To determine the number of solutions for the given system of equations, we can analyze the equations:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
First, we can see if the second equation can be derived from the first one. Notice that the second equation can be transformed to look similar to the first equation:
If we multiply the second equation by 3:
\[ 3(4x - 5y) = 3(6) \] This simplifies to:
\[ 12x - 15y = 18 \]
Now we see that both equations are actually the same:
\[ 12x - 15y = 18 \]
Since both equations are identical, they represent the same line. Therefore, there are infinitely many solutions to this system because every point on the line represented by either equation is a solution.
Conclusion: The answer is infinitely many solutions.