Solve for $x$ and $y$. \[ \left[\begin{array}{cc} -3 & 1 \\ -5 & -5 \end{array}\right]\left[\begin{array}{cc} x & -1 \\ y & 1 \end{array}\right]=\left[\begin{array}{cc} 8 & 4 \\ 0 & 0 \end{array}\right] \]

Step 1 ：We are given the matrix equation: \[\left[\begin{array}{cc}-3 & 1 \\-5 & -5\end{array}\right]\left[\begin{array}{cc}x & -1 \\y & 1\end{array}\right]=\left[\begin{array}{cc}8 & 4 \\0 & 0\end{array}\right]\]

Step 2 ：We can solve for $x$ and $y$ by setting up two equations based on the matrix multiplication.

Step 3 ：The multiplication of two matrices is done element by element. For the first element in the resulting matrix, it's the sum of the products of the corresponding elements in the first row of the first matrix and the first column of the second matrix. For the second element in the first row of the resulting matrix, it's the sum of the products of the corresponding elements in the first row of the first matrix and the second column of the second matrix. The same logic applies to the second row of the resulting matrix.

Step 4 ：So, we can set up the following equations based on the matrix multiplication: \[-3x + y = 8\] and \[-5x - 5y = 0\]

Step 5 ：We can solve these equations to find the values of $x$ and $y$.

Step 6 ：The solution to the system of equations is $x = -2$ and $y = 2$. This means that the values of $x$ and $y$ that satisfy the given matrix equation are $x = -2$ and $y = 2$.

Step 7 ：Final Answer: The solution to the matrix equation is $x = -2$ and $y = 2$. So, the final answer is \(\boxed{x = -2, y = 2}\)