If ` z_(1) = 4 -I and z_(2) = - 3 + 7i` then find
(i) ` z_(1) +z_(2)`
(ii) ` z_(1) -z_(2)`

Answer

To solve the problem, we need to perform operations on the complex numbers \( z_1 \) and \( z_2 \).

Given:
- \( z_1 = 4 - i \)
- \( z_2 = -3 + 7i \)

We will find:
(i) \( z_1 + z_2 \)
(ii) \( z_1 - z_2 \)

### Step-by-step Solution:

**(i) Finding \( z_1 + z_2 \)**

1. Write down the values of \( z_1 \) and \( z_2 \):
\[ z_1 = 4 - i \]
\[ z_2 = -3 + 7i \]

2. Add the real parts and the imaginary parts separately:
\[ z_1 + z_2 = (4 + (-3)) + (-1 + 7)i \]

3. Calculate the real part:
\[ 4 - 3 = 1 \]

4. Calculate the imaginary part:
\[ -1 + 7 = 6 \]

5. Combine the results:
\[ z_1 + z_2 = 1 + 6i \]

**Final result for (i)**:
\[ z_1 + z_2 = 1 + 6i \]

---

**(ii) Finding \( z_1 - z_2 \)**

1. Write down the values of \( z_1 \) and \( z_2 \):
\[ z_1 = 4 - i \]
\[ z_2 = -3 + 7i \]

2. Subtract \( z_2 \) from \( z_1 \):
\[ z_1 - z_2 = (4 - i) - (-3 + 7i) \]

3. Distribute the negative sign:
\[ z_1 - z_2 = 4 - i + 3 - 7i \]

4. Combine the real parts:
\[ 4 + 3 = 7 \]

5. Combine the imaginary parts:
\[ -1 - 7 = -8 \]

6. Combine the results:
\[ z_1 - z_2 = 7 - 8i \]

**Final result for (ii)**:
\[ z_1 - z_2 = 7 - 8i \]

---

### Summary of Results:
- (i) \( z_1 + z_2 = 1 + 6i \)
- (ii) \( z_1 - z_2 = 7 - 8i \)