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We are given a system of linear equations:

\[
\begin{aligned}
1. &\quad x_1 + 2x_2 + 3x_3 + 4x_4 = 32, \\
2. &\quad 2x_1 - x_2 + 2x_3 - x_4 = 3.
\end{aligned}
\]

To solve this using **Gaussian elimination**, we will follow these steps:

### Step 1: Set up the augmented matrix
We can represent the system of equations as an augmented matrix:

\[
\begin{pmatrix}
1 & 2 & 3 & 4 & | & 32 \\
2 & -1 & 2 & -1 & | & 3
\end{pmatrix}
\]

### Step 2: Perform row operations to reduce the matrix to upper triangular form
We aim to make the matrix upper triangular by eliminating the elements below the pivot in the first column.

- Start by subtracting 2 times the first row from the second row to eliminate the \(2x_1\) term in the second equation.

\[
R_2 \to R_2 - 2R_1
\]

The augmented matrix becomes:

\[
\begin{pmatrix}
1 & 2 & 3 & 4 & | & 32 \\
0 & -5 & -4 & -9 & | & -61
\end{pmatrix}
\]

### Step 3: Back substitution
Now we have an upper triangular matrix, and we can solve by back substitution.

1. From the second row:
   \[
   -5x_2 - 4x_3 - 9x_4 = -61
   \]
   This gives the equation:
   \[
   x_2 = \frac{-61 + 4x_3 + 9x_4}{-5}
   \]

2. From the first row:
   \[
   x_1 + 2x_2 + 3x_3 + 4x_4 = 32
   \]
   Substituting \(x_2\) into this equation, we can solve for \(x_1\).

This system is underdetermined (fewer equations than variables), so it will have a family of solutions where \(x_3\) and \(x_4\) can take on free values. From here, you can assign arbitrary values to \(x_3\) and \(x_4\), then compute the corresponding values of \(x_1\) and \(x_2\).

Would you like me to solve for specific values of \(x_3\) and \(x_4\) or explain further details?

Here are five follow-up questions:
1. Would you like to specify values for \(x_3\) and \(x_4\) to compute a particular solution?
2. Do you need help interpreting the meaning of free variables in the solution?
3. Shall we

Solve the following system of linear equations using the Gaussian elimination method: 
1. x1 + 2x2 + 3x3 + 4x4 = 32 
2. 2x1 - x2 + 2x3 - x4 = 3

Learn how to solve a system of linear equations with 2 equations and 4 variables using the Gaussian elimination method. This step-by-step guide will take you through setting up an augmented matrix and performing row operations to reduce the system. Suitable for advanced high school students or undergraduate level.

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