To solve the equation $E = IZ$ for the missing variable $Z$, given the values of $I$ and $E$, follow these steps:

Given:

• $I = 7 + 5i$ (current)
• $E = 4 + 8i$ (voltage)

Equation to solve:

$E = IZ$

Step 1: Isolate $Z$

$Z = \frac{E}{I}$

Step 2: Substitute the given values into the equation

$Z = \frac{4 + 8i}{7 + 5i}$

Step 3: Multiply the numerator and the denominator by the conjugate of the denominator

The conjugate of $7 + 5i$ is $7 - 5i$. So, multiply both the numerator and the denominator by $7 - 5i$:

$Z = \frac{(4 + 8i)(7 - 5i)}{(7 + 5i)(7 - 5i)}$

Step 4: Expand the numerator and the denominator

• Numerator: $(4 + 8i)(7 - 5i) = 4(7) - 4(5i) + 8i(7) - 8i(5i)$ Simplifying this: $28 - 20i + 56i - 40i^2 = 28 + 36i - 40(-1) = 28 + 36i + 40 = 68 + 36i$

• Denominator: $(7 + 5i)(7 - 5i) = 7^2 - (5i)^2 = 49 - 25(-1) = 49 + 25 = 74$

Step 5: Write the final expression for $Z$

$Z = \frac{68 + 36i}{74} = \frac{68}{74} + \frac{36i}{74} = \frac{34}{37} + \frac{18i}{37}$

So, the impedance $Z$ is: $\boxed{Z = \frac{34}{37} + \frac{18i}{37}}$

Would you like further details or have any additional questions?

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Here are 5 related questions to further explore the topic:

1. How can you interpret the real and imaginary parts of impedance in physical terms?
2. What would be the effect on $Z$ if $I$ and $E$ were purely real numbers?
3. How does the concept of phasors relate to complex numbers in electrical engineering?
4. Can you calculate the magnitude and phase angle of the impedance $Z$?
5. How would you solve for $I$ if $Z$ and $E$ were given?

Tip: Remember that in AC circuits, the real part of the impedance represents resistance, while the imaginary part represents reactance.