You have the following system of equations:

1. $x + 2y - 5z + 4w = 0$
2. $2x - 3y + 2z + 3w = 0$
3. $4x - 7y + z - 6w = 0$

This is a system of linear equations. We can solve it using methods like substitution, elimination, or matrix methods (such as Gaussian elimination or using the inverse of a matrix). Let's use the matrix approach to solve it.

First, we express this system as a matrix equation $A \cdot X = B$, where:

1 & 2 & -5 & 4 \\ 2 & -3 & 2 & 3 \\ 4 & -7 & 1 & -6 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \\ z \\ w \end{pmatrix}, \quad B = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$$ To find the solution $$X$$, we can perform Gaussian elimination or use matrix methods. I will proceed with Gaussian elimination to solve for the variables. Let me calculate the solution for you. numpy as np # Coefficient matrix A A = np.array([[1, 2, -5, 4], [2, -3, 2, 3], [4, -7, 1, -6]]) # Right-hand side matrix B (zero vector) B = np.array([0, 0, 0]) # Solve the system of equations A * X = B # Since B is the zero vector, we are looking for the null space solution = np.linalg.lstsq(A, B, rcond=None)[0] solutionThe solution to the system of equations is: $$x = 0, \quad y = 0, \quad z = 0, \quad w = 0$$ This means the only solution to this system is the trivial solution where all variables are zero. Would you like more details on how the Gaussian elimination process works, or do you have any other questions about this system? Here are some follow-up questions you may find helpful: 1. How does Gaussian elimination work step by step? 2. What is the significance of a system having a trivial solution? 3. How do we interpret a system with infinitely many solutions? 4. What happens if the determinant of the coefficient matrix is zero? 5. How can we check the consistency of a system of linear equations? **Tip:** If you're solving a system of linear equations and the right-hand side is the zero vector, the system is homogeneous, and it will always have at least the trivial solution.