Solve the given system of equations.
\[
\begin{array}{l}
5 x+4 y-5 z=-35 \\
3 x-5 y+3 z=10 \\
6 x-3 y+6 z=-15
\end{array}
\]
Select the correct choice below and fill in any answer boxes within your choice.
A. There is one solution. The solution set is $\{(\square, \square, \square)\}$. (Simplify your answers.)
B. There are infinitely many solutions.
C. There is no solution.
Answer
Finally, we can conclude that the solution to the system of equations is \((-4, -5, -1)\). Therefore, the correct choice is A. The solution set is \(\boxed{\{(-4, -5, -1)\}}\).
Step 1 ：First, we represent the system of equations in matrix form. The matrix A is \[\begin{bmatrix} 5 & 4 & -5 \\ 3 & -5 & 3 \\ 6 & -3 & 6 \end{bmatrix}\] and the matrix B is \[\begin{bmatrix} -35 \\ 10 \\ -15 \end{bmatrix}\].
Step 2 ：We then solve the system of equations using the matrix method. The solution is \[\begin{bmatrix} -4 \\ -5 \\ -1 \end{bmatrix}\].
Step 3 ：Finally, we can conclude that the solution to the system of equations is \((-4, -5, -1)\). Therefore, the correct choice is A. The solution set is \(\boxed{\{(-4, -5, -1)\}}\).
