To determine the number of solutions for the system of equations, we can solve the equations simultaneously.
The given equations are:
- \( 6x + 2y = -18 \)
- \( 3x + y = 24 \)
Let's first rewrite the first equation in a simpler form. We can divide the entire equation by 2:
\[ 3x + y = -9 \]
Now we can compare the two equations:
- \( 3x + y = -9 \)
- \( 3x + y = 24 \)
Since both equations are in the form \( 3x + y = \text{constant} \), we can see that the left sides are identical, but the right sides are different:
- From the first equation, \( y = -9 - 3x \).
- From the second equation, \( y = 24 - 3x \).
Since both equations have the same left-hand side but yield different right-hand sides, this implies that the two lines represented by these equations are parallel.
Therefore, the system of equations has no solutions (the lines never intersect).
In conclusion, the system of equations has 0 solutions.