To determine the number of solutions for the given system of equations, we can analyze the two equations you provided:
- \( 3x + 2y = 8 \)
- \( -6x - 4y = 8 \)
First, let's rewrite the second equation in a more manageable form. Notice that you can multiply the first equation by \(-2\) to compare it with the second equation:
Multiplying \( 3x + 2y = 8 \) by \(-2\):
\[ -6x - 4y = -16 \]
Now we have:
- \( -6x - 4y = -16 \) (from the first equation modified)
- \( -6x - 4y = 8 \) (the second equation)
Now, we can see that the left sides of both equations are the same:
\[ -6x - 4y = -16 \] \[ -6x - 4y = 8 \]
Since the left sides are identical, but the right sides are different (−16 ≠ 8), this means that the two equations represent parallel lines.
Parallel lines do not intersect, which means there are no solutions to this system of equations.
Thus, the final answer is:
The system has no solutions.