Consistency of a linear system

The system AX = B is consistent if and only if rank A = rank [A|B].

[A|B] = [1  2 −1 |  3]
        [2  3  1 | 10]
        [3 −1 −7 |  1]

Step 1. R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁:

[1  2 −1 |   3]
[0 −1  3 |   4]
[0 −7 −4 |  −8]

Step 2. R₃ → R₃ − 7R₂:

[1  2 −1 |   3]
[0 −1  3 |   4]
[0  0 −25 | −36]

Consistency

rank A = 3 = rank [A|B] = number of unknowns. The system is consistent with a unique solution (|A| = 25 ≠ 0).

Solution (back substitution)

• −25z = −36 ⇒ z = 36/25
• −y + 3z = 4 ⇒ y = 3(36/25) − 4 = (108 − 100)/25 = 8/25
• x + 2y − z = 3 ⇒ x = 3 − 16/25 + 36/25 = (75 + 20)/25 = 95/25 = 19/5

Check (eq. 3): 3(19/5) − 8/25 − 7(36/25) = (285 − 8 − 252)/25 = 1 ✓.

Answer: x = 19/5, y = 8/25, z = 36/25.