Let's solve each problem step by step.
Problem 1:
\[3(2i + 7) = 6i + 21\]
Distributing on the left side: \[6i + 21 = 6i + 21\]
Both sides are equal, so the equation holds for all values of \( i \). The answer is true for all \( i \).
Problem 2:
\[3i + 5 = 2i - 7\]
Subtracting \(2i\) from both sides: \[ 3i - 2i + 5 = -7 \] This simplifies to: \[i + 5 = -7\]
Subtracting 5 from both sides: \[i = -12\]
So, the solution is \(i = -12\).
Problem 3:
\[3i + 2i - 7 = 18\]
Combining like terms: \[5i - 7 = 18\]
Adding 7 to both sides: \[5i = 25\]
Dividing both sides by 5: \[i = 5\]
So, the solution is \(i = 5\).
Problem 4:
\[3(2i - 7) = 15\]
Distributing on the left side: \[6i - 21 = 15\]
Adding 21 to both sides: \[6i = 36\]
Dividing both sides by 6: \[i = 6\]
So, the solution is \(i = 6\).
Problem 5:
\[3i + 5 = 3i + 7\]
Subtracting \( 3i \) from both sides: \[5 = 7\]
This is not true, so the equation has no solution.
To summarize:
- Problem 1: True for all \( i \)
- Problem 2: \( i = -12 \)
- Problem 3: \( i = 5 \)
- Problem 4: \( i = 6 \)
- Problem 5: No solution.