Step 1: Perform the matrix addition on the left side of the equation. $$ \begin{bmatrix} -2 & x & 3 \\ z+3 & 4 & -y \end{bmatrix} + \begin{bmatrix} 2 & 1-x & -2 \\ -3 & 4+x & y \end{bmatrix} $$ Add the corresponding elements: $$ \begin{bmatrix} -2+2 & x+(1-x) & 3+(-2) \\ (z+3)+(-3) & 4+(4+x) & -y+y \end{bmatrix} $$ Simplify the elements: $$ \begin{bmatrix} 0 & 1 & 1 \\ z & 8+x & 0 \end{bmatrix} $$ Step 2: Equate the resulting matrix with the matrix on the right side of the given equation. $$ \begin{bmatrix} 0 & 1 & 1 \\ z & 8+x & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 1 \\ 2 & 8+x & 4 \end{bmatrix} $$ Step 3: Compare the corresponding elements of the two matrices to form a system of equations. From the element in row 1, column 1: $$ 0 = 0 $$ From the element in row 1, column 2: $$ 1 = 1 $$ From the element in row 1, column 3: $$ 1 = 1 $$ From the element in row 2, column 1: $$ z = 2 $$ From the element in row 2, column 2: $$ 8+x = 8+x $$ From the element in row 2, column 3: $$ 0 = 4 $$ The last equation, $0=4$, is a contradiction. This means that there are no values of $x$, $y$, and $z$ that can satisfy all the conditions of the given matrix equation simultaneously. Therefore, there is no solution for $x$, $y$, and $z$. The final answer is $\boxed{\text{No solution exists for } x, y, \text{ and } z}$.