Perform the indicated operation and give your answer in the form `x+yi`, where x and y are real numbers and `i= sqrt(-1)`
`(5-3i)/(6+i)`

Answer

To solve the expression \((5 - 3i)/(6 + i)\) and express it in the form \(x + yi\), we will follow these steps:

### Step 1: Multiply the numerator and denominator by the conjugate of the denominator
The conjugate of \(6 + i\) is \(6 - i\). Thus, we multiply both the numerator and denominator by \(6 - i\):

\[ \frac{(5 - 3i)(6 - i)}{(6 + i)(6 - i)} \]

### Step 2: Simplify the denominator
Now, we simplify the denominator:

\[ (6 + i)(6 - i) = 6^2 - i^2 = 36 - (-1) = 36 + 1 = 37 \]

### Step 3: Expand the numerator
Next, we expand the numerator:

\[ (5 - 3i)(6 - i) = 5 \cdot 6 + 5 \cdot (-i) - 3i \cdot 6 - 3i \cdot (-i) \]
\[ = 30 - 5i - 18i + 3i^2 \]
Since \(i^2 = -1\), we substitute:

\[ = 30 - 5i - 18i + 3(-1) = 30 - 5i - 18i - 3 = 27 - 23i \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{27 - 23i}{37} \]

### Step 5: Split into real and imaginary parts
We can express this as:

\[ \frac{27}{37} - \frac{23}{37}i \]

Thus, in the form \(x + yi\):

\[ x = \frac{27}{37}, \quad y = -\frac{23}{37} \]

### Final Answer
The final answer is:

\[ \frac{27}{37} - \frac{23}{37}i \]

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