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The given system of equations is:

\[
\text{System A:}
\begin{cases}
x = 8 + 4y \\
-x + 4y = -8
\end{cases}
\]

### Step 1: Simplify the second equation
Start by substituting \( x = 8 + 4y \) into the second equation \( -x + 4y = -8 \).

\[
- (8 + 4y) + 4y = -8
\]

Distribute the negative sign:

\[
-8 - 4y + 4y = -8
\]

Simplify:

\[
-8 = -8
\]

This is a true statement, meaning that the system does not lead to a contradiction. Therefore, the system does **not** have a unique solution but instead has **infinitely many solutions**.

### Step 2: Solve for the dependent equation
Since the equation simplifies to a true statement, we know that \( x \) is dependent on \( y \). From the first equation \( x = 8 + 4y \), we can express \( x \) in terms of \( y \), or equivalently, solve for \( y \) in terms of \( x \).

Let's rewrite the system. From the equation \( x = 8 + 4y \), we can solve for \( y \):

\[
y = \frac{x - 8}{4}
\]

Thus, the solutions are determined by the equation \( y = \frac{x - 8}{4} \), meaning the system has infinitely many solutions that lie on this line.

### Final Answer:
- The system has infinitely many solutions.
- The equation they satisfy is \( y = \frac{x - 8}{4} \).

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Would you like further clarification, or do you have any specific questions on this?

Here are 5 related questions for practice:
1. What does it mean for a system of equations to have infinitely many solutions?
2. How would you verify if two equations are equivalent?
3. How can you graph the solution of the system on the coordinate plane?
4. What happens when a system of equations has no solution?
5. What are the possible outcomes when solving a system of linear equations?

**Tip:** In a system of linear equations, if both equations simplify to the same line, the system will have infinitely many solutions because the two equations represent the same line geometrically.

The given system of equations is:

System A: 
 x = 8 + 4y 
 -x + 4y = -8

Solve the system of equations x = 8 + 4y and -x + 4y = -8. This problem covers how to determine if a system has infinitely many solutions by simplifying the equations and solving for y. Suitable for Grades 8-10 students.

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