If `( x + iy) ( 2 - 3 i) = 4 + i ` then

Options

A. x = 1, y = 2
B. x = `5/ 12`, y =`14/13`
C. x = 3, y = 4
D. none of these

Answer

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

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

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

This simplifies to:

\[ 2x - 3xi + 2iy - 3i^2y \]

Since \(i^2 = -1\), we can replace \(-3i^2y\) with \(3y\):

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

Thus, we have:

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

### Step 2: Set Equal to the Right Side
Now we set this equal to the right-hand side of the equation:

\[ (2x + 3y) + (-3x + 2y)i = 4 + i \]

### Step 3: Equate Real and Imaginary Parts
From this equation, we can equate the real and imaginary parts:

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

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

### Step 4: Solve the System of Equations
Now we have a system of linear equations. We can solve these equations simultaneously.

From equation (1):
\[ 2x + 3y = 4 \implies 2x = 4 - 3y \implies x = 2 - \frac{3}{2}y \quad (3) \]

Substituting equation (3) into equation (2):
\[ -3(2 - \frac{3}{2}y) + 2y = 1 \]

Expanding this gives:
\[ -6 + \frac{9}{2}y + 2y = 1 \]

Combining like terms:
\[ -6 + \frac{9}{2}y + \frac{4}{2}y = 1 \]
\[ -6 + \frac{13}{2}y = 1 \]

Adding 6 to both sides:
\[ \frac{13}{2}y = 7 \]

Multiplying both sides by \(\frac{2}{13}\):
\[ y = \frac{14}{13} \]

### Step 5: Substitute Back to Find x
Now we substitute \(y\) back into equation (3) to find \(x\):
\[ x = 2 - \frac{3}{2} \cdot \frac{14}{13} \]
\[ x = 2 - \frac{21}{13} \]
\[ x = \frac{26}{13} - \frac{21}{13} = \frac{5}{13} \]

### Final Solution
Thus, the values of \(x\) and \(y\) are:
\[ x = \frac{5}{13}, \quad y = \frac{14}{13} \]