Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

To determine the number of solutions for the given system of equations:

1. Write the equations down:

   \[ 6x + 5y = -2 \quad (1) \] \[ -30x - 25y = 10 \quad (2) \]

2. Simplify the second equation:

   The second equation can be simplified by multiplying it by -1 to make it easier to compare with the first equation:

   \[ 30x + 25y = -10 \quad (3) \]

3. Align the equations:

   Now we have: \[ 6x + 5y = -2 \quad (1) \] \[ 30x + 25y = -10 \quad (3) \]

4. Check if one equation is a multiple of the other:

   We can see that multiplying equation (1) by 5 gives: \[ 5(6x + 5y) = 5(-2) \implies 30x + 25y = -10 \] This is exactly what we have in equation (3). This means that equation (3) is just a multiple of equation (1).

5. Conclusion:

   Since both equations represent the same line (they are equivalent), the system has infinitely many solutions. All points \((x, y)\) that satisfy \(6x + 5y = -2\) will also satisfy the second equation.

Therefore, the system of equations has infinitely many solutions.