Let's solve each equation one by one and match them to the appropriate solutions.
1. **Equation**: \( 2i + 3 = 2i - 5 \)
Subtract \( 2i \) from both sides:
\[ 3 = -5 \]
This is a false statement, indicating that there are no values of \( i \) that satisfy the equation.
- **Match**: 4). no solution
2. **Equation**: \( 3(2i - 7) = 6i - 21 \)
Distributing on the left side:
\[ 6i - 21 = 6i - 21 \]
The left side equals the right side for any value of \( i \). Therefore, there are infinite solutions.
- **Match**: 2). infinite solutions
3. **Equation**: \( 3i + 2i - 4 = 18i \)
Combine like terms:
\[ 5i - 4 = 18i \]
Subtract \( 5i \) from both sides:
\[ -4 = 13i \]
Now solve for \( i \):
\[ i = -\frac{4}{13} \]
- **Match**: 6). i = -4 over 13
Now we have matched the equations with their solutions:
1. **4** (no solution)
2. **2** (infinite solutions)
3. **6** (i = -4 over 13)