Consistency of a linear system

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.

[A|B] = [5  3  7 | 4]
        [3 26  2 | 9]
        [7  2 10 | 5]

Row reduction

R₂ → 5R₂ − 3R₁ and R₃ → 5R₃ − 7R₁:

[5   3   7 |  4]
[0 121 −11 | 33]
[0 −11   1 | −3]

Divide R₂ by 11:

[5   3   7 |  4]
[0  11  −1 |  3]
[0 −11   1 | −3]

R₃ → R₃ + R₂:

[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 ✓.