The problem involves solving a system of linear equations using the elimination method. The given equations are:

1. $-8x - y = -6$
2. $-4x + y = -6$

Let me solve this step-by-step:

Step 1: Add the equations

Adding the two equations together eliminates $y$:

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

Simplify:

$-8x - 4x = -12$

$-12x = -12$

Step 2: Solve for $x$

Divide both sides by $-12$:

$x = 1$

Step 3: Substitute $x = 1$ into one equation

Substitute $x = 1$ into the first equation:

$-8(1) - y = -6$

Simplify:

$-8 - y = -6$

Add 8 to both sides:

$-y = 2$

Multiply through by $-1$:

$y = -2$

Final Solution:

The solution to the system of equations is:

$x = 1, \, y = -2$

Would you like further details or have any questions?
Here are 5 related questions for practice:

1. Solve another system using elimination: $3x + 2y = 12$ and $2x - 2y = 6$.
2. What happens if elimination doesn't immediately work? What alternative method can you use?
3. How can graphing confirm the solution to a system of linear equations?
4. Solve the same system using substitution and compare results.
5. What is the determinant of the coefficient matrix for this system?

Tip: Always double-check your solution by substituting the values of $x$ and $y$ back into the original equations.