Consistency of a linear system

The system is x + 9y − z = 27, x − 8y + 16z = 10, 2x + y + 15z = 37. It is consistent if rank A = rank [A|B].

[A|B] = [1  9 −1 | 27]
        [1 −8 16 | 10]
        [2  1 15 | 37]

Step 1. R₂ → R₂ − R₁, R₃ → R₃ − 2R₁:

[1   9  −1 |  27]
[0 −17  17 | −17]
[0 −17  17 | −17]

Step 2. R₃ → R₃ − R₂, and divide R₂ by −17:

[1  9 −1 | 27]
[0  1 −1 |  1]
[0  0  0 |  0]

Rank and consistency

• Rank of the augmented matrix [A|B] = 2
• Rank of A = 2

The ranks are equal, so the system is consistent. Since rank 2 < 3 unknowns, there are infinitely many solutions (one free variable).

Solution

Let z = k.

• y − z = 1 ⇒ y = 1 + k
• x + 9y − z = 27 ⇒ x = 27 − 9(1 + k) + k = 18 − 8k

Answer: x = 18 − 8k, y = 1 + k, z = k, for any real k.

Check with k = 1: (10, 2, 1): 10 + 18 − 1 = 27 ✓; 10 − 16 + 16 = 10 ✓; 20 + 2 + 15 = 37 ✓.