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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?

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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).

Determine if the following system of equations has no solutions, infinitely many solutions, or exactly one solution.

2x + y = 4
4x + 2y = 8

Learn how to determine the number of solutions for the system of linear equations 2x + y = 4 and 4x + 2y = 8. This example covers key algebraic concepts like systems of equations and explains why this system has infinitely many solutions. Suitable for Grades 8-10 students.

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