Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $1/x+1/y+1/z=0$ 
Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $\frac1x+\frac1y+\frac1z=0$

My thinking was that since the numbers are integers, then there can't be $2$ negative values that cancel out the positive or $2$ positive numbers to cancel the negative. One integer would have to cancel out the second, and the third wouldn't make the sum zero. Am I right?

Alternatively, denote: $y=ax,z=abx$, then:
$$\begin{cases}x+ax+abx=0 \\ \frac{1}{x}+\frac{1}{ax}+\frac{1}{abx}=0\end{cases} \Rightarrow \begin{cases}1+a+ab=0 \\ 1+\frac{1}{a}+\frac{1}{ab}=0\end{cases} \Rightarrow \begin{cases}a=-\frac{1}{b+1} \\ b^2+b+1=0\end{cases} \Rightarrow \emptyset.$$
