3x + 6y - 2z + 9 w = 6 -5x + 4y -5z -6w = 5 - 3x + 8y + 2 z-3w=3 - 4x + l0y + 3z - 9w= 9

You have a system of linear equations. Let's rewrite the equations for clarity:

1. \( 3x + 6y - 2z + 9w = 6 \)
2. \( -5x + 4y - 5z - 6w = 5 \)
3. \( -3x + 8y + 2z - 3w = 3 \)
4. \( -4x + 10y + 3z - 9w = 9 \)

To solve this system of equations, we can use methods such as substitution, elimination, or matrix operations (e.g., Gaussian elimination). Here, let's use the matrix method:

1. **Form the coefficient matrix \(A\)**, the variable matrix \(X\), and the constant matrix \(B\)**:
   \[
   A = \begin{pmatrix}
   3 & 6 & -2 & 9 \\
   -5 & 4 & -5 & -6 \\
   -3 & 8 & 2 & -3 \\
   -4 & 10 & 3 & -9
   \end{pmatrix}, \quad
   X = \begin{pmatrix}
   x \\
   y \\
   z \\
   w
   \end{pmatrix}, \quad
   B = \begin{pmatrix}
   6 \\
   5 \\
   3 \\
   9
   \end{pmatrix}
   \]
   
2. **The matrix equation is**:
   \[
   AX = B
   \]
   
3. **To solve for \(X\)** (which contains \(x\), \(y\), \(z\), and \(w\)), we find the inverse of matrix \(A\) (if it exists) and multiply it by matrix \(B\):
   \[
   X = A^{-1}B
   \]

Let me calculate the solution for you.The solution to the system of equations is:

\[
\begin{aligned}
x & = 2.582 \\
y & = 1.172 \\
z & = -1.164 \\
w & = -1.234 \\
\end{aligned}
\]

These values satisfy all four equations in the system.

Would you like more details or have any questions?

Here are 5 related questions you might consider:

1. How would you solve a similar system with three variables instead of four?
2. What are the advantages of using matrix methods for solving linear systems?
3. Can you solve this system using substitution or elimination methods?
4. How can you verify that the solution is correct?
5. What are the implications if the determinant of matrix \(A\) is zero?

**Tip:** Always check the determinant of the coefficient matrix before attempting to solve a system of linear equations using the matrix inverse method. If the determinant is zero, the system has either no solution or infinitely many solutions.

Learn how to solve a system of linear equations using the matrix method with step-by-step calculations. This problem involves four equations with four variables: x, y, z, and w. The solution includes detailed matrix operations and verification steps. Suitable for high school students studying linear algebra.

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