Let's solve the given system of linear equations:

\[
\begin{aligned}
-3x - y &= 3 \quad \text{(Equation 1)} \\
6x + 2y &= 1 \quad \text{(Equation 2)}
\end{aligned}
\]

### Step 1: Simplify Equation 2
Let's simplify the second equation by dividing everything by 2:

\[
3x + y = \frac{1}{2} \quad \text{(Equation 3)}
\]

Now we have the system:

\[
\begin{aligned}
-3x - y &= 3 \quad \text{(Equation 1)} \\
3x + y &= \frac{1}{2} \quad \text{(Equation 3)}
\end{aligned}
\]

### Step 2: Add Equations 1 and 3
Let's add Equation 1 and Equation 3 to eliminate \( y \):

\[
(-3x - y) + (3x + y) = 3 + \frac{1}{2}
\]

This simplifies to:

\[
0 = \frac{7}{2}
\]

### Conclusion
Again, we get a contradiction: \( 0 = \frac{7}{2} \), which is impossible. This means the system of equations is **inconsistent** and has no solution.

This indicates that the two lines represented by these equations are parallel and do not intersect.

Would you like more details or have any questions?

Here are some related questions:
1. What are the implications of an inconsistent system in real-world scenarios?
2. How can you use the graphical method to confirm the solution of a system of equations?
3. What is the role of the determinant in solving systems of linear equations?
4. How do you find the point of intersection for consistent systems of linear equations?
5. Can a system of linear equations have exactly two solutions?
6. What are the properties of parallel lines in a coordinate plane?
7. How does changing the coefficients of the equations affect the solution?
8. What is the difference between consistent and inconsistent systems?

**Tip:** Always verify your solutions by substituting the values back into the original equations to check for consistency.

Learn how to solve the system of linear equations -3x - y = 3 and 6x + 2y = 1 step-by-step. Understand why this system is inconsistent and how to interpret parallel lines in a coordinate plane. Suitable for students in Grades 9-12.

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