Which one of the following is not a basic solution of system of linear…

Key insight: For a system with 2 equations (m = 2) and 3 variables (n = 3), a basic solution is obtained by setting n − m = 1 variable to zero and solving the remaining 2×2 system.

Find basic solutions by setting one variable to zero and solving:

• Set x₃ = 0: solve x₁ + 2x₂ = 4 and 2x₁ + x₂ = 5. This gives x₂ = 1 and x₁ = 2, so the basic solution is (x₁,x₂,x₃) = (2, 1, 0).

• Set x₂ = 0: solve x₁ + x₃ = 4 and 2x₁ + 5x₃ = 5. Solving yields x₃ = -1 and x₁ = 5, so the basic solution is (5, 0, -1).

• Set x₁ = 0: solve 2x₂ + x₃ = 4 and x₂ + 5x₃ = 5. Solving gives x₃ = 2/3 and x₂ = 5/3, so the basic solution is (0, 5/3, 2/3).

Conclusion: The three basic solutions are (2, 1, 0), (5, 0, -1), and (0, 5/3, 2/3). The triplet (-1, 2, 1) is a solution of the system but not a basic solution because none of its components is zero, whereas a basic solution must have exactly one variable set to zero for this system.

Therefore, the triple x₁ = -1, x₂ = 2, x₃ = 1 is not a basic solution.