Let's solve the system of equations using the elimination method:

The given system is:

1. $x + 7y = -11$
2. $2x + 4y = 8$

Step 1: Multiply the first equation by 2 to align the coefficients of $x$.

$2(x + 7y) = 2(-11)$

This gives:

$2x + 14y = -22$

Now we have the system:

1. $2x + 14y = -22$
2. $2x + 4y = 8$

Step 2: Subtract the second equation from the first equation.

$(2x + 14y) - (2x + 4y) = -22 - 8$

Simplifying:

$(2x - 2x) + (14y - 4y) = -30$

This simplifies to:

$10y = -30$

Step 3: Solve for $y$.

$y = \frac{-30}{10} = -3$

Step 4: Substitute $y = -3$ into one of the original equations to solve for $x$.

Using the first equation $x + 7y = -11$:

$x + 7(-3) = -11$

$x - 21 = -11$

$x = -11 + 21$

$x = 10$

Final Solution:

The solution to the system of equations is:

$(x, y) = (10, -3)$

Let me know if you'd like more details or have any questions!

Related questions:

1. What would happen if we used substitution instead of elimination?
2. How would the solution change if the system were inconsistent?
3. How can we check the solution by substituting back into the original equations?
4. What would the system look like if we had $x - 7y = -11$ and $2x + 4y = 8$?
5. What if the system had more than two variables?

Tip: When solving by elimination, make sure to adjust coefficients so you can eliminate one variable easily!