WLOG \(x\le y \le z\) we have:

\(\dfrac{7}{10}=\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\ge\dfrac{1}{z}+\dfrac{1}{z}+\dfrac{1}{z}=\dfrac{3}{z}\)

\(\Rightarrow\dfrac{7}{10}\ge\dfrac{3}{z}\Rightarrow z\ge\dfrac{30}{7}\approx4,28\)

Because \(z\) is positive integer id est \(z\ge 5\)

Because \(x+y+z\) is smallest value should find \(z\) is smallest value

\(\Rightarrow z=5\Rightarrow x=3;y=6\) (wrong because \(x\le y \le z\))

\(\Rightarrow z=6\Rightarrow y=5\Rightarrow x=3\) (right because \(x\le y \le z\))

We only find 1 smallest value satisfy, should \(z=7;z=8;... \) wrong

Hence \(x+y+z=3+5+6=14\)