Determine whether the two functions are inverses.
$$w(x)=\frac{-2}{x-5}\text{ and }z(x)=\frac{-2+5x}{x}$$

Step 1 ：To determine if two functions are inverses of each other, we need to check if the composition of the two functions results in the identity function. The identity function is a function that always returns the same value that was used as its argument. In other words, for all x, f(x) = x.

Step 2 ：So, we need to check if $w(z(x))=x$ and $z(w(x))=x$. If both are true, then w and z are inverses of each other.

Step 3 ：Let's start by calculating $w(z(x))$.

Step 4 ：The result of $w(z(x))$ is x, which is the identity function. This means that the first condition for the functions to be inverses of each other is met.

Step 5 ：Now, let's calculate $z(w(x))$ to check the second condition.

Step 6 ：The result of $z(w(x))$ is also x, which is the identity function. This means that the second condition for the functions to be inverses of each other is also met.

Step 7 ：Since both conditions are met, we can conclude that the functions w and z are inverses of each other.

Step 8 ：$\boxed{{\displaystyle \text{The functions }w(x)=\frac{-2}{x-5}\text{ and }z(x)=\frac{-2+5x}{x}\text{ are inverses of each other.}}}$