Solve by Elimination/Addition
\[
\left\{\begin{array}{l}
3 x+6 y=-6 \\
9 x+18 y=-18
\end{array}\right.
\]
No solution
Infinite number of solutions

Step 1 ：The given system of equations is: \[\left\{\begin{array}{l} 3 x+6 y=-6 \\ 9 x+18 y=-18 \end{array}\right.\]

Step 2 ：The first step in solving a system of equations by elimination is to make the coefficients of one of the variables the same in both equations. In this case, the coefficients of both x and y in the second equation are three times the coefficients of x and y in the first equation. This suggests that the two equations might actually be the same, just scaled differently. If this is the case, then there would be an infinite number of solutions, as any solution to one equation would also be a solution to the other.

Step 3 ：To confirm this, we can divide the second equation by 3 and see if it becomes the same as the first equation.

Step 4 ：The two equations are indeed the same when the second equation is scaled down by a factor of 3. This means that any solution to one equation is also a solution to the other, so there are an infinite number of solutions to the system of equations.

Step 5 ：Final Answer: The system of equations has an \(\boxed{\text{infinite number of solutions}}\).