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Let's solve the problem by working through each part step by step.

### Given Complex Numbers:
- \( z_1 = 2 + 3i \)
- \( z_2 = 2 - 3i \)

### 3.1.1: \( z_1 + z_2 \)
To add two complex numbers, add the real parts and the imaginary parts separately.

\[
z_1 + z_2 = (2 + 3i) + (2 - 3i)
\]

\[
= (2 + 2) + (3i - 3i)
\]

\[
= 4 + 0i = 4
\]

So, \( z_1 + z_2 = 4 \).

### 3.1.2: \( z_1 - z_2 \)
To subtract two complex numbers, subtract the real parts and the imaginary parts separately.

\[
z_1 - z_2 = (2 + 3i) - (2 - 3i)
\]

\[
= (2 - 2) + (3i + 3i)
\]

\[
= 0 + 6i = 6i
\]

So, \( z_1 - z_2 = 6i \).

### 3.1.3: \( 2z_1 + 3z_2 \)
First, calculate \( 2z_1 \) and \( 3z_2 \) separately.

\[
2z_1 = 2 \times (2 + 3i) = 4 + 6i
\]
\[
3z_2 = 3 \times (2 - 3i) = 6 - 9i
\]

Now, add the two results:

\[
2z_1 + 3z_2 = (4 + 6i) + (6 - 9i)
\]

\[
= (4 + 6) + (6i - 9i)
\]

\[
= 10 - 3i
\]

So, \( 2z_1 + 3z_2 = 10 - 3i \).

---

### Summary of Results:
1. \( z_1 + z_2 = 4 \)
2. \( z_1 - z_2 = 6i \)
3. \( 2z_1 + 3z_2 = 10 - 3i \)

Do you need more details or have any questions?

---

### Related Questions:
1. What is the magnitude of the complex number \( z_1 \)?
2. How would you find the conjugate of \( z_2 \)?
3. Can you express \( z_1 \) in polar form?
4. What is the result of multiplying \( z_1 \times z_2 \)?
5. How do you divide two complex numbers, say \( z_1 \div z_2 \)?

### Tip:
When adding or subtracting complex numbers, always handle the real and imaginary components separately to avoid confusion.

Given two complex numbers, z_1 = 2 + 3i and z_2 = 2 - 3i, find the following: 1. z_1 + z_2, 2. z_1 - z_2, 3. 2z_1 + 3z_2.

Learn how to solve operations with complex numbers, including addition, subtraction, and scalar multiplication. The problem involves z_1 = 2 + 3i and z_2 = 2 - 3i. Solutions cover addition (z_1 + z_2 = 4), subtraction (z_1 - z_2 = 6i), and scalar multiplication (2z_1 + 3z_2 = 10 - 3i). Suitable for Grades 9-12.

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