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To determine which system of equations is inconsistent, we need to analyze each set of linear equations. A system is inconsistent if it has no solution, meaning the lines representing the equations are parallel and never intersect.

Let's go through each option one by one.

### Option 1:
\[
\begin{aligned}
2x + 8y &= 6 \\
5x + 20y &= 2
\end{aligned}
\]
- Divide the second equation by 5: 
  \[
  x + 4y = \frac{2}{5}
  \]
- This makes the system:
  \[
  \begin{aligned}
  2x + 8y &= 6 \\
  x + 4y &= \frac{2}{5}
  \end{aligned}
  \]
  The ratios of the coefficients of \(x\) and \(y\) are proportional, but the constants (right-hand sides) are not. This means the lines are parallel but do not intersect, making this system **inconsistent**.

### Option 2:
\[
\begin{aligned}
5x + 4y &= -14 \\
3x + 6y &= 6
\end{aligned}
\]
This system needs further analysis:
- Multiply the first equation by 3 and the second by 5:
  \[
  \begin{aligned}
  15x + 12y &= -42 \\
  15x + 30y &= 30
  \end{aligned}
  \]
  The equations are not proportional, meaning the system might have a solution and is **not inconsistent**.

### Option 3:
\[
\begin{aligned}
x + 2y &= 3 \\
4x + 6y &= 5
\end{aligned}
\]
- Multiply the first equation by 4:
  \[
  4x + 8y = 12
  \]
  Compare this with the second equation:
  \[
  4x + 6y = 5
  \]
  These lines are not parallel, so the system is **not inconsistent**.

### Option 4:
\[
\begin{aligned}
3x - 2y &= 2 \\
6x - 4y &= 4
\end{aligned}
\]
- The second equation is simply a multiple of the first equation, so these two lines are coincident, meaning they have infinitely many solutions. This system is **not inconsistent**.

### Conclusion:
The system in **Option 1** is inconsistent.

Would you like more details or have any questions? Here are five relative questions to expand on this:

1. How can you determine if two lines are parallel by comparing their equations?
2. What happens to a system of equations if the two lines are coincident?
3. Can a system have more than one solution, and how is this represented graphically?
4. How do you solve a system of equations using substitution or elimination?
5. What are the implications of having an inconsistent system in real-world problems?

**Tip:** To check if a system of equations is inconsistent, always compare the ratios of the coefficients and constants. If the coefficients are proportional but the constants are not, the system is inconsistent.

Which system of equations is inconsistent?

Learn how to determine which system of linear equations is inconsistent by analyzing the relationship between the slopes and intercepts of the equations. This algebra problem covers key concepts in identifying parallel lines and inconsistent systems, suitable for students in Grades 7-9.

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