To analyze both systems and determine the number of solutions for each, we need to check the relationships between the equations in each system.
System 1:
- \( 4x + 3y = -1 \)
- \( 12x + 9y = -21 \)
First, notice that the second equation can be simplified. We can divide the entire equation by 3:
\[ \frac{12x + 9y}{3} = \frac{-21}{3} \quad \Rightarrow \quad 4x + 3y = -7 \]
Now we have the system:
- \( 4x + 3y = -1 \)
- \( 4x + 3y = -7 \)
These two equations clearly describe parallel lines (same left-hand side, different right-hand sides), which means they will never intersect. Therefore, System 1 has no solution.
System 2:
- \( y = 2x + 3 \)
- \( 4x - 2y = -6 \)
Let's substitute \( y \) from the first equation into the second equation. Replacing \( y \) with \( 2x + 3 \):
\[ 4x - 2(2x + 3) = -6 \]
Distributing the -2 yields:
\[ 4x - 4x - 6 = -6 \]
This simplifies to:
\[ -6 = -6 \]
This is always true, indicating that the two equations describe the same line. Therefore, System 2 has an infinite number of solutions.
Conclusion:
- System I has no solution.
- System II has an infinite number of solutions.
Thus, the correct statement is: System I has no solution and System II has an infinite number of solutions.