Find x and y

Given:

\[ 2\begin{bmatrix} 1 & 3 \\ 0 & x \end{bmatrix} + \begin{bmatrix} y & 0 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 5 & 6 \\ 1 & 8 \end{bmatrix} \]

Step 1: Multiply the First Matrix

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

Step 2: Add Matrices

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

Step 3: Compare with RHS

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

Step 4: Equate Elements

\[ 2 + y = 5 \Rightarrow y = 3 \]

\[ 2x + 2 = 8 \Rightarrow x = 3 \]

Final Answer:

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