\ \ (x+y)^2-2xy=2xy+4 \\
36=4xy+4 \ \ => \ \ xy=8 \\
\left \{ {{xy=8} \atop {x+y=6}} \right. \ \ => \ \ (6-y)y=8 \ \ => \ \ \left \{ {{(2,4)} \atop {(4,2)}} \right. " alt="a^2+b^2=a^2+2ab-2ab+b^2=(a+b)^2-2ab \\
x^2+y^2=2(xy+2) \ \ <=> \ \ (x+y)^2-2xy=2xy+4 \\
36=4xy+4 \ \ => \ \ xy=8 \\
\left \{ {{xy=8} \atop {x+y=6}} \right. \ \ => \ \ (6-y)y=8 \ \ => \ \ \left \{ {{(2,4)} \atop {(4,2)}} \right. " align="absmiddle" class="latex-formula">
\ \ \frac{5}{xy} = \frac{5}{6} \ \ => \ \ xy=6 \\
\left \{ {{x^2-y^2=5} \atop {xy=6}} \right. \ \ => \ \ x= \frac{6}{y} \\
\frac{36}{y^2}-y^2=5 \ \ => \ \ y^4+5y^2-36=0 \ \ => \\
=> (y^2+9)(y^2-4)=0 \ \ => \ \ y \in \{-2,2\} \ \=> \ \ \left \{ {{(-3,-2)} \atop {(3,2)}} \right. " alt=" \frac{x^2-y^2}{xy}= \frac{5}{6} \ \=> \ \ \frac{5}{xy} = \frac{5}{6} \ \ => \ \ xy=6 \\
\left \{ {{x^2-y^2=5} \atop {xy=6}} \right. \ \ => \ \ x= \frac{6}{y} \\
\frac{36}{y^2}-y^2=5 \ \ => \ \ y^4+5y^2-36=0 \ \ => \\
=> (y^2+9)(y^2-4)=0 \ \ => \ \ y \in \{-2,2\} \ \=> \ \ \left \{ {{(-3,-2)} \atop {(3,2)}} \right. " align="absmiddle" class="latex-formula">