System of Linear Equations — Consistency and Inconsistency

System of Linear Equations — Consistency and Inconsistency.

Definition of a System. A system of linear equations is a set of two or more linear equations involving the same variables. Example: a₁x + b₁y + c₁z = d₁; a₂x + b₂y + c₂z = d₂ Matrix form: A X = B, where A = coefficient matrix, X = variable matrix, B = constants matrix..

Classification of Systems. Consistent System: At least one solution → Independent (Unique) → Dependent (Infinite) Inconsistent System: No solution.

Rank Method for Consistency. Let A = coefficient matrix and [A|B] = augmented matrix. • If rank(A) = rank([A|B]) = n → Unique Solution • If rank(A) = rank([A|B]) < n → Infinite Solutions • If rank(A) ≠ rank([A|B]) → No Solution.

Example 1 – Unique Solution. System: x + y + z = 6; 2x + 3y + 4z = 14; x + 2y + 3z = 10 rank(A) = rank([A|B]) = 3 = n ✅ Consistent system with unique solution.

Example 2 – Infinite Solutions. System: x + y + z = 3; 2x + 2y + 2z = 6 rank(A) = rank([A|B]) < n ✅ Consistent system with infinite solutions.

Example 3 – Inconsistent System. System: x + y + z = 3; x + y + z = 4 rank(A) ≠ rank([A|B]) ❌ Inconsistent system — no solution.

Geometric Interpretation. For 3-variable systems: • Planes intersect at a single point → Unique solution • Planes meet along a line → Infinite solutions • Planes are parallel → No solution.

Summary & Learning Outcomes. Summary Table: rank(A) = rank([A|B]) = n → Unique solution rank(A) = rank([A|B]) < n → Infinite solutions rank(A) ≠ rank([A|B]) → No solution Learning Outcomes: • Represent linear systems using matrices • Apply the rank test for consistency • Analyze solutions geometrically.

Outro. Thank you for watching this lecture on Systems of Linear Equations — Consistency and Inconsistency. Prepared by: Dr. Debasish Majumder Department of Basic Science and Humanities JIS College of Engineering.