To determine the number of solutions for the given system of equations, we can analyze the equations:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
Let's see if one equation is a multiple of the other.
We can simplify the second equation:
Multiply the second equation by 3:
\[ 3(4x - 5y) = 3(6) \implies 12x - 15y = 18 \]
Now, we can see that the first equation is indeed the same as the transformed second equation. Thus, we have:
\[ 12x - 15y = 18 \quad \text{(first equation)} \] \[ 12x - 15y = 18 \quad \text{(transformed second equation)} \]
Since both equations represent the same line, this means there are infinitely many solutions.
The correct response is:
infinitely many solutions