To determine how many solutions the system of equations has, we can solve them using substitution or elimination.
First, let's rewrite the system of equations:
- \( 6x + 2y = -18 \)
- \( 3x + y = 24 \)
We can start by solving the second equation for \( y \):
From equation 2: \[ y = 24 - 3x \]
Now we can substitute this expression for \( y \) into the first equation:
\[ 6x + 2(24 - 3x) = -18 \]
Now, distribute the 2: \[ 6x + 48 - 6x = -18 \]
Notice that \( 6x \) and \( -6x \) cancel each other out, leaving: \[ 48 = -18 \]
This statement is false, which means there are no values of \( x \) and \( y \) that satisfy both equations simultaneously.
Therefore, the system of equations has no solutions.
In summary, the system is inconsistent.