To solve the system of equations using elimination, we need to first rewrite the equations into a standard linear form:

Given equations:

1. $-3y - x + 9 = 0$
2. $0 = 3y + 12 + 8x$

Rewrite the equations:

1. $-3y - x + 9 = 0$ becomes: $-3y - x = -9$
2. $0 = 3y + 12 + 8x$ becomes: $3y + 8x = -12$

Now, we have the system: $-3y - x = -9$ $3y + 8x = -12$

Prepare for elimination:

Add the two equations together to eliminate $y$: $(-3y - x) + (3y + 8x) = -9 + (-12)$ $- x + 8x = -21$ $7x = -21$

Solve for $x$: $x = -3$

Substitute $x = -3$ into one of the original equations to find $y$:

Substitute into $-3y - x = -9$: $-3y - (-3) = -9$ $-3y + 3 = -9$ $-3y = -12$ $y = 4$

Solution:

The solution is: $x = -3, \, y = 4$

Would you like further clarification or details? Below are related questions to explore further:

1. How does the elimination method work, and why is it effective?
2. Can substitution also solve this system, and how would it compare?
3. How can we verify the solution to ensure it's correct?
4. What if the equations had no solution or infinitely many solutions?
5. How does the elimination method extend to systems with more than two variables?

Tip: Always double-check solutions by substituting them back into the original equations!