solve the system of equations x+y-2z+3w=0 ,x-2y+z-w=0,4x+y-5z+8w=0,5x-7y+2z-w=0

We are given the system of four equations with four unknowns:

\[
\begin{aligned}
1) & \quad x + y - 2z + 3w = 0 \\
2) & \quad x - 2y + z - w = 0 \\
3) & \quad 4x + y - 5z + 8w = 0 \\
4) & \quad 5x - 7y + 2z - w = 0 \\
\end{aligned}
\]

Let's solve this system using the method of substitution or matrix techniques like Gaussian elimination. I'll compute the solution step-by-step.

### Step 1: Write the system in matrix form \( AX = B \)
This system can be written as \( AX = B \), where \( A \) is the coefficient matrix, \( X \) is the column vector of variables, and \( B \) is the column vector of constants (all zeros in this case).

The augmented matrix is:

\[
\begin{pmatrix}
1 & 1 & -2 & 3 & | & 0 \\
1 & -2 & 1 & -1 & | & 0 \\
4 & 1 & -5 & 8 & | & 0 \\
5 & -7 & 2 & -1 & | & 0 \\
\end{pmatrix}
\]

### Step 2: Apply Gaussian elimination

I'll now perform Gaussian elimination to reduce the matrix and solve for the variables.

Let me proceed with the computation.The solution to the system of equations is:

\[
x = 0, \quad y = 0, \quad z = 0, \quad w = 0
\]

This means that the only solution is the trivial solution where all variables are zero.

Would you like to dive into more details about the solution process or explore related questions?

Here are 5 related questions you could explore:
1. What is Gaussian elimination, and how does it work in solving systems of equations?
2. How can we use matrix inversion to solve systems of equations?
3. What are the conditions for a system of equations to have a unique solution, infinite solutions, or no solution?
4. How does the rank of a matrix relate to the solutions of a system of linear equations?
5. Can a system of linear equations have non-trivial solutions when the right-hand side is zero?

**Tip:** Always check the determinant of the coefficient matrix to ensure whether a unique solution exists.

Learn how to solve the system of linear equations x + y - 2z + 3w = 0, x - 2y + z - w = 0, 4x + y - 5z + 8w = 0, 5x - 7y + 2z - w = 0 using Gaussian elimination. This solution covers key concepts in linear algebra, including matrix methods and the Rank-Nullity Theorem. Suitable for students in Grades 10-12 and university levels.

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