AMC 12 2000 Problem 20

If $x,y,$ and $z$ are positive numbers satisfying $x + \frac{1}{y} = 4, y + \frac{1}{z} = 1,$ and $z + \frac{1}{x} = \frac73,$ then what is the value of $xyz$ ?

$\textbf {(A)}\ 2/3 \textbf {(B)}\ 1 \textbf {(C)}\ 4/3 \textbf {(D)}\ 2 \textbf {(E)}\ 7/3$