If the sum of the matrices `[{:(x),(x),(y):}],[{:(y),(y),(z):}]and[{:(z),(0),(0):}] "is the matrix" [{:(10),(5),(5):}]`, then what is the value of y ?

Options

A. -5
B. 0
C. 5
D. 10

Explanation

To solve the problem, we need to find the value of \( y \) given the sum of the matrices:

\[ \begin{bmatrix} x & x & y \\ y & y & z \\ z & 0 & 0 \end{bmatrix} + \begin{bmatrix} y & y & z \\ \end{bmatrix} + \begin{bmatrix} z & 0 & 0 \end{bmatrix} = \begin{bmatrix} 10 & 5 & 5 \end{bmatrix} \]

### Step 1: Set up the equation by adding the corresponding elements of the matrices.

The sum of the matrices can be expressed as follows:

\[ \begin{bmatrix} x + y + z & x + y & y + z \\ \end{bmatrix} = \begin{bmatrix} 10 & 5 & 5 \end{bmatrix} \]

### Step 2: Create equations based on the equality of corresponding elements.

From the equality of the matrices, we can derive the following equations:

1. \( x + y + z = 10 \) (from the first column)
2. \( x + y = 5 \) (from the second column)
3. \( y + z = 5 \) (from the third column)

### Step 3: Solve the equations.

From equation (2), we can express \( x \) in terms of \( y \):

\[ x = 5 - y \]

Now, substitute \( x \) into equation (1):

\[ (5 - y) + y + z = 10 \]

This simplifies to:

\[ 5 + z = 10 \]

Thus, we find:

\[ z = 10 - 5 = 5 \]

### Step 4: Substitute \( z \) back into equation (3) to find \( y \).

Now substitute \( z = 5 \) into equation (3):

\[ y + 5 = 5 \]

This gives us:

\[ y = 5 - 5 = 0 \]

### Conclusion

The value of \( y \) is:

\[ \boxed{0} \]