If x +iy `=(3+5i)/(7-6i)` then y is equal to

Options

A. `(9)/(85)`
B. `-(9)/(85)`
C. `(53)/(85)`
D. `-(53)/(85)`

Answer

To find the value of \( y \) from the equation \( x + iy = \frac{3 + 5i}{7 - 6i} \), we will convert the right-hand side into Cartesian form \( a + bi \).

### Step-by-Step Solution:

1. **Write the equation**:
\[ x + iy = \frac{3 + 5i}{7 - 6i} \]

2. **Multiply the numerator and denominator by the conjugate of the denominator**:
The conjugate of \( 7 - 6i \) is \( 7 + 6i \). Thus, we multiply both the numerator and denominator by \( 7 + 6i \):
\[ x + iy = \frac{(3 + 5i)(7 + 6i)}{(7 - 6i)(7 + 6i)} \]

3. **Calculate the denominator**:
\[ (7 - 6i)(7 + 6i) = 7^2 + 6^2 = 49 + 36 = 85 \]

4. **Calculate the numerator**:
\[ (3 + 5i)(7 + 6i) = 3 \cdot 7 + 3 \cdot 6i + 5i \cdot 7 + 5i \cdot 6i \]
\[ = 21 + 18i + 35i + 30(-1) = 21 + 53i - 30 = -9 + 53i \]

5. **Combine the results**:
Now we can write:
\[ x + iy = \frac{-9 + 53i}{85} \]
This can be separated into real and imaginary parts:
\[ x + iy = -\frac{9}{85} + \frac{53}{85}i \]

6. **Identify \( y \)**:
From the equation \( x + iy = -\frac{9}{85} + \frac{53}{85}i \), we can see that:
\[ y = \frac{53}{85} \]

### Final Answer:
Thus, the value of \( y \) is:
\[ y = \frac{53}{85} \]