If `2[{:(,3,x),(,0,1):}]+3[{:(,1,3),(,y,2):}]=[{:(,z,-7),(,15,8):}]`, find the values of x, y and z.

Answer

To solve the equation \(2\begin{pmatrix} 3 & x \\ 0 & 1 \end{pmatrix} + 3\begin{pmatrix} 1 & 3 \\ y & 2 \end{pmatrix} = \begin{pmatrix} z & -7 \\ 15 & 8 \end{pmatrix}\), we will follow these steps:

### Step 1: Multiply each matrix by its respective scalar
First, we will multiply each element of the first matrix by 2 and the second matrix by 3.

\[ 2\begin{pmatrix} 3 & x \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 \cdot 3 & 2 \cdot x \\ 2 \cdot 0 & 2 \cdot 1 \end{pmatrix} = \begin{pmatrix} 6 & 2x \\ 0 & 2 \end{pmatrix} \]

\[ 3\begin{pmatrix} 1 & 3 \\ y & 2 \end{pmatrix} = \begin{pmatrix} 3 \cdot 1 & 3 \cdot 3 \\ 3 \cdot y & 3 \cdot 2 \end{pmatrix} = \begin{pmatrix} 3 & 9 \\ 3y & 6 \end{pmatrix} \]

### Step 2: Add the resulting matrices
Now, we will add the two resulting matrices:

\[ \begin{pmatrix} 6 & 2x \\ 0 & 2 \end{pmatrix} + \begin{pmatrix} 3 & 9 \\ 3y & 6 \end{pmatrix} = \begin{pmatrix} 6 + 3 & 2x + 9 \\ 0 + 3y & 2 + 6 \end{pmatrix} = \begin{pmatrix} 9 & 2x + 9 \\ 3y & 8 \end{pmatrix} \]

### Step 3: Set the resulting matrix equal to the right-hand side matrix
Now we set the resulting matrix equal to the matrix on the right-hand side:

\[ \begin{pmatrix} 9 & 2x + 9 \\ 3y & 8 \end{pmatrix} = \begin{pmatrix} z & -7 \\ 15 & 8 \end{pmatrix} \]

### Step 4: Equate corresponding elements
From the equality of the matrices, we can equate the corresponding elements:

1. \(9 = z\)
2. \(2x + 9 = -7\)
3. \(3y = 15\)
4. \(8 = 8\) (This is always true and does not provide new information)

### Step 5: Solve for \(z\)
From the first equation:
\[ z = 9 \]

### Step 6: Solve for \(x\)
From the second equation:
\[ 2x + 9 = -7 \implies 2x = -7 - 9 \implies 2x = -16 \implies x = \frac{-16}{2} = -8 \]

### Step 7: Solve for \(y\)
From the third equation:
\[ 3y = 15 \implies y = \frac{15}{3} = 5 \]

### Final Values
Thus, the values are:
- \(x = -8\)
- \(y = 5\)
- \(z = 9\)