You are given the following system of linear equations:

1. $7x - 4y = -11$
2. $7x + 4y = -59$

We will solve this system using the elimination method.

Step 1: Add the two equations to eliminate $y$.

$(7x - 4y) + (7x + 4y) = -11 + (-59)$ This simplifies to:

$7x + 7x = -70$ $14x = -70$

Step 2: Solve for $x$.

$x = \frac{-70}{14} = -5$

Step 3: Substitute $x = -5$ into one of the original equations. Let's use the first one:

$7(-5) - 4y = -11$ $-35 - 4y = -11$

Step 4: Solve for $y$.

Add 35 to both sides:

$-4y = -11 + 35$ $-4y = 24$

Now, divide both sides by -4:

$y = \frac{24}{-4} = -6$

Final Answer:

$x = -5, \quad y = -6$

Verifying the solution:

Substitute $x = -5$ and $y = -6$ into the second equation:

$7(-5) + 4(-6) = -59$ $-35 - 24 = -59$ This is correct.

Thus, the solution to the system of equations is $\boxed{x = -5, y = -6}$.

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Would you like more details on this method or have any other questions?

Here are some related questions to explore:

1. How do you solve a system of equations using substitution?
2. What is the geometric interpretation of solving a system of linear equations?
3. Can the elimination method be used for non-linear systems?
4. What happens if the two lines represented by the equations are parallel?
5. How do you solve a system of three equations with three variables?

Tip: Always double-check your final solution by substituting the values back into both original equations.