The answer is yes; here's why:

A multiple of 7 can be written 7$k$, or 7$n$, or whatever, where $k$, $n$, etc. are whole numbers.

So if $x + y = z$, and $x$ and $y$ are multiples of 7, we can say $x$ = 7$k$ and $y$ = $7$n.

Then $z = x + y = 7k + 7n = 7(k + n)$, and $z$ is also a multiple of 7.

The argument is the same for the other cases, since $x = z - y$ and $y = z - x$ .