To determine how many solutions the system of equations has, we can analyze the two equations.
The given system of equations is:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
First, let's check if the second equation is a multiple of the first equation. We can multiply the second equation by 3 to see if it matches the first equation:
\[ 3(4x - 5y) = 3(6) \]
This simplifies to:
\[ 12x - 15y = 18 \]
Since after multiplying the second equation by 3, it becomes identical to the first equation, this indicates that both equations describe the same line.
To conclude about the solutions:
- Since the two equations represent the same line, there are infinitely many solutions, as every point on that line is a solution to both equations.
The correct response is:
infinitely many solutions.