Multiplying all three equations, we get $x^2 y^2 z^2 = 82944.$  Since $x,$$y,$ and $z$ are positive, $xyz = \sqrt{82944} = 288.$  Then
\begin{align*}
x &= \frac{xyz}{yz} = \frac{288}{72} = 4, \\
y &= \frac{xyz}{xz} = \frac{288}{48} = 6, \\
z &= \frac{xyz}{xy} = \frac{288}{24} = 12.
\end{align*}Hence, $x + y + z = \boxed{22}.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
If $x$, $y$, and $z$ are positive with $xy = 24$, $xz = 48$, and $yz = 72$, then find $x + y + z.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
22
If $x$, $y$, and $z$ are positive with $xy = 24$, $xz = 48$, and $yz = 72$, then find $x + y + z.$
test