Problem 11.2. Solve the system

$$
\left\lvert\, \begin{aligned}
& \left(4^{\sqrt{x^{2}+x}}+7.2^{\sqrt{x^{2}+x}}-1\right) \sin (\pi y)=7|\sin (\pi y)| \\
& x^{2}+4 x+y^{2}=0
\end{aligned}\right.
$$

Aleksandar Ivanov
11.2. The system is defined for $x \in(-\infty,-1] \cup[0,+\infty)$ and every $y$. We consider three cases.

Case 1. Let $\sin (\pi y)>0$. The first equation gives

$$
4^{\sqrt{x^{2}+x}}+7.2^{\sqrt{x^{2}+x}}-1=7
$$

and setting $t=2^{\sqrt{x^{2}+x}}>0$ we obtain the equation $t^{2}+7 t-8=0$ with roots $t_{1}=1$ and $t_{2}=-8$. Since $t>0$, it follows that $2^{\sqrt{x^{2}+x}}=1$, whence $x^{2}+x=0$, i.e. $x=0$ and $x=-1$. For $x=0$ we get $y=0$ and for $x=-1$ we have $y= \pm \sqrt{3}$. Since $\sin 0=0, \sin (\sqrt{3} \pi)0$, the only solution of the system in this case is $x=-1, y=-\sqrt{3}$.

Case 2. Let $\sin (\pi y)=0$, i.e. $y$ is an integer. From the second equation we obtain $4-y^{2} \geq 0$, whence $y=0, \pm 1, \pm 2$. For $y=0$ we get $x=0,-4$; for $y= \pm 1$ we find $x=-2 \pm \sqrt{3}$ and for $y= \pm 2$ we have $x=-2$. Since $-10$ we obtain the equation $t^{2}+7 t+6=0$ with roots $t_{1}=-1$ and $t_{2}=-6$. Since $t>0$, the system has no solutions in this case.