\[ \text{Given matrix } A \text{ has } x \text{ rows and } x+5 \text{ columns.} \] \[ \text{Matrix } B \text{ has } y \text{ rows and } 11 - y \text{ columns.} \] \[ \text{Since } AB \text{ and } BA \text{ exist,} \] \[ \text{The number of columns of } A \text{ must be equal to the number of rows of } B. \] \[ \Rightarrow x+5 = y. \] \[ \text{Similarly, the number of columns of } B \text{ must be equal to the number of rows of } A. \] \[ \Rightarrow x = 11 - y. \] Solving the system of equations: \[ x + 5 = y \] \[ x = 11 - y \] Substituting \( x = 11 - y \) in the first equation: \[ (11 - y) + 5 = y \] \[ 16 - y = y \] \[ 16 = 2y \] \[ y = 8 \] Substituting \( y = 8 \) in \( x = 11 - y \): \[ x = 11 - 8 = 3 \] Thus, the values of \( x \) and \( y \) are: \[ \boxed{3 \text{ and } 8} \]