$M$ Question 9 Given the system of equations: \[ \left\{\begin{array}{l} x+y+z=0 \\ y-3 z=11 \\ 2 x+y+5 z=-10 \end{array}\right. \] (a) Determine the type of system: $\bigcirc$ inconsistent $\bigcirc$ dependent (b) If your answer is dependent, find the complete solution. Write $x, y$, and $z$ as functions of $t$, where $z=t$. If your answer is inconsistent, write DNE in the box for all three variables. \[ \begin{array}{l} x= \\ y= \\ z= \end{array} \]

Step 1 ：The matrix representation of the system of equations is: \[ \begin{bmatrix} 1 & 1 & 1 & 0 \ 0 & 1 & -3 & 11 \ 2 & 1 & 5 & -10 \end{bmatrix} \]

Step 2 ：Start the Gaussian elimination by subtracting 2 times the first row from the third row. This will give us a new third row and the updated matrix will be: \[ \begin{bmatrix} 1 & 1 & 1 & 0 \ 0 & 1 & -3 & 11 \ 0 & -1 & 3 & -10 \end{bmatrix} \]

Step 3 ：Next, add the second row to the third row to get a new third row. The updated matrix will be: \[ \begin{bmatrix} 1 & 1 & 1 & 0 \ 0 & 1 & -3 & 11 \ 0 & 0 & 0 & 1 \end{bmatrix} \]

Step 4 ：From the last row of the updated matrix, we can see that 0 = 1, which is not possible. Therefore, the system of equations is inconsistent.

Step 5 ：Final Answer: The system of equations is \(\boxed{\text{inconsistent}}\). The values of \(x\), \(y\), and \(z\) do not exist, so they are \(\boxed{\text{DNE}}\).