x+y+z+w=2
2x-y+2z-w=-5
3x+2y+3z+4w=7
x-2y-3z+2w=5
solve these equations by gauss jordan method

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.

\[
\begin{pmatrix}
1 & 1 & 1 & 1 & | & 2 \\
2 & -1 & 2 & -1 & | & -5 \\
3 & 2 & 3 & 4 & | & 7 \\
1 & -2 & -3 & 2 & | & 5
\end{pmatrix}
\]

### 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.

solve these equations by gauss jordan method: x+y+z+w=2, 2x-y+2z-w=-5, 3x+2y+3z+4w=7, x-2y-3z+2w=5

Learn how to solve a system of linear equations using the Gauss-Jordan elimination method. The system includes x + y + z + w = 2, 2x - y + 2z - w = -5, and more. This step-by-step guide covers key concepts in linear algebra, including matrix row reduction and finding the solution via RREF.

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