We are given the system of linear equations:

$-x - 4y = 2 \quad \text{(1)}$ $x + 4y = -2 \quad \text{(2)}$

Step 1: Add the two equations.

By adding equations (1) and (2), we eliminate $y$:

$(-x - 4y) + (x + 4y) = 2 + (-2)$ $0 = 0$

This result indicates that the two equations are dependent, meaning they represent the same line. Therefore, there are infinitely many solutions, and $x$ and $y$ are dependent on each other.

Step 2: Express one variable in terms of the other.

From equation (2), solve for $x$:

$x + 4y = -2$ $x = -2 - 4y$

Thus, $x$ is expressed in terms of $y$. For any value of $y$, we can calculate $x$ using this equation.

Final Answer:

The system has infinitely many solutions. The solution set is: $x = -2 - 4y \quad \text{for any value of } y.$

Let me know if you would like a specific example or further clarification!

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Here are 5 related questions:

1. How do you solve systems of linear equations using substitution?
2. What are the different methods to solve a system of equations?
3. How do you interpret the result when the system of equations has infinitely many solutions?
4. What is the graphical representation of a system of dependent equations?
5. Can systems of equations have no solution? How is that identified?

Tip: When two equations lead to a result like $0 = 0$, this usually means the lines are the same, leading to infinite solutions.