https://file.aiquickdraw.com/mathbot/file/1725594034990mbh13k23.jpg

The problem in the image asks to determine if the system of equations has no solution, infinitely many solutions, or exactly one solution. The system of equations is:

\[
\begin{aligned}
1) \quad 2x - 6y = 8 \\
2) \quad 4x - 12y = -6
\end{aligned}
\]

Let's analyze these equations step by step to determine the nature of the solution.

### Step 1: Check if the equations are multiples of each other.
We can multiply the first equation by 2 to compare it with the second equation:

\[
2(2x - 6y) = 2 \times 8 \implies 4x - 12y = 16
\]

Now, compare this with the second equation:

\[
4x - 12y = -6
\]

### Step 2: Compare both equations.
We now have:
- \( 4x - 12y = 16 \)
- \( 4x - 12y = -6 \)

These two equations have the same left-hand side but different right-hand sides (16 vs -6), which means they are contradictory. Therefore, the system of equations is **inconsistent** and **has no solution**.

### Conclusion:
The system of equations has **no solution**.

Would you like a deeper explanation or details about any of the steps? Here are some related questions to deepen your understanding:

1. What does it mean for two lines to be parallel and how does that relate to having no solutions?
2. How can we tell if a system of equations has infinitely many solutions?
3. Could the system have been manipulated differently to simplify the solution process?
4. What are the methods to solve systems of equations (e.g., substitution, elimination, graphing)?
5. How does this problem change if the right-hand sides were the same?

**Tip:** Always check if equations are multiples of each other when determining whether a system has no solutions, infinitely many, or exactly one.

Determine if the following system of equations has no solution, infinitely many solutions, or exactly one solution:
1) 2x - 6y = 8
2) 4x - 12y = -6

Analyze the system of linear equations 2x - 6y = 8 and 4x - 12y = -6 to determine whether the system has no solution, infinitely many solutions, or exactly one solution. Suitable for students in Grades 8-10.

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