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

Options

A. `9/85`
B. `-9/(85)`
C. `(53)/(85)`
D. None of these

Explanation

To solve the problem \( x + i y = \frac{3 + 5i}{7 - 6i} \), we need to find the value of \( y \).

### Step-by-Step Solution:

1. **Rationalize the Denominator**:
We start with the expression:
\[ x + i y = \frac{3 + 5i}{7 - 6i} \]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( 7 + 6i \):
\[ x + i y = \frac{(3 + 5i)(7 + 6i)}{(7 - 6i)(7 + 6i)} \]

2. **Calculate the Denominator**:
The denominator can be calculated using the formula \( a^2 - b^2 \):
\[ (7 - 6i)(7 + 6i) = 7^2 - (6i)^2 = 49 - (-36) = 49 + 36 = 85 \]

3. **Calculate the Numerator**:
Now, we calculate the numerator:
\[ (3 + 5i)(7 + 6i) = 3 \cdot 7 + 3 \cdot 6i + 5i \cdot 7 + 5i \cdot 6i \]
This expands to:
\[ 21 + 18i + 35i + 30i^2 \]
Since \( i^2 = -1 \), we have:
\[ 30i^2 = 30(-1) = -30 \]
Therefore, the numerator becomes:
\[ 21 - 30 + (18i + 35i) = -9 + 53i \]

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

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

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