Example 1. Solve the system of equations
$$
\left\{\begin{array}{l}
3 x+2 y-6=0, \\
2(x+2 y)+5(x-3)=0 .
\end{array}\right.
$$
Solve the following system of equations:
$$
A:\left\{\begin{array}{l}
(x+2 y)+2(x-3)=0, \\
2(x+2 y)+5(x-3)=0 .
\end{array}\right.
$$
$A$ can be transformed into a homogeneous linear system of equations with $x+2 y, x-3$ as unknowns, whose coefficient determinant $C=1 \neq 0$. Therefore, $A$ has only the trivial solution.
Thus, $\left\{\begin{array}{l}x+2 y=0, \\ x-3=0 .\end{array}\right.$
Hence, $\left\{\begin{array}{l}x=3, \\ y=-\frac{3}{2} .\end{array}\right.$