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### Given:
1. \( z_1 = x + 2i \)
2. \( z_2 = 4 + \sqrt{3} y \cdot i \)
3. You are told that \( z_1 = z_2 \), and you are provided the value \( x = 4 \) and \( y = \frac{2}{\sqrt{3}} \).

### Step-by-Step Solution:

Since \( z_1 = z_2 \), we can equate the real and imaginary parts.

From the first equation \( z_1 = x + 2i \):
- Real part: \( x \)
- Imaginary part: \( 2i \)

From the second equation \( z_2 = 4 + \sqrt{3} y \cdot i \):
- Real part: \( 4 \)
- Imaginary part: \( \sqrt{3} y \cdot i \)

### 1. Equate the Real Parts:
Since \( z_1 = z_2 \), the real part of \( z_1 \) must equal the real part of \( z_2 \). Therefore:
\[ x = 4 \]

This confirms the given value of \( x = 4 \).

### 2. Equate the Imaginary Parts:
The imaginary part of \( z_1 \) is \( 2i \), and the imaginary part of \( z_2 \) is \( \sqrt{3} y \cdot i \). Therefore:
\[
2 = \sqrt{3} y
\]

Solve for \( y \):
\[
y = \frac{2}{\sqrt{3}}
\]

This matches the given value of \( y = \frac{2}{\sqrt{3}} \).

### Conclusion:
Both values of \( x = 4 \) and \( y = \frac{2}{\sqrt{3}} \) are correct.

Would you like further details on any specific part?

### Related Questions:
1. What are complex numbers and how are they used in solving equations?
2. How can we simplify expressions involving square roots, like \( \sqrt{3} \)?
3. What are real and imaginary parts of complex numbers?
4. How do we solve equations involving both real and imaginary components?
5. What does it mean geometrically when two complex numbers are equal?

### Tip:
When equating complex numbers, always separate real and imaginary parts for easier comparison and solution.

Solve the equation z1 = z2 where z1 = x + 2i and z2 = 4 + √3 * y * i. Given x = 4 and y = 2 / √3, find the values of x and y.

Learn how to solve equations involving complex numbers by equating their real and imaginary parts. In this problem, z1 = z2, with z1 = x + 2i and z2 = 4 + √3 * y * i, and given x = 4 and y = 2 / √3. Follow a step-by-step solution to find the values of x and y.

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