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Let's solve the given system of linear equations step by step using:

1. **Cramer's Rule**.
2. **Inverse Matrix Method**.

The system of equations is:

\[
\begin{aligned}
x + 2y - 2z &= -1 \\
2x + 3y &= 5 \\
2x - 2y + 5z &= 10
\end{aligned}
\]

### (a) **Cramer's Rule**:
Cramer's Rule uses determinants to solve a system of linear equations. The system can be represented in matrix form as \( A \mathbf{x} = \mathbf{b} \), where:

\[
A = \begin{pmatrix}
1 & 2 & -2 \\
2 & 3 & 0 \\
2 & -2 & 5
\end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} -1 \\ 5 \\ 10 \end{pmatrix}
\]

The solutions for \( x \), \( y \), and \( z \) are given by:

\[
x = \frac{\det(A_x)}{\det(A)}, \quad y = \frac{\det(A_y)}{\det(A)}, \quad z = \frac{\det(A_z)}{\det(A)}
\]

Where \( A_x \), \( A_y \), and \( A_z \) are the matrices formed by replacing the respective columns of \( A \) with \( \mathbf{b} \).

Let me calculate these determinants for you.For part (a) **Cramer's Rule**, the determinants are:

- \( \det(A) = 15 \)
- \( \det(A_x) = 15 \)
- \( \det(A_y) = 15 \)
- \( \det(A_z) = 30 \)

Thus, the solution is:

\[
x = \frac{15}{15} = 1, \quad y = \frac{15}{15} = 1, \quad z = \frac{30}{15} = 2
\]

### (b) **Inverse Matrix Method**:
The solution to the system can also be found using the inverse of matrix \( A \), as follows:

\[
\mathbf{x} = A^{-1} \mathbf{b}
\]

Let me now calculate \( A^{-1} \) and find the solution.For part (b) **Inverse Matrix Method**, the solution is:

\[
x = 1, \quad y = 1, \quad z = 2
\]

Both methods yield the same result:

\[
x = 1, \quad y = 1, \quad z = 2
\]

Do you want further details on either method or have any questions?

Here are 5 related questions:
1. How can Cramer's Rule be applied to larger systems of equations?
2. What is the geometric interpretation of solving a system using matrices?
3. Can Cramer's Rule be used when the determinant of \( A \) is zero?
4. How is the inverse matrix method affected by ill-conditioned matrices?
5. What are the limitations of using the inverse matrix method for solving systems?

**Tip**: Always check the determinant of a matrix before applying Cramer's Rule or the inverse matrix method. If it's zero, the system may not have a unique solution.

Solve the system of equations: x + 2y - 2z = -1, 2x + 3y = 5, 2x - 2y + 5z = 10 using Cramer's Rule and the inverse matrix method.

Learn how to solve the linear system of equations x + 2y - 2z = -1, 2x + 3y = 5, 2x - 2y + 5z = 10 using both Cramer's Rule and the inverse matrix method. This example covers key linear algebra concepts such as determinants and matrix inversion.

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