Step 1 :The given system of equations is: \[\begin{array}{r} 6 x+3 y=0 \\ -18 x-9 y=0 \end{array}\]
Step 2 :We can see that the second equation is just the first equation multiplied by -3. This means that the two equations are not independent, they are the same line.
Step 3 :Therefore, there are infinitely many solutions to this system of equations.
Step 4 :We can also see that both equations equal to 0, which means the solution to the system of equations is the point where the line intersects with the origin (0,0).
Step 5 :The solution for the system of equations is: \[\{x: -y/2\}\]
Step 6 :This confirms our initial thought that the system of equations has infinitely many solutions. The solution is given in terms of y, which means any value of y will give a corresponding value of x, hence there are infinitely many solutions.
Step 7 :The solution also confirms that the point (0,0) is a solution to the system of equations, as when y=0, x also equals 0.
Step 8 :Final Answer: \[\boxed{\text{There are infinitely many solutions and the equations are dependent. The solution set is all points on the line defined by the equation } 6x + 3y = 0. \text{ This includes the point at the origin, (0,0).}}\]