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Match the equation with its solution.

3i + 2i - 4 = 18i ______________
2i + 3 = 2i - 5 _______________
3(2i - 7) = 6i - 21 _____________
Options: Infinite Solutions, i = 8, i = 4/13, i = 5, No Solution

3 answers

Let's solve each equation one by one and match them with the options given.

  1. Equation: \(3i + 2i - 4 = 18i\)

    Combine the terms on the left side:

    \[(3i + 2i) - 4 = 18i\]

    \[5i - 4 = 18i\]

    Now, isolate \(i\):

    \[-4 = 18i - 5i\]

    \[-4 = 13i\]

    Therefore,

    \[i = -\frac{4}{13}\]

    This does not match any of the options directly, so we will move on.

  2. Equation: \(2i + 3 = 2i - 5\)

    Subtract \(2i\) from both sides:

    \[3 = -5\]

    This is not true, which indicates:

    No Solution.

  3. Equation: \(3(2i - 7) = 6i - 21\)

    Distributing the left side:

    \[6i - 21 = 6i - 21\]

    This simplifies to an identity, meaning both sides are equal for all values of \(i\):

    Infinite Solutions.

Now let's compile the matches:

  • \(3i + 2i - 4 = 18i\) \( \rightarrow \) (none of the options align with -4/13)
  • \(2i + 3 = 2i - 5\) \( \rightarrow \) No Solution
  • \(3(2i - 7) = 6i - 21\) \( \rightarrow \) Infinite Solutions

Thus, the best matches we have found are:

  • \(3i + 2i - 4 = 18i\) - No direct match
  • \(2i + 3 = 2i - 5\) - No Solution
  • \(3(2i - 7) = 6i - 21\) - Infinite Solutions

To summarize in terms of the matching process:

  • 3i + 2i - 4 = 18i -> No matched solution
  • 2i + 3 = 2i - 5 -> No Solution
  • 3(2i - 7) = 6i - 21 -> Infinite Solutions
are you sure this is correct

Let's reevaluate each equation carefully to ensure accurate solutions.

  1. Equation: \(3i + 2i - 4 = 18i\)

    Combine the terms on the left side:

    \[ 5i - 4 = 18i \]

    Now, isolate \(i\):

    \[ -4 = 18i - 5i \]

    \[ -4 = 13i \]

    Therefore,

    \[ i = -\frac{4}{13} \]

    Note: This solution matched mathematically, but the value does not correspond with the provided options.

  2. Equation: \(2i + 3 = 2i - 5\)

    Subtract \(2i\) from both sides:

    \[ 3 = -5 \]

    This statement is false, indicating that:

    No Solution.

  3. Equation: \(3(2i - 7) = 6i - 21\)

    Expanding the left side:

    \[ 6i - 21 = 6i - 21 \]

    This simplifies to an identity, implying that both sides are equal for all values of \(i\):

    Infinite Solutions.

Conclusion:

With careful reevaluation, the matching is as follows:

  • \(3i + 2i - 4 = 18i\) \( \rightarrow \) (i = -4/13, not an option)
  • \(2i + 3 = 2i - 5\) \( \rightarrow \) No Solution
  • \(3(2i - 7) = 6i - 21\) \( \rightarrow \) Infinite Solutions

Therefore, the correct matches are:

  • No Solution for \(2i + 3 = 2i - 5\)
  • Infinite Solutions for \(3(2i - 7) = 6i - 21\)

The original equation \(3i + 2i - 4 = 18i\) does yield \(i = -\frac{4}{13}\), but it does not match any provided options.

Thank you for prompting me to double-check! Please let me know if you need any further clarification.