To determine how many solutions the system of equations has, we can first analyze the given equations:
- \( 12x - 15y = 18 \)
- \( 4x - 5y = 6 \)
We'll use the elimination method to find out if the equations are consistent and how many solutions they have.
Step 1: Simplify the equations if necessary
Let's simplify the second equation:
The equation \( 4x - 5y = 6 \) can be multiplied by 3 to align it with the coefficients in the first equation:
\[ 3(4x - 5y) = 3(6) \]
This gives us:
\[ 12x - 15y = 18 \]
Step 2: Compare the equations
Now we have:
- \( 12x - 15y = 18 \)
- \( 12x - 15y = 18 \)
Since both equations are identical, all points that satisfy one equation will satisfy the other. Hence, there are infinitely many solutions.
Conclusion
The system of equations has infinitely many solutions.