We are given the system of four equations:

$1) \; x + y + z + w = 2$ $2) \; 2x - y + 2z - w = -5$ $3) \; 3x + 2y + 3z + 4w = 7$ $4) \; x - 2y - 3z + 2w = 5$

To solve this system using the Gauss-Jordan elimination method, we first convert the system into an augmented matrix, perform row operations to achieve reduced row echelon form (RREF), and then extract the solutions.

Step 1: Set up the augmented matrix.

1 & 1 & 1 & 1 & | & 2 \\ 2 & -1 & 2 & -1 & | & -5 \\ 3 & 2 & 3 & 4 & | & 7 \\ 1 & -2 & -3 & 2 & | & 5 \end{pmatrix}$$ ### Step 2: Apply row operations to achieve the reduced row echelon form (RREF). I will perform these calculations to find the RREF. Let me do the row reductions.The reduced row echelon form (RREF) of the augmented matrix is: $$\begin{pmatrix} 1 & 0 & 0 & 0 & | & 0 \\ 0 & 1 & 0 & 0 & | & 1 \\ 0 & 0 & 1 & 0 & | & -1 \\ 0 & 0 & 0 & 1 & | & 2 \end{pmatrix}$$ ### Step 3: Interpret the solution from the RREF. From this matrix, we can read off the solutions for $$x$$, $$y$$, $$z$$, and $$w$$: $$x = 0, \quad y = 1, \quad z = -1, \quad w = 2$$ Thus, the solution to the system of equations is: $$(x, y, z, w) = (0, 1, -1, 2)$$ Would you like further details on any of these steps or have any questions? --- Here are 5 related questions you might explore: 1. How can I solve a system of equations using matrix inversion? 2. What are the key differences between Gauss-Jordan and Gaussian elimination? 3. How does the reduced row echelon form relate to the solution of a system of equations? 4. Can I apply the Gauss-Jordan method to inconsistent or dependent systems? 5. How can I solve a system of nonlinear equations? **Tip:** Always double-check the rank of the matrix when solving systems of equations to ensure you don't encounter inconsistencies or dependent equations.