Find 'x' and 'y', if `2[(1,3),(0,x)]+[(y,0),(1,2)]=[(5,6),(1,8)]`

Answer

To solve the equation \( 2\begin{pmatrix} 1 & 3 \\ 0 & x \end{pmatrix} + \begin{pmatrix} y & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 1 & 8 \end{pmatrix} \), we will follow these steps:

### Step 1: Multiply the first matrix by 2
We start by multiplying each element of the matrix \( \begin{pmatrix} 1 & 3 \\ 0 & x \end{pmatrix} \) by 2.

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

**Hint:** Remember to multiply each element of the matrix by the scalar.

### Step 2: Write the equation with the multiplied matrix
Now we can rewrite the equation with the multiplied matrix:

\[ \begin{pmatrix} 2 & 6 \\ 0 & 2x \end{pmatrix} + \begin{pmatrix} y & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 1 & 8 \end{pmatrix} \]

### Step 3: Add the two matrices on the left side
Next, we add the two matrices on the left side:

\[ \begin{pmatrix} 2 + y & 6 + 0 \\ 0 + 1 & 2x + 2 \end{pmatrix} = \begin{pmatrix} 2 + y & 6 \\ 1 & 2x + 2 \end{pmatrix} \]

### Step 4: Set the resulting matrix equal to the right side
Now we set the resulting matrix equal to the right side:

\[ \begin{pmatrix} 2 + y & 6 \\ 1 & 2x + 2 \end{pmatrix} = \begin{pmatrix} 5 & 6 \\ 1 & 8 \end{pmatrix} \]

### Step 5: Equate the corresponding elements
From the equality of matrices, we can equate the corresponding elements:

1. \( 2 + y = 5 \)
2. \( 6 = 6 \) (This is always true and does not provide new information)
3. \( 1 = 1 \) (This is also always true)
4. \( 2x + 2 = 8 \)

### Step 6: Solve for \( y \)
From the first equation \( 2 + y = 5 \):

\[ y = 5 - 2 = 3 \]

### Step 7: Solve for \( x \)
From the fourth equation \( 2x + 2 = 8 \):

\[ 2x = 8 - 2 = 6 \\ x = \frac{6}{2} = 3 \]

### Final Answer
Thus, the values of \( x \) and \( y \) are:

\[ x = 3, \quad y = 3 \]