As the given matrices are equal, their corresponding elements are also equal. Comparing the corresponding elements, we get:

$x+y=6$,  $x y=8$,   $5+z=5$

Now, $5+z=5 \Rightarrow z=0$

Using $(x-y)^{2}=(x+y)^{2}-4 x y,$   we get

$\Rightarrow(x-y)^{2}=36-32=4$

$\Rightarrow $  $x-y=\pm 2$

When $x-y=2$ and $x+y=6,$ we get $x=4$ and $y=2$

When $x-y=-2$ and $x+y=6$ we get $x=2$ and $y=4$

$\therefore $    $x=4$,  $y=2,$ and $z=0$ or $x=2$,  $y=4,$ and $z=0$