$解:因为\frac {xy}{x+y}=-2$
$所以\frac {x+y}{xy}=\frac {1}{x}+\frac {1}{y}=-\frac {1}{2}$
$同理可得:\frac {1}{y}+\frac {1}{z}=\frac {3}{4},\frac {1}{x}+\frac {1}{z}=-\frac {3}{4}$
$所以\frac {1}{x}+\frac {1}{y}+\frac {1}{z}=(-\frac {1}{2}+\frac {3}{4}-\frac {3}{4})÷2=-\frac {1}{4}$
$所以原式=\frac {1}{\frac {1}{x}+\frac {1}{y}+\frac {1}{z}}=-4$