6.098. $\left\{\begin{array}{l}x^{3}+y^{3}=7 \\ x y(x+y)=-2\end{array}\right.$
Solution.

Rewrite the system of equations as

$$
\left\{\begin{array} { l } 
{ ( x + y ) ( x ^ { 2 } - x y + y ^ { 2 } ) = 7 , } \\
{ x y ( x + y ) = - 2 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
(x+y)\left((x+y)^{2}-3 x y\right)=7 \\
x y(x+y)=-2
\end{array}\right.\right.
$$

Let $\left\{\begin{array}{l}x+y=u, \\ x y=v,\end{array}\right.$ then the system of equations has the form

$$
\left\{\begin{array}{l}
u\left(u^{2}-3 v\right)=7, \\
u v=-2
\end{array} \Rightarrow v=-\frac{2}{u}\right.
$$

From the first equation of the obtained system, we find

$$
u\left(u^{2}+\frac{6}{u}\right)=7 \Leftrightarrow u^{3}+6=7, u^{3}=1
$$

from which $u=1$. Then $v=-\frac{2}{1}=-2$. Further,

$$
\left\{\begin{array} { l } 
{ x + y = 1 , } \\
{ x y = - 2 }
\end{array} \Leftrightarrow \left(\left\{\begin{array}{l}
x_{1}=2, \\
y_{1}=-1
\end{array} \text { or }\left\{\begin{array}{l}
x_{2}=-1, \\
y_{2}=2 .
\end{array}\right)\right.\right.\right.
$$

Answer: $(2 ;-1)(-1 ; 2)$