<s>[INST] Solve system of equations $\frac{x}{y}+\frac{y}{x}=\frac{10}{3}$, $x^2-y^2=8$ 
Solve the system:
  $$\begin{array}{l}\dfrac{x}{y}+\dfrac{y}{x}=\dfrac{10}{3} \\
 x^2-y^2=8\end{array}$$

First, we have $x \ne 0$ and $y \ne 0$. We can rewrite the first equation as $$\dfrac{x^2+y^2}{xy}=\dfrac{10}{3}$$
What should I do next?
 [/INST] Denote: $\frac xy=a$. Then from the first equation:
$$a+\frac1a=\frac{10}{3} \Rightarrow 3a^2-10a+1=0 \Rightarrow a=\frac13,3 \Rightarrow \\
x=3y,y=3x$$
Plug these into the second equation:
$$(3y)^2-y^2=8\Rightarrow y=\pm 1 \Rightarrow x=\pm 3\\
x^2-(3x)^2=8 \Rightarrow x^2=-1 \Rightarrow \emptyset.$$
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