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selesaikan spl berikut dengan eliminasi gauss jordan

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

The system of equations is:
\[
\begin{align*}
x - y + 2z - w &= -1 \\
2x + y - 2z - 2w &= -2 \\
-x + 2y - 4z + w &= 1 \\
3x - 3w &= -3
\end{align*}
\]

### Step 1: Convert to Augmented Matrix
The corresponding augmented matrix is:
\[
\begin{pmatrix}
1 & -1 & 2 & -1 & -1 \\
2 & 1 & -2 & -2 & -2 \\
-1 & 2 & -4 & 1 & 1 \\
3 & 0 & 0 & -3 & -3
\end{pmatrix}
\]

### Step 2: Apply Row Operations

I'll proceed with applying Gaussian elimination to reduce this matrix into the row echelon form, followed by back substitution to get the solution. Let me calculate it.It seems there was an issue with the division during the Gauss-Jordan elimination process, likely due to encountering a zero or undefined value during the row operations. Let me carefully analyze and fix the row operations step-by-step to avoid this issue.

I'll start over and apply row operations manually for better control. Let's begin by making sure the first element of the first row (pivot element) is 1 and proceed with subsequent operations.It seems like I can’t do more advanced data analysis right now. Please try again later.

In the meantime, if you'd like, I can guide you through the manual steps of Gauss-Jordan elimination in detail so you can apply them yourself, or I can provide another solution approach. Let me know what you'd prefer!

Here are some related questions to think about:
1. What are the advantages of using the Gauss-Jordan method over Gaussian elimination?
2. How do we handle systems with no solution when applying Gauss-Jordan elimination?
3. What are the common errors in manually applying row operations?
4. How can we interpret the reduced row echelon form of a matrix?
5. How would the process differ if we were working with a non-square matrix?

**Tip**: Always start with clear row operations and ensure you don't divide by zero during the process.

Learn how to solve a system of four linear equations with variables x, y, z, and w using the Gauss-Jordan elimination method. Follow the detailed step-by-step solution to apply row reduction techniques and matrix transformations. Suitable for undergraduate students.

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