For $x$ directly proportional to $y$ and $z$, and inversely proportional to $w$, if $x=4$ when $(w,y,z)=(6,8,5)$, what is $x$ when $(w,y,z)=(4,10,9)$? 
Given that $x$ is directly proportional to $y$, and $z$ and is inversely proportional to $w$, and that $x = 4$ when $(w,y,z) = (6,8,5)$, what is $x$ when $(w,y,z)=(4,10,9)$?

Part of the answer says that

Because x is inversely proportional to $w$, when all other variables are constant, $xw$ is constant. Similarly, when the other two variables are constant, each of $\frac{x}{y}$ and $\frac{x}{z}$ is constant. We can combine all these by saying $\frac{xw}{yz}$ is constant.

I don't understand what "when all other variables are constant" or "when the other two variables are constant" means.
I also need an intuitive explanation as to how / why $xw$, $\frac{x}{y}$, and $\frac{x}{z}$ are combined together.
 A: Obviously you are dealing with so called combined proportions.
Your quantity $x$ depends on $3$ other quantities $w,y,z$.
Quantity $x$ is proportional to $y$. This means, if you vary only quantity $y$, then $x$ depends proportionally on $y$. You could write it as $x= y\cdot a$, where $a$ is a constant, which still depends on $w$ and $z$.
Now, $x$ is also proportional to $z$. If you vary only quantity $z$, then using the intermediate result from above, you can write $x = y\cdot a = y\cdot z \cdot b$, where $b$ is a constant, which still depends on $w$.
Finally, $x$ is inversely proportional to $w$. Hence, if you vary only $w$, then $x$ depends inversely proportionally on $w$. Using the result of the paragraph before you can write $x = y\cdot z \cdot b = \boxed{\frac{yz}{w}\cdot c}$ with a constant, which is independent of $w,y,z$.
So, with the first set of values you calculate $c$:
$$4 = c\frac{8\cdot 5}{6}\Leftrightarrow c= \frac 35$$
Now, you can calculate $x$ for the second set of values:
$$x = \frac 35 \frac{10\cdot 9}{4} = \frac{27}{2}$$
