To classify the system of equations, we can follow these steps:
1. Write the system of equations in a standard form:
[tex]\[
\begin{array}{c}
3x = -6 - y \\
4 + y = -3x - 1
\end{array}
\][/tex]
2. Rewrite both equations to isolate [tex]\(y\)[/tex]:
[tex]\[
\text{For the first equation \(3x = -6 - y\):}
\][/tex]
[tex]\[
y = -6 - 3x
\][/tex]
[tex]\[
\text{For the second equation \(4 + y = -3x - 1\):}
\][/tex]
[tex]\[
y = -3x - 5
\][/tex]
Now our equations are:
[tex]\[
\begin{array}{c}
y = -3x - 6 \\
y = -3x - 5
\end{array}
\][/tex]
3. Compare the slopes and intercepts of the lines. We observe that both equations have the same slope (-3), but different y-intercepts (-6 and -5).
4. Conclusion: Since the slopes are the same and the y-intercepts are different, the lines are parallel and do not intersect.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{parallel}} \][/tex]
