The system of equations provided is:

1. $4x - y = 6$
2. $8x - 2y = 12$

Step 1: Check for dependency

We can check if one equation is a multiple of the other. Rewrite the second equation:

$8x - 2y = 12$

Divide the entire equation by 2:

$4x - y = 6$

This is identical to the first equation, meaning the second equation is a multiple of the first. Therefore, both equations represent the same line.

Step 2: Analyze the solutions

Since the two equations are identical, every solution of the first equation is also a solution of the second equation. This implies that there are infinitely many solutions.

Final Answer:

The system has infinitely many solutions.

Do you want additional details or have further questions? Here are 5 related questions to deepen your understanding:

1. How can you graphically represent systems with infinitely many solutions?
2. What happens if the equations are inconsistent instead of dependent?
3. Can you express the solution set of this system parametrically?
4. How do you solve systems like these using substitution or elimination methods?
5. How can you check for infinitely many solutions algebraically in other systems?

Tip: Always simplify the equations in a system to see if they are multiples of each other to quickly identify dependent systems!