We are given the equation:

\[
3x + yi = 5x + 1 + 2i
\]

### Step 1: Separate the real and imaginary parts.

- The real part is \( 3x \) on the left-hand side and \( 5x + 1 \) on the right-hand side.
- The imaginary part is \( yi \) on the left-hand side and \( 2i \) on the right-hand side.

### Step 2: Equating the real and imaginary parts.

#### Real part:
\[
3x = 5x + 1
\]
Solving for \( x \):
\[
3x - 5x = 1 \implies -2x = 1 \implies x = -\frac{1}{2}
\]

#### Imaginary part:
\[
y = 2
\]

### Final Solution:
\[
x = -\frac{1}{2}, \quad y = 2
\]

Would you like more details on any of these steps?

Here are some related questions:
1. How do you solve complex number equations in general?
2. What are the rules for equating real and imaginary parts?
3. How can complex numbers be used in physical applications?
4. What is the geometric representation of a complex number?
5. Can you explain the polar form of complex numbers?

**Tip:** When dealing with complex equations, always split them into real and imaginary components for easier solving.

Learn how to solve the complex number equation 3x + yi = 5x + 1 + 2i. This problem involves separating real and imaginary parts, equating them, and solving for x and y. Suitable for students in Grades 9-12, covering core algebraic concepts.

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