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We are given complex numbers \( z_1 = 3 - i \), \( z_2 = 2 - 3i \), and \( z_3 = -6 + i \), with a relation for \( Z \) expressed as:

\[
Z = z_2 + \frac{z_1 + z_3}{z_1 \cdot z_3}
\]

We need to find \( E = I \cdot Z \) when \( I = 2 - 4i \).

### Step 1: Find \( z_1 + z_3 \)
\[
z_1 + z_3 = (3 - i) + (-6 + i) = -3
\]

### Step 2: Find \( z_1 \cdot z_3 \)
\[
z_1 \cdot z_3 = (3 - i)(-6 + i) = 3(-6 + i) - i(-6 + i)
\]
\[
= -18 + 3i + 6i + 1 = -17 + 9i
\]

### Step 3: Calculate \( \frac{z_1 + z_3}{z_1 \cdot z_3} \)
\[
\frac{-3}{-17 + 9i}
\]

To simplify, multiply the numerator and denominator by the conjugate of the denominator:
\[
\frac{-3}{-17 + 9i} \cdot \frac{-17 - 9i}{-17 - 9i} = \frac{-3(-17 - 9i)}{(-17)^2 + (9)^2}
\]
\[
= \frac{51 + 27i}{289 + 81} = \frac{51 + 27i}{370}
\]
\[
= \frac{51}{370} + \frac{27i}{370}
\]

### Step 4: Calculate \( Z \)
\[
Z = z_2 + \frac{z_1 + z_3}{z_1 \cdot z_3}
\]
\[
Z = (2 - 3i) + \left( \frac{51}{370} + \frac{27i}{370} \right)
\]
\[
= \left( 2 + \frac{51}{370} \right) + \left( -3i + \frac{27i}{370} \right)
\]
\[
= \frac{740 + 51}{370} + i\left( \frac{-1110 + 27}{370} \right)
\]
\[
= \frac{791}{370} + i\left( \frac{-1083}{370} \right)
\]
\[
Z = 2.138 + i(-2.927)
\]

### Step 5: Find \( E = I \cdot Z \)
We know \( I = 2 - 4i \), so:
\[
E = (2 - 4i)(2.138 - 2.927i)
\]
Using distribution:
\[
E = 2(2.138 - 2.927i) - 4i(2.138 - 2.927i)
\]
\[
E = 4.276 - 5.854i - 4i(2.138 - 2.927i)
\]
\[
= 4.276 - 5.854i - 8.552i + 11.708
\]
\[
E = (4.276 + 11.708) + (-5.854 - 8.552)i
\]
\[
E = 15.984 - 14.406i
\]

Thus, \( E = 15.984 - 14.406i \).

---

Would you like further details on any step or additional explanation?

Here are 5 related questions for practice:
1. Calculate \( z_1 \cdot z_2 \) and simplify.
2. Find \( |Z| \) (the modulus of \( Z \)).
3. What is the argument (angle) of \( I \) in polar form?
4. Express \( Z \) in polar form.
5. Calculate \( E \) if \( I = 1 + i \).

**Tip:** When multiplying complex numbers, remember to use the distributive property carefully and combine like terms (real and imaginary parts).

Misalkan z1 = 3 - i, z2 = 2 - 3i, z3 = -6 + i, dan Z = z2 + (z1 + z3)/(z1 * z3), tentukan E = I * Z bila diketahui I = 2 - 4i.

Learn how to solve a complex number equation involving z1 = 3 - i, z2 = 2 - 3i, and z3 = -6 + i, and calculate E = I * Z, where I = 2 - 4i. This step-by-step solution covers algebraic operations and the use of conjugates to simplify complex division.

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