Overview
This lecture explains how to classify systems of two equations as consistent/inconsistent and dependent/independent, and provides methods to determine the number of solutions.
Types of Solutions in Systems of Equations
- If a system has only one solution, it is consistent and independent.
- If a system has many (infinite) solutions, it is consistent and dependent.
- If a system has no solution, it is inconsistent and independent.
Determining the Number of Solutions
- One solution: Solving yields specific values for each variable (e.g., x = 2, y = 3).
- No solution: Solving results in a false statement (e.g., 2 = 5).
- Many solutions: Solving results in an identity (e.g., 0 = 0 or x = x).
Example 1: One Solution
- Equations: 3x + y = 17 and 4x - y = 18.
- Adding the equations gives 7x = 35, so x = 5.
- Substituting x gives y = 2.
- Result: Only one solution (5, 2); system is consistent and independent.
Example 2: Many Solutions
- Equations: 2x + 4y = 8 and x + 2y = 4.
- Elimination yields 0 = 0.
- Result: Many solutions; system is consistent and dependent.
Example 3: No Solution
- Equations: 3x + 2y = 5 and 6x + 4y = 8.
- Elimination yields 0 = –2.
- Result: No solution; system is inconsistent and independent.
Key Terms & Definitions
- Consistent — system with at least one solution.
- Inconsistent — system with no solutions.
- Independent — exactly one solution.
- Dependent — infinitely many solutions.
Action Items / Next Steps
- Practice solving systems to identify if they are consistent/inconsistent and dependent/independent.
- Review more examples to reinforce distinguishing solution types.