Discussion
The following system of equations :
x + z = 3
x − y − z = 1
−x + y = 1
Answer: B
The system is consistent with a unique solution.
The coefficient matrix and its determinant are
\[ \begin{aligned} &\begin{vmatrix} 1 & 0 & 1 \\ 1 & -1 & -1 \\ -1 & 1 & 0 \end{vmatrix} \\ &= 1(0 + 1) - 0 + 1(1 - 1) = 1 \end{aligned} \]
A non-zero determinant means the matrix is invertible, so there is exactly one solution. Solving: from the third equation \(y = x + 1\); then the second gives \(z = x - y - 1 = -2\), and the first gives \(x = 5\), \(y = 6\).
A linear system can never have exactly three solutions (it has 0, 1 or infinitely many), and the non-zero determinant rules out both 'infinitely many' and 'not consistent'.
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