Solve the following system of equations:

$$
\begin{aligned}
& \lg \left(\frac{x^{2}}{y^{3}}\right)=1 \\
& \lg \left(x^{2} y^{3}\right)=7
\end{aligned}
$$
The domain of interpretation for the task is the set of numbers $x \neq 0, y>0$. Many incorrectly sought the solution on the set of numbers $x>0, y>0$. The correct solution is:

$$
x= \pm 100, \quad y=10
$$

In many papers, the following incorrect reasoning appeared: from $y^{6}=10^{6}$, it follows from the equality of the exponents that $y=10$. This is not true, because $(-10)^{6}=10^{6}$!