Let xy = 7 and x + y = -6
Using the second equation we have y = -6 - x
So....substituting this into the first equation, we have
x (-6 - x) = 7 simplify
-6x - x^2 = 7 rearrange
x^2 + 6x + 7 = 0 using the onsite calculator we have
$${{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{6}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{7}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\sqrt{{\mathtt{2}}}}{\mathtt{\,-\,}}{\mathtt{3}}\\
{\mathtt{x}} = {\sqrt{{\mathtt{2}}}}{\mathtt{\,-\,}}{\mathtt{3}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{4.414\: \!213\: \!562\: \!373\: \!095}}\\
{\mathtt{x}} = -{\mathtt{1.585\: \!786\: \!437\: \!626\: \!905}}\\
\end{array} \right\}$$
So these are the two interchangeable answers for x and y.......
And as Annabelle1DirectionS found, there are no integer solutions......!!!!