I have to find all $x,y$ pairs for the following:

$\frac{3x-4y}{xy}=-8$,

$\frac{2x+7y}{xy}=43$

The hint states that I should rewrite the left-sides of the equations. There are two routes I can see being taken here. First one would involve removing the denominators, the second one would be writing these fractions in their reciprocal form like $\frac{3}{y}+\frac{4}{x}$. For the sake of simplicity, I'll just remove the denominator by multiplying both sides.

This gives me the following:

$3x-4y = -8xy$,

$2x+7y = 43xy$

At this point it seems that I can't substitute any variable nicely without fractions. That means that I should go for the elimination of $x$.

$6x + 21y = 138xy$

$-$

$6x - 8y = -16xy$

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$29y = 154xy \therefore 29 = 154x \therefore \frac{29}{154}=x$ Now that does not look right. I get the same result by just splitting up the fractions...