The system of equations x + y + z = 3 , x - y + z = 1 , 2x + 2z = 5 is

The correct answer is Option (3) → inconsistent

Write the system as AX = B with:

    \(\displaystyle A = \begin{bmatrix}1 & 1 & 1\\ 1 & -1 & 1\\ 2 & 0 & 2\end{bmatrix}\)

    \(|A| = 1(-2 - 0) - 1(2 - 2) + 1(0 + 2)\)

    \(|A| = -2 - 0 + 2 = 0\)

Since \(|A| = 0\), there is no unique solution: the system has either no solution or infinitely many.

Adding the first two equations:

    \(2x + 2z = 4\)

But the third equation says \(2x + 2z = 5\). Both cannot hold, so the system has no solution; it is inconsistent.