To determine how many solutions the system of equations has, we can analyze the two equations given:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
A common method is to simplify the equations and compare their forms.
First, observe the first equation: \[ 12x - 15y = 18 \] We can simplify this by dividing all terms by 3: \[ 4x - 5y = 6 \]
Now we see that the first equation, after simplification, actually becomes identical to the second equation: \[ 4x - 5y = 6 \]
Since both equations are the same, they represent the same line in a two-dimensional space. Therefore, there are infinitely many solutions, as any point on this line is a solution to the system.
In conclusion, the system has infinitely many solutions.