To determine the number of solutions for the given system of equations, we can analyze the two equations:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
First, we can try to manipulate the second equation to see if it can be made to look similar to the first one.
The second equation can be multiplied by 3 to make the coefficients of \( x \) and \( y \) match that of the first equation:
\[ 3(4x - 5y) = 3(6) \] \[ 12x - 15y = 18 \]
Now, we see that:
- The first equation is \( 12x - 15y = 18 \).
- The transformed second equation is \( 12x - 15y = 18 \).
Since both equations are identical, they represent the same line in the Cartesian plane. Therefore, every point on this line is a solution to the system.
Thus, the system of equations has infinitely many solutions.