Solving with Elimination Quick Check

To determine how many solutions the system of equations has, we can use the elimination method. The system of equations is:

1. \( 12x - 15y = 18 \)
2. \( 4x - 5y = 6 \)

First, let's manipulate the second equation to make it easier to compare with the first equation. We can multiply the second equation by 3:

\[
3(4x - 5y) = 3(6) \implies 12x - 15y = 18
\]

Now we have:

1. \( 12x - 15y = 18 \)
2. \( 12x - 15y = 18 \)

Since both equations are identical, they represent the same line. This means that there are infinitely many solutions, as every point on that line is a solution to the system.

Thus, the answer is:

**infinitely many solutions**.