Verified Answer :

  • Given \(x = 3 + 8i\) and \(y = 7 - i\), we identify the product \(xy = 29 + 53i\).
  • Using algebraic manipulation, we establish the following matches:
  • \(-8 - 41i = y - 5x\)
  • \(58 + 106i = 2xy\)
  • \(-15 + 19i = 2x - 3y\)
  • \(-29 - 53i = -xy\)
  • Each target complex number is now correctly associated with its corresponding expression in terms of \(x\) and \(y\).

Explanation

  1. Analysis of the Problem

We are given the complex numbers \(x = 3 + 8i\) and \(y = 7 - i\). We need to match four specific target complex numbers to expressions involving \(x\) and \(y\). These target values are \(-8 - 41i\), \(58 + 106i\), \(-15 + 19i\), and \(-29 - 53i\). Our strategy involves calculating basic arithmetic operations like addition, subtraction, multiplication, and scalar multiples, then comparing these results against the target values to identify the correct matches.

  1. Matching Expressions to Target Values

First, let us calculate the product \(xy\) to assist in identifying other expressions:
\[xy = (3 + 8i)(7 - i) = 21 - 3i + 56i - 8i^2 = 21 + 53i + 8 = 29 + 53i.\]
With this product in hand, we can easily match the targets involving \(xy\):

  1. Target \(-29 - 53i = -(29 + 53i) = -xy\).
  2. Target \(58 + 106i = 2(29 + 53i) = 2xy\).

Now, we look for combinations involving subtractions:

  1. Target \(-8 - 41i\): Testing \(y - 5x\), we get \((7 - i) - 5(3 + 8i) = 7 - i - 15 - 40i = -8 - 41i\). This is a match.
  2. Target \(-15 + 19i\): Testing \(2x - 3y\), we get \(2(3 + 8i) - 3(7 - i) = 6 + 16i - 21 + 3i = -15 + 19i\). This is also a match.
  1. Finalizing the Results

Based on the calculations above, the expressions corresponding to each target value are:

  • \(-8 - 41i = y - 5x\)
  • \(58 + 106i = 2xy\)
  • \(-15 + 19i = 2x - 3y\)
  • \(-29 - 53i = -xy\)

These pairings cover all four target values successfully.

Examples

Understanding the arithmetic of complex numbers is essential for fields like electrical engineering, where impedance in AC circuits is represented by complex numbers. For example, if you have two components in a circuit with different impedance values \(x\) and \(y\), you might need to determine the total impedance of combinations like series connections or parallel arrangements, which involve operations exactly like these: adding, subtracting, and multiplying complex values. Mastery of these operations ensures that you can model and solve for electrical behavior accurately.

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