https://file.aiquickdraw.com/mathbot/file/1734879231796fi1h8spg.jpg

The system of linear equations is:

\[
\begin{aligned}
3x_1 + 2x_2 - 5x_3 + x_4 &= 14, \\
-4x_1 + 3x_2 + 4x_3 - x_4 &= 1, \\
5x_1 + 5x_2 + 2x_3 + 3x_4 &= 18, \\
-2x_1 - 7x_2 + 6x_3 - x_4 &= -29.
\end{aligned}
\]

We will solve this using the **Gauss elimination method**, which involves forming the augmented matrix and reducing it to row echelon form to find the solution. Let me proceed with the calculations. numpy as np

# Define the augmented matrix for the system of equations
aug_matrix = np.array([
    [3, 2, -5, 1, 14],
    [-4, 3, 4, -1, 1],
    [5, 5, 2, 3, 18],
    [-2, -7, 6, -1, -29]
], dtype=float)

# Perform Gaussian elimination
def gaussian_elimination(matrix):
    rows, cols = matrix.shape
    for i in range(rows):
        # Make the diagonal element 1 (if it's not already)
        if matrix[i, i] != 0:
            matrix[i] = matrix[i] / matrix[i, i]
        
        # Eliminate entries below the pivot
        for j in range(i + 1, rows):
            matrix[j] = matrix[j] - matrix[j, i] * matrix[i]

    # Back substitution to find the solution
    for i in range(rows - 1, -1, -1):
        for j in range(i - 1, -1, -1):
            matrix[j] = matrix[j] - matrix[j, i] * matrix[i]
    
    return matrix

# Apply Gaussian elimination to the augmented matrix
reduced_matrix = gaussian_elimination(aug_matrix.copy())

# Extract the solution (last column of the reduced matrix)
solution = reduced_matrix[:, -1]
reduced_matrix, solutionThe system of equations has been solved using Gaussian elimination. The solution is:

\[
\begin{aligned}
x_1 &= -63, \\
x_2 &= -13, \\
x_3 &= -17, \\
x_4 &= 144.
\end{aligned}
\]

Let me know if you'd like a detailed step-by-step explanation of the Gaussian elimination process!

### Related Questions:
1. How does Gaussian elimination compare to other methods, such as Cramer's rule or LU decomposition?
2. Can Gaussian elimination handle inconsistent or dependent systems of equations? 
3. What are the computational complexities of Gaussian elimination for large matrices?
4. How does partial or full pivoting improve the stability of Gaussian elimination?
5. Could this system be solved more efficiently using matrix algebra in Python?

### Tip:
When solving systems of equations, always check for singular matrices (determinant = 0) to avoid division errors or inconsistent solutions!

Learn how to solve a system of linear equations using Gaussian elimination. This step-by-step solution involves row reduction, augmented matrices, and back substitution. Suitable for high school and college students studying linear algebra.

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