The system of equations you have is:

1. $x + 2y = 0$
2. $x - y = -6$

Let's solve it step by step.

Step 1: Solve for one variable

From the second equation, solve for $x$: $x = y - 6$

Step 2: Substitute into the first equation

Substitute $x = y - 6$ into the first equation: $(y - 6) + 2y = 0$ Simplify: $y - 6 + 2y = 0$ $3y - 6 = 0$ $3y = 6$ $y = 2$

Step 3: Substitute $y = 2$ into $x = y - 6$

Now substitute $y = 2$ into the equation $x = y - 6$: $x = 2 - 6 = -4$

So, $x = -4$.

Answer: The value of $x$ is $\boxed{-4}$, which corresponds to option (a).

Let me know if you'd like further explanation or if you have any other questions!

Additional Questions:

1. How would you solve this system using substitution versus elimination?
2. What would happen if the coefficients of $x$ and $y$ were different in both equations?
3. How do you solve for $y$ in a system of linear equations?
4. What is the graphical representation of this system of equations?
5. How can you check if the solution is correct?

Tip: Always check your solution by substituting it back into the original equations!