Choice A is correct. The perimeter of a triangle is the sum of the lengths of its three sides. The triangle shown has side lengths $x$, $y$, and $z$. It's given that the triangle has a perimeter of $22$ units. Therefore, $x+y+z=22$. If $x=9$ units and $y=7$ units, the value of $z$, in units, can be found by substituting $9$ for $x$ and $7$ for $y$ in the equation $x+y+z=22$, which yields $9+7+z=22$, or $16+z=22$. Subtracting $16$ from both sides of this equation yields $z=6$. Therefore, if $x=9$ units and $y=7$ units, the value of $z$, in units, is $6$.

Choice B is incorrect. This is the value of $y$, in units, not the value of $z$, in units.

Choice C is incorrect. This is the value of $x$, in units, not the value of $z$, in units.

Choice D is incorrect. This is the value of $x+y$, in units, not the value of $z$, in units.