To determine how many solutions the system of equations has, we can analyze the two equations:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
First, we can simplify equation 2. Notice that if we multiply the entire second equation by 3, we get:
\[ 3(4x - 5y) = 3(6) \] \[ 12x - 15y = 18 \]
Now we see that both equations are actually the same:
- \( 12x - 15y = 18 \) (original equation 1)
- \( 12x - 15y = 18 \) (transformed equation 2)
Since both equations represent the same line, the system has infinitely many solutions, as every point on the line is a solution to both equations.
Therefore, the correct response is:
infinitely many solutions.