Write the augmented matrix of the system and use the matrix method to solve the system. \( \begin{array}{l}-x+4 y+3 z=-16 \\ \begin{array}{r}x+y+z=10 \\ x+y+z=0\end{array} \\ \text { Select the correct choice and, if necessary, fill in the answer boxes to complete your choice. } \\ \text { A. This system has exactly one solution. The solution is ( } \\ \text { (Type integers or decimals.) } \\ \text { B. This system has infinitely many solutions of the form } \\ \text { (Type integers or decimals.) } \\ \text { C. This system has no solution. }\end{array} \) ?, z), where \( z \) is any real number.

To solve the system of equations using an augmented matrix, we first represent the equations in matrix form. The augmented matrix for the system is: \[ \begin{bmatrix} -1 & 4 & 3 & | & -16 \\ 1 & 1 & 1 & | & 10 \\ 1 & 1 & 1 & | & 0 \end{bmatrix} \] Now we can apply Gaussian elimination to solve for \(x\), \(y\), and \(z\). First, we can subtract the second row from the third row, which gives us: \[ \begin{bmatrix} -1 & 4 & 3 & | & -16 \\ 1 & 1 & 1 & | & 10 \\ 0 & 0 & 0 & | & -10 \end{bmatrix} \] The last row indicates that there is no solution because it states \(0 = -10\). Thus, we conclude that: C. This system has no solution.