To solve the system of equations and determine whether it has no solutions, infinitely many solutions, or exactly one solution, we will analyze the two given equations:

1. $2x + y = 4$
2. $4x + 2y = 8$

Step 1: Simplify the second equation

Let's simplify the second equation by dividing through by 2: $4x + 2y = 8 \quad \Rightarrow \quad 2x + y = 4$

Now, we can see that the second equation is the same as the first equation. Therefore, both equations are identical.

Step 2: Interpretation

Since both equations represent the same line, the system of equations does not represent two different lines but a single line repeated. This means there are infinitely many solutions because any point on this line satisfies both equations.

Conclusion:

The system of equations has infinitely many solutions.

Would you like more details on this method or any other questions related to the solution?

Here are 5 related questions:

1. How can you tell if a system of linear equations has no solution?
2. What is the graphical interpretation of infinitely many solutions for a system of equations?
3. How do you solve a system of equations using substitution or elimination methods?
4. What are the conditions for a system of linear equations to have exactly one solution?
5. Can you provide an example of a system of equations with no solutions?

Tip: To determine the number of solutions in a system of linear equations, check if the equations represent parallel lines (no solutions), the same line (infinite solutions), or intersect at one point (exactly one solution).