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$\\Solution:\\ Given:\\ 2^x=3^y=6^{-z}\\\\ Let,\ 2^x=3^y=6^{-z}=k\\\\ \Rightarrow 2^x=k\\\\ \Rightarrow k^{\frac1x}=2\\\\ Similarly,\ \ \ k^{\frac1y}=3\ \ \ \ and\ \ \ k^{\frac1{-z}}=6\\ \\Now\\\\ \Rightarrow k^{\frac1{-z}}=2\times3\\\\ \Rightarrow k^{\frac1{-z}}=k^{\frac1x}\times k^{\frac1y}\\\\ \Rightarrow k^{\frac1{-z}}=k^{\frac1x+{\frac1y}}\ \ \ \ [\because a^m\times b^m=a^{m+n}]\\\\ Bases\ are\ equal\ then\ the\ exponents\ are\ also\ equal\\\\ \Rightarrow -\frac{1}{z}=\frac1x+\frac1y\\\\ \therefore \frac1x+\frac1y+\frac{1}{z}=0$

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