solve the system  2x-4y+5z=7, -3x-2y+2z=10,-2x+3y-2z=14

We are solving the system of linear equations:

1. \( 2x - 4y + 5z = 7 \)
2. \( -3x - 2y + 2z = 10 \)
3. \( -2x + 3y - 2z = 14 \)

### Solution
We will use the **matrix method** to find the solution, representing the system in the form \( A\mathbf{x} = \mathbf{b} \), where:

\[
A = \begin{bmatrix} 
2 & -4 & 5 \\ 
-3 & -2 & 2 \\ 
-2 & 3 & -2 
\end{bmatrix}, \quad
\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad
\mathbf{b} = \begin{bmatrix} 7 \\ 10 \\ 14 \end{bmatrix}.
\]

To solve, \( \mathbf{x} = A^{-1}\mathbf{b} \).

#### Step 1: Compute \( A^{-1} \)
The inverse of \( A \) is computed as \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \), where \( \det(A) \) is the determinant of \( A \), and \( \text{adj}(A) \) is the adjugate matrix.

#### Step 2: Solve for \( \mathbf{x} \)
Perform \( A^{-1} \mathbf{b} \) to find the solution.

I'll compute the values explicitly. numpy as np

# Define the coefficient matrix A and the constant vector b
A = np.array([[2, -4, 5],
              [-3, -2, 2],
              [-2, 3, -2]])
b = np.array([7, 10, 14])

# Solve the system using matrix inversion
solution = np.linalg.solve(A, b)
solutionThe solution to the system of equations is:

\[
x = -2.90, \quad y = 9.52, \quad z = 10.17
\]

### Final Answer:
- \( x \approx -2.90 \)
- \( y \approx 9.52 \)
- \( z \approx 10.17 \)

Would you like a detailed explanation of the steps, or do you have any questions? 

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### Related Questions:
1. How can the determinant of a matrix help identify if a system has a unique solution?
2. What are the advantages of using matrix methods over substitution or elimination methods?
3. How does changing one equation affect the solution of the system?
4. Can you use the Cramer's Rule to solve this system?
5. What are the computational complexities of solving linear systems using matrices?

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### Tip:
For large systems, numerical methods like LU decomposition or iterative methods (e.g., Jacobi or Gauss-Seidel) are more efficient than direct matrix inversion.

solve the system 2x-4y+5z=7, -3x-2y+2z=10, -2x+3y-2z=14

Learn how to solve the system of linear equations 2x-4y+5z=7, -3x-2y+2z=10, and -2x+3y-2z=14 using matrix methods. Explore concepts like matrix inversion, determinants, and linear algebra techniques. Suitable for high school and undergraduate students.

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