To determine the characteristics of the system of equations:
[tex]\[
\begin{cases}
6x + y = -30 \\
36x + 6y = 24
\end{cases}
\][/tex]
we will perform the following analysis:
### Step 1: Check for Independence of Equations
To check if the equations are independent or describe the same line, we can simplify both equations or analyze their coefficients.
Starting with the first equation:
[tex]\[
6x + y = -30
\][/tex]
Now consider the second equation:
[tex]\[
36x + 6y = 24
\][/tex]
Notice that the second equation can be divided by 6 to simplify it:
[tex]\[
\frac{36x + 6y}{6} = \frac{24}{6}
\][/tex]
[tex]\[
6x + y = 4
\][/tex]
So, the system simplifies to:
[tex]\[
\begin{cases}
6x + y = -30 \\
6x + y = 4
\end{cases}
\][/tex]
### Step 2: Compare the Simplified Equations
The simplified system of equations is:
[tex]\[
\begin{cases}
6x + y = -30 \\
6x + y = 4
\end{cases}
\][/tex]
We can observe that both simplified equations have the same left-hand side but different right-hand sides. This indicates that they are inconsistent because [tex]\(6x + y\)[/tex] cannot simultaneously be both [tex]\(-30\)[/tex] and [tex]\(4\)[/tex]. Therefore, there is no solution to this system since the two lines are parallel and never intersect.
### Conclusion
Given our analysis:
- The system does not have a unique solution.
- The lines represented by these equations are parallel and do not intersect.
Therefore, the correct statement about the system of equations is:
"If this system were graphed, the lines would be parallel."
