<s>[INST] $\frac{x}{y}-\frac{y}{x}=\frac56$ and $x^2-y^2=5$ 
Solve the system: $$\begin{array}{|l}
 \dfrac{x}{y}-\dfrac{y}{x}=\dfrac{5}{6} \\ x^2-y^2=5 \end{array}$$

First, we have $x,y \ne 0$. Let's write the first equation as:
$$\dfrac{x}{y}-\dfrac{y}{x}=\dfrac{5}{6} \Leftrightarrow \dfrac{x^2-y^2}{xy}=\dfrac{5}{6}$$
We have $x^2-y^2=5$, therefore $xy=6$. What to do next?
 [/INST] $xy = 6 \implies y= \dfrac6x$ . So , 
$$x^2-y^2 = 5 \implies x^2 - \dfrac{36}{x^2} =5$$
Taking $x^2 = a$ , we get :
$$a^2-36=5a\implies (a-9)(a+4) = 0$$
We have $a=9\implies x=\pm3$ and $y = \pm2$
And we have $a=-4 \implies x = \pm 2i$  and $y=\pm 3i$
So the solutions are $(x,y) = (3,2)$  , $(x,y) = (-3,-2)$ , 
$(2i,-3i)$ and $(x,y) = (-2i,3i)$
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