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We are given two matrices, \( A \) and \( B \), and asked to find the value of \( x^2 + y^2 + z^2 \).

Matrix \( A \) is:
\[
A = \begin{pmatrix} 7 & x \\ y & 1 \end{pmatrix}
\]
Matrix \( B \) is:
\[
B = \begin{pmatrix} x + y & z + 2 \\ 3 & y - z \end{pmatrix}
\]
Since the matrices are equal, corresponding elements must be equal. We can set up the following system of equations:

1. \( 7 = x + y \) (from the first row, first column).
2. \( x = z + 2 \) (from the first row, second column).
3. \( y = 3 \) (from the second row, first column).
4. \( 1 = y - z \) (from the second row, second column).

### Step 1: Solve for \( y \)
From equation (3), we know:
\[
y = 3
\]

### Step 2: Solve for \( z \)
Substitute \( y = 3 \) into equation (4):
\[
1 = 3 - z
\]
Solving for \( z \):
\[
z = 2
\]

### Step 3: Solve for \( x \)
Substitute \( y = 3 \) into equation (1):
\[
7 = x + 3
\]
Solving for \( x \):
\[
x = 4
\]

### Step 4: Calculate \( x^2 + y^2 + z^2 \)
Now that we have \( x = 4 \), \( y = 3 \), and \( z = 2 \), we can calculate:
\[
x^2 + y^2 + z^2 = 4^2 + 3^2 + 2^2 = 16 + 9 + 4 = 29
\]

Thus, the value of \( x^2 + y^2 + z^2 \) is \( \boxed{29} \).

Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:
1. How do you check if two matrices are equal?
2. What is the importance of systems of equations in solving matrix problems?
3. How can matrices be used in solving linear transformations?
4. How would this problem change if the matrices were of higher dimensions?
5. How can you represent matrices geometrically?

**Tip:** When solving matrix equality problems, always match corresponding elements to set up your system of equations effectively.

Si la matriz A = (7, x; y, 1) es igual a la matriz B = (x + y, z + 2; 3, y - z), hallar el valor de x^2 + y^2 + z^2.

Learn how to solve a system of equations derived from the equality of two matrices to find the value of x^2 + y^2 + z^2. This problem involves matrix operations and algebraic techniques suitable for Grades 10-12.

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