Match the equation with its solution(s).(5 points)

Question

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3i+5=2i−7
3(2i−7)=15
3i+5=3i+7
3i+2i−7=18
3(2i+7)=6i+21

Answer 1

To match the equations with their solutions, we need to solve each equation and determine which one corresponds to which solution.

1. For the equation:
   \(3i + 5 = 2i - 7\)
   Rearranging gives:
   \(3i - 2i = -7 - 5\)
   \(i = -12\)
   Solution: \(i = -12\)

2. For the equation:
   \(3(2i - 7) = 15\)
   Expanding:
   \(6i - 21 = 15\)
   Rearranging gives:
   \(6i = 15 + 21\)
   \(6i = 36\)
   \(i = 6\)
   Solution: \(i = 6\)

3. For the equation:
   \(3i + 5 = 3i + 7\)
   Rearranging gives:
   \(5 = 7\)
   This is inconsistent (no solution)

4. For the equation:
   \(3i + 2i - 7 = 18\)
   Simplifying gives:
   \(5i - 7 = 18\)
   Rearranging gives:
   \(5i = 18 + 7\)
   \(5i = 25\)
   \(i = 5\)
   Solution: \(i = 5\)

5. For the equation:
   \(3(2i + 7) = 6i + 21\)
   Expanding:
   \(6i + 21 = 6i + 21\)
   This is always true (an identity, infinitely many solutions)

Now we can match:

1. \(3i + 5 = 2i - 7\) → \(i = -12\)
2. \(3(2i - 7) = 15\) → \(i = 6\)
3. \(3i + 5 = 3i + 7\) → No solution
4. \(3i + 2i - 7 = 18\) → \(i = 5\)
5. \(3(2i + 7) = 6i + 21\) → Infinitely many solutions

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