Find the values of x,y,z from the following matrix equation : `[{:(x+y,x+y+z),(2x+y,0):}]=[{:(3,6),(5,0):}]`

Explanation

To find the values of \( x, y, z \) from the matrix equation

\[ \begin{pmatrix} x+y & x+y+z \\ 2x+y & 0 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 5 & 0 \end{pmatrix} \]

we can equate the corresponding elements of the matrices.

### Step 1: Set up the equations
From the matrix equation, we can derive the following equations:

1. \( x + y = 3 \) (from the first element of the first row)
2. \( x + y + z = 6 \) (from the second element of the first row)
3. \( 2x + y = 5 \) (from the first element of the second row)
4. \( 0 = 0 \) (from the second element of the second row, which is trivially true)

### Step 2: Solve the first equation for \( y \)
From equation (1):
\[ y = 3 - x \tag{1} \]

### Step 3: Substitute \( y \) into the third equation
Substituting equation (1) into equation (3):
\[ 2x + (3 - x) = 5 \]
This simplifies to:
\[ 2x + 3 - x = 5 \]
\[ x + 3 = 5 \]
\[ x = 5 - 3 \]
\[ x = 2 \]

### Step 4: Substitute \( x \) back to find \( y \)
Now substitute \( x = 2 \) back into equation (1):
\[ y = 3 - 2 \]
\[ y = 1 \]

### Step 5: Substitute \( x \) and \( y \) into the second equation to find \( z \)
Now substitute \( x = 2 \) and \( y = 1 \) into equation (2):
\[ x + y + z = 6 \]
\[ 2 + 1 + z = 6 \]
\[ 3 + z = 6 \]
\[ z = 6 - 3 \]
\[ z = 3 \]

### Final Solution
Thus, the values of \( x, y, z \) are:
\[ x = 2, \quad y = 1, \quad z = 3 \]