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Let's analyze the systems of equations to determine whether they are independent, inconsistent, or dependent.

### First System:
\[
\begin{cases}
2x + y = 3 \\
4x + 2y = 6
\end{cases}
\]

1. Multiply the first equation by 2 to see if it's proportional to the second equation:
\[
2(2x + y) = 2 \cdot 3 \quad \Rightarrow \quad 4x + 2y = 6
\]

This shows that both equations are equivalent, meaning the system represents the same line. Therefore, the system is **dependent** (infinite solutions).

### Second System:
\[
\begin{cases}
2x + 3y = 1 \\
4x + 6y = 2
\end{cases}
\]

2. Again, multiply the first equation by 2:
\[
2(2x + 3y) = 2 \cdot 1 \quad \Rightarrow \quad 4x + 6y = 2
\]

This shows that the second equation is also a multiple of the first. Thus, this system is **dependent** as well (infinite solutions).

### Answers:
1. **Dependent**
2. **Dependent**

Would you like further details or have any other questions?

### 5 Related Questions:
1. How can you graphically interpret dependent systems of equations?
2. What is the difference between inconsistent and independent systems of equations?
3. How do you solve a system of linear equations using substitution or elimination?
4. What conditions make a system of linear equations inconsistent?
5. How can you identify if a system has a unique solution?

### Tip:
When the equations in a system are proportional (one is a multiple of the other), the system is dependent, meaning the two equations represent the same line.

Determine whether the following system of linear equations is independent, inconsistent, or dependent:
System 1:
2x + y = 3
4x + 2y = 6

System 2:
2x + 3y = 1
4x + 6y = 2

Learn how to determine whether a system of linear equations is dependent, inconsistent, or independent. This problem analyzes two systems of equations to show that both are dependent, meaning they represent the same line. Suitable for students in Grades 9-12.

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