We are given the system of linear equations:

\[
1. \quad x + 4y = 6
\]
\[
2. \quad \frac{1}{2}x + 2y = 3
\]

### Step 1: Eliminate fractions
To simplify the second equation, we can multiply both sides by 2 to eliminate the fraction:

\[
2 \times \left( \frac{1}{2}x + 2y = 3 \right)
\]
This gives:
\[
x + 4y = 6
\]

### Step 2: Analyze the system
Now, the system of equations looks like this:

\[
x + 4y = 6
\]
\[
x + 4y = 6
\]

Both equations are identical, meaning they represent the same line. Therefore, this system has **infinitely many solutions**. Any pair \((x, y)\) that satisfies one equation will satisfy the other.

### Conclusion:
The system has infinitely many solutions because the two equations are dependent (they represent the same line).

Would you like to explore specific values of \(x\) and \(y\) or need further details?

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Here are some follow-up questions for deeper understanding:

1. How do you find the general solution of a system with infinitely many solutions?
2. What is the graphical interpretation of a system with infinitely many solutions?
3. How can we express the solution set of this system parametrically?
4. What happens when two equations in a system represent parallel but distinct lines?
5. How do you use substitution or elimination in solving systems of linear equations?

**Tip:** When two equations in a system are identical, the system is said to have infinitely many solutions. They represent the same line.

Explore how to solve the system of linear equations X + 4y = 6 and 1/2x + 2y = 3. Learn the concept of dependent systems, how to eliminate fractions, and why this system has infinitely many solutions. This problem is ideal for students in Grades 8-10 studying algebra and systems of equations.

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