The system of equations x - 3y + 5z = 4 2x - 6y + 10z = 11 3x - 9y +…

The correct answer is Option (3) → inconsistent, with no solution

The coefficient matrix is

    \( \displaystyle A = \begin{bmatrix}1 & -3 & 5\\ 2 & -6 & 10\\ 3 & -9 & 15\end{bmatrix} \)

Its second and third rows are 2 times and 3 times the first row. A determinant with proportional rows is zero, so \(|A| = 0\) and there is no unique solution.

Write \(u = x - 3y + 5z\). The three equations say

    \( u = 4 \)
    \( \displaystyle 2u = 11,\ \text{that is } u = \frac{11}{2} \)
    \( 3u = 12,\ \text{that is } u = 4 \)

The same expression cannot equal both 4 and \(\frac{11}{2}\), so the system is inconsistent: it has no solution.