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

The given system is:

\[
\begin{aligned}
1. & \quad x + 4y = 11 \\
2. & \quad -5x - 4y = -23
\end{aligned}
\]

### Step 1: Add the equations to eliminate \(y\).
Since the coefficients of \(y\) are opposites (+4 and -4), we can add the two equations to eliminate \(y\):

\[
(x + 4y) + (-5x - 4y) = 11 + (-23)
\]

Simplifying this:

\[
x - 5x + 4y - 4y = 11 - 23
\]

\[
-4x = -12
\]

### Step 2: Solve for \(x\).
Now, solve for \(x\):

\[
x = \frac{-12}{-4} = 3
\]

### Step 3: Substitute \(x = 3\) back into one of the original equations to find \(y\).
We'll use the first equation \(x + 4y = 11\):

\[
3 + 4y = 11
\]

Subtract 3 from both sides:

\[
4y = 8
\]

Divide by 4:

\[
y = 2
\]

### Final Answer:
The solution to the system of equations is \((x, y) = (3, 2)\).

Do you want any more details or have any other questions? 

Here are some related questions:
1. What if the coefficients of \(y\) were not opposites? How would you approach elimination then?
2. How do you handle systems where both variables have non-zero coefficients?
3. How does the elimination method compare to substitution in terms of solving speed?
4. What are the common pitfalls when using the elimination method?
5. How can you verify that the solution to a system of equations is correct?
6. How would you solve a system of equations graphically?
7. Can you use the elimination method for nonlinear systems of equations?
8. How does the determinant of the coefficient matrix relate to solving linear systems?

**Tip:** Always double-check your final solution by substituting it back into the original equations to ensure accuracy.

Learn how to solve the system of linear equations x + 4y = 11 and -5x - 4y = -23 using the elimination method. This problem covers key algebraic concepts, including linear equations and systems of equations. Suitable for students in Grades 7-9.

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