https://r2.erweima.ai/mathbot/image/17271963009586yh7ku1i.jpg

Let's break down and solve the problem step by step.

### Given:
1. \( Z_1 I_1 + Z_2 I_1 = E_1 \)
   
   Simplifying:
   \[
   I_1 (Z_1 + Z_2) = E_1
   \]

   Therefore:
   \[
   I_1 = \frac{E_1}{Z_1 + Z_2}
   \]

2. \( Z_1 = 4.8 + j4.5 \)
3. \( Z_2 = 9.5 - j5.8 \)
4. \( E_1 = 0.5 - j1.0 \)

### Step 1: Add \( Z_1 \) and \( Z_2 \)
\[
Z_1 + Z_2 = (4.8 + j4.5) + (9.5 - j5.8)
\]
\[
Z_1 + Z_2 = (4.8 + 9.5) + j(4.5 - 5.8)
\]
\[
Z_1 + Z_2 = 14.3 - j1.3
\]

### Step 2: Compute \( I_1 \)
\[
I_1 = \frac{E_1}{Z_1 + Z_2} = \frac{0.5 - j1.0}{14.3 - j1.3}
\]

To perform this division of complex numbers, we'll multiply both the numerator and denominator by the conjugate of the denominator:
\[
I_1 = \frac{(0.5 - j1.0)(14.3 + j1.3)}{(14.3 - j1.3)(14.3 + j1.3)}
\]

### Step 3: Simplify the denominator
\[
(14.3 - j1.3)(14.3 + j1.3) = 14.3^2 - (j1.3)^2 = 204.49 + 1.69 = 206.18
\]

### Step 4: Simplify the numerator
\[
(0.5 - j1.0)(14.3 + j1.3) = (0.5)(14.3) + (0.5)(j1.3) - (j1.0)(14.3) - (j1.0)(j1.3)
\]
\[
= 7.15 + j0.65 - j14.3 + 1.3
\]
\[
= (7.15 + 1.3) + j(0.65 - 14.3)
\]
\[
= 8.45 - j13.65
\]

### Step 5: Final result for \( I_1 \)
\[
I_1 = \frac{8.45 - j13.65}{206.18}
\]
\[
I_1 = 0.041 + j(-0.066)
\]

### Imaginary Part of \( I_1 \)
The imaginary part of \( I_1 \) is:
\[
\boxed{-0.066}
\]

---

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

Here are some related questions to explore further:
1. How do you perform division of complex numbers?
2. Why do we multiply by the conjugate when dividing complex numbers?
3. Can we solve for the magnitude of \( I_1 \) from the above result?
4. How would the result change if the values of \( Z_1 \), \( Z_2 \), or \( E_1 \) were different?
5. What is the significance of the real and imaginary parts of \( I_1 \) in electrical circuits?

**Tip:** Always use the conjugate when dividing complex numbers to simplify the denominator to a real number.

¿Cuál es la parte imaginaria de I1? Z1I1 + Z2I1 = E1, Z1=4.8+j4.5, Z2=9.5+j(-5.8), E1=0.5+j(-1.0)

Step-by-step solution to find the imaginary part of I1 in the complex equation Z1I1 + Z2I1 = E1, where Z1=4.8+j4.5, Z2=9.5-j5.8, and E1=0.5-j1.0. This involves division of complex numbers and use of conjugates.

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