3x - 2y - z = 3 )
• Eliminate ( z ):
  • Combine equation pairs to isolate variables.
• Solution: ( (\frac{1}{2}, 4, -6) ).
• Classification: Consistent and independent.

Missing Variable Strategy

• If a variable is missing from equations, treat it as already eliminated.
• Continue elimination with other pairs.

Example: Missing Variable

• Equations:
  • ( a - 3c = 6 )
  • ( b + 2c = 2 )
  • ( 7a - b - 5c = 14 )
• Steps:
  • Eliminate using remaining equations.
  • Solve for remaining variables.
• Solution: ( (3, 4, -1) ).
• Classification: Consistent and independent.

Infinite Solutions Example

• Equations:
  • ( x + y + z = 1 )
  • ( 2x + y + z = -1 )
  • ( 3x - 2y - z = 3 )
• Steps:
  • Use elimination to find dependent equations.
  • Express solution in terms of a free variable.
• Solution Set: ( (x, 2x + 1, -3x) ).
• Classification: Consistent and dependent.

Inconsistent System Example

• Equations:
  • ( x + z = 0 )
  • ( x + y + 2z = 3 )
  • ( -y - z = -2 )
• Steps:
  • Eliminate to get conflicting equations.
• Conclusion: No solutions.
• Classification: Inconsistent.

Summary

• Consistent systems have solutions (one or infinite).
• Inconsistent systems have no solutions.
• Independent means exactly one solution.
• Dependent means infinitely many solutions.

Next Topics

• Nonlinear systems of equations.