Verified Answer :

  • Sum the real parts and imaginary parts of the known complex terms: \((7 + 5) + (-3 + 9 + 13)i = 12 + 19i\).
  • Isolate the unknown variable \(x\) by subtracting this sum from the right side of the equation: \(x = (10 - 5i) - (12 + 19i)\).
  • Combine the real parts \(10 - 12 = -2\) and imaginary parts \(-5 - 19 = -24\).
  • Final Answer: \(\boxed{-2 - 24i}\).

Explanation

  1. Analyzing the Equation

We are given the equation \(x + (7 - 3i) + (5 + 9i) + 13i = 10 - 5i\), where \(x\) represents a complex number. Our objective is to determine the value of \(x\). Recall that the sum of complex numbers is performed by adding their real and imaginary parts separately: \((a + bi) + (c + di) = (a + c) + (b + d)i\).

  1. Simplifying the Left Side

To isolate \(x\), we first sum the known complex terms on the left side of the equation.
The real parts are: \(7 + 5 = 12\).
The imaginary parts are: \(-3 + 9 + 13 = 19\).
Thus, the equation simplifies to:
\[x + (12 + 19i) = 10 - 5i\]

  1. Solving for x

Now, we subtract the complex number \((12 + 19i)\) from both sides of the equation to find \(x\):
\[x = (10 - 5i) - (12 + 19i)\]
Grouping the real and imaginary parts:
\[x = (10 - 12) + (-5 - 19)i\]
\[x = -2 - 24i\]
The missing term is \(-2 - 24i\).

Examples

Complex numbers are vital in electrical engineering, particularly in AC circuit analysis. When calculating the impedance of a circuit, engineers use complex numbers to represent both resistance (real part) and reactance (imaginary part). Finding a "missing" impedance value in a series of connected components, much like the one solved here, allows for precise system tuning and safety design, ensuring components can handle the calculated current and voltage loads.

Related Questions