Consistency of a linear system

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

[A|B] = [1  2 −1 |  0]
        [2  3  1 | 10]
        [3 −1 −7 |  1]

Step 1. R₂ → R₂ − 2R₁, R₃ → R₃ − 3R₁:

[1  2 −1 |  0]
[0 −1  3 | 10]
[0 −7 −4 |  1]

Step 2. R₃ → R₃ − 7R₂:

[1  2 −1 |   0]
[0 −1  3 |  10]
[0  0 −25 | −69]

Consistency

rank A = 3 = rank [A|B] = number of unknowns. (Also |A| = 25 ≠ 0.) So the system is consistent and has a unique solution.

Solution

• −25z = −69 ⇒ z = 69/25
• −y + 3z = 10 ⇒ y = 3(69/25) − 10 = (207 − 250)/25 = −43/25
• x + 2y − z = 0 ⇒ x = −2y + z = 86/25 + 69/25 = 155/25 = 31/5

Check (eq. 3): 3(31/5) − (−43/25) − 7(69/25) = (465 + 43 − 483)/25 = 25/25 = 1 ✓.

Answer: x = 31/5, y = −43/25, z = 69/25.