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Evaluate $\frac{x}{yz} + \frac{y}{xz} + \frac{z}{y}$ given that $x + y + z = 4$ and $xyz = -60$ and $xy + xz + yz = −17$.

I made this expression have a common denominator $\frac{x}{yz} + \frac{y}{xz} + \frac{z}{y}$, which turned out to be $xyz=-60$. However, I tried using all other combinations for the two other formulas but nothing worked

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    Are you sure, that there is $\frac{z}{y}$, not $\frac{z}{xy}$? – 2017-01-17
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    Not sure if this will be useful, but the sums/products are coefficients of a cubic. Again, not sure if that observation is helpful is any way... – 2017-01-17
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    @JaroslawMatlak You must be right, because only then there is a unique result. – 2017-01-17

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Subsituting $z=4-x-y$ we obtain th following two equations:

$$ - x^2 - xy + 4x - y^2 + 4y + 17=0 $$

$$ - x^2y - xy^2 + 4xy + 60=0 $$ Taking resultants we see that $(x,y)=(-4,3),(-4,5),(3,-4),(3,5),(5,-4),(5,3)$. Then we have $$ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{y}=\frac{5}{4},\frac{-1}{12},\ldots, $$ but much nicer, always $$ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy}=-\frac{5}{6} $$