The system of equations \(x + 2y - z = 2\), \(2x + y + z = 4\), \(3x - y + 2z = 4\) has
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → a unique solution
The system is AX = B. When \(|A| \ne 0\), A is invertible and \(X = A^{-1}B\) is the unique solution.
\(\displaystyle |A| = \begin{vmatrix}1 & 2 & -1\\ 2 & 1 & 1\\ 3 & -1 & 2\end{vmatrix}\)
\(|A| = 1(2 + 1) - 2(4 - 3) + (-1)(-2 - 3)\)
\(|A| = 3 - 2 + 5 = 6 \ne 0\)
So the system is consistent with exactly one solution. (It is x = y = z = 1: \(1 + 2 - 1 = 2\), \(2 + 1 + 1 = 4\), \(3 - 1 + 2 = 4\).)
Why the other options are incorrect
Option (1) no solution — Incorrect. A system with \(|A| \ne 0\) always has a solution; here x = y = z = 1 works.
Option (2) infinitely many solutions — Incorrect. That can happen only when \(|A| = 0\); here \(|A| = 6\).
Option (4) only the solution \(x = y = z = 0\) — Incorrect. x = y = z = 0 gives 0 on the left, not 2, so it is not a solution at all.