https://r2.erweima.ai/mathbot/image/1725749147071k8rzmfh4.jpg

The image contains an equality between two matrices. The matrices are given as:

\[
\begin{bmatrix}
x + 2 & 8 & -3 \\
1 & 2y & 2z \\
7 & -2 & y + 2
\end{bmatrix}
=
\begin{bmatrix}
2x + 6 & 8 & -3 \\
1 & 18 & -8 \\
7 & -2 & 11
\end{bmatrix}
\]

To find the values of \( x \), \( y \), and \( z \), we need to equate the corresponding elements from both matrices:

1. For the element in the first row, first column:  
   \[ x + 2 = 2x + 6 \]

2. For the element in the second row, second column:  
   \[ 2y = 18 \]

3. For the element in the second row, third column:  
   \[ 2z = -8 \]

4. For the element in the third row, third column:  
   \[ y + 2 = 11 \]

### Solving for \( x \), \( y \), and \( z \):

1. **Solving for \( x \):**  
   \[ x + 2 = 2x + 6 \implies 2 = x + 6 \implies x = -4 \]

2. **Solving for \( y \):**  
   From \( 2y = 18 \):  
   \[ y = \frac{18}{2} = 9 \]

3. **Solving for \( z \):**  
   From \( 2z = -8 \):  
   \[ z = \frac{-8}{2} = -4 \]

4. **Verifying the value of \( y \):**  
   From \( y + 2 = 11 \):  
   \[ 9 + 2 = 11 \] (which is correct)

### Final Solution:
- \( x = -4 \)
- \( y = 9 \)
- \( z = -4 \)

Would you like further details, or do you have any questions?

### Related Questions:
1. How to solve systems of equations using matrix methods?
2. What are the properties of matrix equality?
3. How to perform matrix multiplication?
4. How to find the determinant of a matrix?
5. What is the inverse of a matrix, and how is it calculated?

### Tip:
When working with matrix equalities, always equate corresponding elements to form equations that can be solved for unknown variables.

Learn how to solve a matrix equality problem by finding the values of x, y, and z. This tutorial covers core algebraic concepts, including equating corresponding matrix elements and solving linear equations. Suitable for students in Grades 9-12.

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