We are given the system of equations:

$4x + 6y = -8$

$2x + y = 12$

Step 1: Multiply the second equation to align coefficients

We want to eliminate one variable. The second equation has $y$ with a coefficient of 1, while the first has $6y$. To match coefficients, multiply the second equation by 6:

$6(2x + y) = 6(12)$

$12x + 6y = 72$

Now our system is:

$4x + 6y = -8$

$12x + 6y = 72$

Step 2: Subtract the equations

Subtract the first equation from the second:

$(12x + 6y) - (4x + 6y) = 72 - (-8)$

$12x + 6y - 4x - 6y = 72 + 8$

$8x = 80$

$x = 10$

Step 3: Solve for $y$

Substituting $x = 10$ into the second equation:

$2(10) + y = 12$

$20 + y = 12$

$y = 12 - 20$

$y = -8$

Final Answer:

$\boxed{(10, -8)}$

Would you like a detailed explanation of any step?

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Related Questions:

1. Solve another system using elimination:
   $3x + 2y = 7$ $5x - 4y = -3$
2. How does the elimination method compare to substitution?
3. Can you solve this system using substitution instead of elimination?
4. What happens if a system has no solution or infinite solutions?
5. How can you use matrices to solve this system?

Tip: Always check your solution by plugging the values back into the original equations!