Find x and y from Matrix Equation

Given:

\[ \begin{bmatrix} 1 & 0 \\ y & 5 \end{bmatrix} + 2\begin{bmatrix} x & 0 \\ 1 & -2 \end{bmatrix} = I \]

\[ I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

Step 1: Multiply the Second Matrix

\[ 2\begin{bmatrix} x & 0 \\ 1 & -2 \end{bmatrix} = \begin{bmatrix} 2x & 0 \\ 2 & -4 \end{bmatrix} \]

Step 2: Add the Matrices

\[ \begin{bmatrix} 1 & 0 \\ y & 5 \end{bmatrix} + \begin{bmatrix} 2x & 0 \\ 2 & -4 \end{bmatrix} = \begin{bmatrix} 1 + 2x & 0 \\ y + 2 & 1 \end{bmatrix} \]

Step 3: Compare with Identity Matrix

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

Step 4: Equate Corresponding Elements

\[ 1 + 2x = 1 \Rightarrow x = 0 \]

\[ y + 2 = 0 \Rightarrow y = -2 \]

Final Answer:

\[ x = 0, \quad y = -2 \]