To solve the system of linear equations:

$-4x - 2y = -12$ $4x + 8y = -24$

we can solve by either substitution or elimination. I'll proceed with the elimination method here.

Step 1: Eliminate One Variable

To eliminate one of the variables, we can add the two equations together. Notice that the coefficients of $x$ are opposites:

1. $-4x - 2y = -12$
2. $4x + 8y = -24$

Adding these two equations:

$(-4x + 4x) + (-2y + 8y) = -12 + (-24)$

$0x + 6y = -36$

$6y = -36$

Step 2: Solve for $y$

Divide both sides by 6:

$y = -6$

Step 3: Substitute $y$ Back to Find $x$

Now that we have $y = -6$, substitute it into one of the original equations to solve for $x$. Let's use the first equation:

$-4x - 2y = -12$

Substituting $y = -6$:

$-4x - 2(-6) = -12$

$-4x + 12 = -12$

Subtract 12 from both sides:

$-4x = -24$

Divide both sides by -4:

$x = 6$

Final Solution

The solution to the system is:

$x = 6, \quad y = -6$

Verification

We can substitute $x = 6$ and $y = -6$ into the second equation to verify:

$4x + 8y = -24$

$4(6) + 8(-6) = 24 - 48 = -24$

This matches the original equation, confirming that the solution is correct.

Summary

$x = 6, \quad y = -6$

Would you like more details or have any questions?

Related Questions:

1. How do you solve a system of linear equations by substitution?
2. Can a system of linear equations have no solution or infinitely many solutions?
3. What is the graphical interpretation of solving two linear equations?
4. How do you determine if two lines are parallel or intersecting based on their equations?
5. What is the determinant method (Cramer's rule) for solving a system of equations?

Tip: When adding or subtracting equations to eliminate a variable, always ensure that the coefficients are opposite or identical to simplify the process.