To determine the number of solutions for the given system of equations, we can use the method of substitution or elimination. Alternatively, we can convert the equations to slope-intercept form and check if the lines are parallel, identical, or intersecting.
Let's start with the given equations:
- \( 6x + 2y = -18 \)
- \( 3x + y = 24 \)
Step 1: Simplify the first equation.
We can divide the entire first equation by 2 to simplify it: \[ 3x + y = -9 \]
Now, we have a new system of equations:
- \( 3x + y = -9 \) (equation 1)
- \( 3x + y = 24 \) (equation 2)
Step 2: Compare the equations.
Now, we will analyze both equations:
- Equation 1: \( 3x + y = -9 \)
- Equation 2: \( 3x + y = 24 \)
Both equations have the same left-hand side, \( 3x + y \), but their right-hand sides are different (\(-9\) and \(24\)). This means that these two lines are parallel and will never intersect.
Conclusion
Since the two equations represent parallel lines, there are no solutions to the system of equations.
Thus, the answer is that the system of equations has 0 solutions.