I want to find the value of $x+y+z$, but I don't have any direct information about it. I do have some information about the products of pairs of these variables, so maybe I can use that somehow. One idea is to try to eliminate one of the variables and express the others in terms of it. For example, if I divide the first equation by the second, I get $y/z = 24/48 = 1/2$. This means that $y = z/2$, or $z = 2y$. Now I can substitute this into the third equation and get $y(2y) = 72$, or $y^2 = 36$. Since $y$ is positive, I can take the positive square root and get $y = 6$. Then, $z = 2y = 12$. To find $x$, I can use the first equation and get $x = 24/y = 24/6 = 4$. Now I have the values of $x$, $y$, and $z$, so I can add them up to get $x+y+z = 4+6+12 = 22$. 
If $x$, $y$, and $z$ are positive with $xy=24$, $xz = 48$, and $yz=72$, what is the value of $x+y+z$?