To determine the number of solutions for the system of equations:
- 12x - 15y = 18
- 4x - 5y = 6
First, let's manipulate the first equation to see if it can be related to the second equation.
Step 1: Simplify the first equation
We can simplify the first equation by dividing all terms by 3:
\[ \frac{12x}{3} - \frac{15y}{3} = \frac{18}{3} \]
This simplifies to:
\[ 4x - 5y = 6 \]
Step 2: Compare with the second equation
Notice that the second equation is:
\[ 4x - 5y = 6 \]
Both equations are identical. This means they represent the same line in the xy-plane.
Conclusion
Since both equations are identical, the system has infinitely many solutions. The correct response is:
infinitely many solutions.