Find the value of x and y given that `(x + yi) (2-3i)=4+i`

Answer

To solve the equation \((x + yi)(2 - 3i) = 4 + i\), we will follow these steps:

### Step 1: Expand the left-hand side
We start by expanding the left-hand side of the equation:

\[ (x + yi)(2 - 3i) = x \cdot 2 + x \cdot (-3i) + yi \cdot 2 + yi \cdot (-3i) \]

This simplifies to:

\[ 2x - 3xi + 2yi - 3y(i^2) \]

Since \(i^2 = -1\), we can replace \(i^2\) with \(-1\):

\[ 2x - 3xi + 2yi + 3y \]

Now, we can combine the real and imaginary parts:

\[ (2x + 3y) + (-3x + 2y)i \]

### Step 2: Set the real and imaginary parts equal
Now we set the real part and the imaginary part equal to the corresponding parts of the right-hand side \(4 + i\):

1. Real part:
\[ 2x + 3y = 4 \quad \text{(1)} \]

2. Imaginary part:
\[ -3x + 2y = 1 \quad \text{(2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:

1. \(2x + 3y = 4\)
2. \(-3x + 2y = 1\)

We can solve these equations simultaneously. Let's multiply the first equation by 3 to eliminate \(x\):

\[ 6x + 9y = 12 \quad \text{(3)} \]

Now, we can add equation (2) multiplied by 2 to equation (3):

\[ -6x + 4y = 2 \quad \text{(4)} \]

Adding (3) and (4):

\[ (6x - 6x) + (9y + 4y) = 12 + 2 \]
\[ 13y = 14 \]

### Step 4: Solve for \(y\)
Now, we can solve for \(y\):

\[ y = \frac{14}{13} \]

### Step 5: Substitute \(y\) back to find \(x\)
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). Let's use equation (1):

\[ 2x + 3\left(\frac{14}{13}\right) = 4 \]

This simplifies to:

\[ 2x + \frac{42}{13} = 4 \]

To eliminate the fraction, we can multiply the entire equation by 13:

\[ 26x + 42 = 52 \]

Now, isolate \(x\):

\[ 26x = 52 - 42 \]
\[ 26x = 10 \]
\[ x = \frac{10}{26} = \frac{5}{13} \]

### Final Answer
Thus, the values of \(x\) and \(y\) are:

\[ x = \frac{5}{13}, \quad y = \frac{14}{13} \]