Let's analyze the system of equations:

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

Step 1: Check the system visually or algebraically.

First, observe the two equations:

• The first equation is $x + 4y = 4$.
• The second equation is $-x - 4y = -4$.

Step 2: Add both equations together.

We can eliminate the variables by adding the two equations:

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

Simplifying:

$0 = 0$

This results in an identity (0 = 0), which means the two equations are not contradictory and are essentially the same. This indicates that the system has infinitely many solutions.

Step 3: Conclusion.

Since both equations represent the same line, the system has many solutions.

To express this in terms of $y$:

From the first equation $x + 4y = 4$, solve for $x$:

$x = 4 - 4y$

Thus, $x$ is expressed in terms of $y$, and since there are infinite possible values for $y$, there are infinitely many solutions for $x$ and $y$.

---

Would you like to dive deeper into the reasoning behind this, or do you have another question?

Here are some related questions for further exploration:

1. How do we recognize a system with no solution?
2. What is the graphical interpretation of systems with infinite solutions?
3. How can we solve a system of linear equations using substitution?
4. What does it mean when two lines are parallel in a system of equations?
5. How do you determine the number of solutions to a system based on its determinant?

Tip: To quickly check if a system has no solution, a unique solution, or infinite solutions, try adding or subtracting the equations to eliminate variables. If you end up with a contradiction (like $0 = 5$), there's no solution. If you get an identity like $0 = 0$, there are infinitely many solutions.