--- title: "Consistency of a linear system" subject: "Mathematics III" chapter: 1 key: "semester_3/mathematics_iii:consistency-of-a-linear-system-three" source: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system-three site: BCT Engineering --- # Consistency of a linear system Mathematics III · Chapter 1: Determinants and Matrices - Asked in: 2081 Bhadra, 3+2 marks - Video: [Consistency of system of linear equations using rank method](https://www.youtube.com/watch?v=povS4KL3nR8) — maths voice - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/consistency-of-a-linear-system-three - Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-1 ## Question Test for consistency of the system of equations 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5 and solve if found consistent. ## Short answer Reduce the augmented matrix to echelon form; the third row becomes zero, so rank A = rank [A|B] = 2 < 3. The system is consistent with infinitely many solutions. Setting z = k gives y = (3 + k)/11 and x = (7 − 16k)/11. ## Answer A system AX = B is consistent if rank A = rank [A|B]. Here we also have to decide between a unique and an infinite set of solutions. ```text [A|B] = [5 3 7 | 4] [3 26 2 | 9] [7 2 10 | 5] ``` #### Row reduction R₂ → 5R₂ − 3R₁ and R₃ → 5R₃ − 7R₁: ```text [5 3 7 | 4] [0 121 −11 | 33] [0 −11 1 | −3] ``` Divide R₂ by 11: ```text [5 3 7 | 4] [0 11 −1 | 3] [0 −11 1 | −3] ``` R₃ → R₃ + R₂: ```text [5 3 7 | 4] [0 11 −1 | 3] [0 0 0 | 0] ``` #### Consistency rank A = 2 = rank [A|B] < 3 (number of unknowns). The system is **consistent with infinitely many solutions** (one free variable; also |A| = 0). #### Solution Let z = k (any real number). - 11y − k = 3 ⇒ y = (3 + k)/11 - 5x + 3y + 7k = 4 ⇒ 5x = 4 − (9 + 3k)/11 − 7k = (35 − 80k)/11 ⇒ x = (7 − 16k)/11 **Answer: x = (7 − 16k)/11, y = (3 + k)/11, z = k, for any real k.** Check with k = 0: (7/11, 3/11, 0) gives 5x + 3y = 35/11 + 9/11 = 4 ✓. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The ranks of A and [A|B] are A. both 2, so consistent B. 2 and 3, so inconsistent C. both 3, so unique D. both 1 Answer: A. The last row reduces to zero. (Key point: Consistency) 2. Since rank 2 < 3 unknowns, the system has A. a unique solution B. infinitely many solutions with one free variable C. no solution D. exactly two solutions Answer: B. n - r = 1 free variable. (Key point: Type of solution) 3. With z = k, y equals A. (3 - k)/11 B. 11(3 + k) C. (3 + k)/11 D. (k - 3)/11 Answer: C. 11y - k = 3. (Key point: Solution) 4. What does |A| = 0 indicate here? A. A is invertible B. The system is inconsistent C. A is orthogonal D. A is singular, consistent with rank less than 3 Answer: D. Rank less than 3 means zero determinant. 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