Answer
$x=-1, y=5, z=-1, w=-3$
Work Step by Step
Step 1. Write the original system of equations in matrix form $AX=B$:
$\begin{bmatrix} 1 & 2 & 0 & 3\\0 & 1 & 1 & 1\\0 & 1 & 0 & 1 \\1 & 2 & 0 & 2 \end{bmatrix} \begin{bmatrix} x\\y\\z\\w \end{bmatrix}\begin{array}( \\= \\ \\ \end{array} \begin{bmatrix} 0\\1\\2\\3 \end{bmatrix}$
Step 2. Note that matrix A is the same as in Exercise 25. Recall the answer to that question, we have:
$\begin{array}(\\A^{-1}=\\ \\ \end{array} \begin{bmatrix} 0 & 0 & -2 & 1\\-1 & 0 & 1 & 1\\0 & 1 & -1 & 0 \\1 & 0 & 0 & -1 \end{bmatrix} $
Step 3. Use the formula $X=A^{-1}B$, we get the solutions as:
$\begin{array}(\\X=\\ \\ \end{array} \begin{bmatrix} 0 & 0 & -2 & 1\\-1 & 0 & 1 & 1\\0 & 1 & -1 & 0 \\1 & 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} 0\\1\\2\\3 \end{bmatrix} \begin{array}(\\=\\ \\ \end{array} \begin{bmatrix} -1\\5\\-1\\-3 \end{bmatrix}$
Step 4. Conclusion: the solutions for the system of equations are $x=-1, y=5, z=-1, w=-3$
