Find the real numbers x,y such that `( iy+ x)(3+2i) = 1+ i`

Answer

To solve the equation \((iy + x)(3 + 2i) = 1 + i\), we will follow these steps:

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

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

Calculating each term:
- \(iy \cdot 3 = 3iy\)
- \(iy \cdot 2i = 2i^2y = -2y\) (since \(i^2 = -1\))
- \(x \cdot 3 = 3x\)
- \(x \cdot 2i = 2xi\)

Putting it all together, we have:

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

### Step 2: Combine Like Terms
Now, we can combine the real and imaginary parts:

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

### Step 3: Set the Equation Equal to the Right-Hand Side
Now we set the expanded expression equal to the right-hand side of the original equation:

\[ (3x - 2y) + (2x + 3y)i = 1 + i \]

### Step 4: Equate Real and Imaginary Parts
From the equation above, we can equate the real parts and the imaginary parts:

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

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

### Step 5: Solve the System of Equations
Now we have a system of two equations:

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

We can solve these equations simultaneously.

#### Multiply Equation (1) by 3:
\[ 9x - 6y = 3 \quad \text{(3)} \]

#### Multiply Equation (2) by 2:
\[ 4x + 6y = 2 \quad \text{(4)} \]

### Step 6: Add Equations (3) and (4)
Now, we add equations (3) and (4):

\[ (9x - 6y) + (4x + 6y) = 3 + 2 \]

This simplifies to:

\[ 13x = 5 \]

### Step 7: Solve for \(x\)
Now, we can solve for \(x\):

\[ x = \frac{5}{13} \]

### Step 8: Substitute \(x\) back into one of the original equations
We can substitute \(x\) back into equation (1) to find \(y\):

\[ 3\left(\frac{5}{13}\right) - 2y = 1 \]

This simplifies to:

\[ \frac{15}{13} - 2y = 1 \]

### Step 9: Solve for \(y\)
Rearranging gives:

\[ -2y = 1 - \frac{15}{13} \]

Calculating the right-hand side:

\[ 1 = \frac{13}{13} \implies 1 - \frac{15}{13} = \frac{13 - 15}{13} = -\frac{2}{13} \]

So we have:

\[ -2y = -\frac{2}{13} \]

Dividing both sides by -2:

\[ y = \frac{1}{13} \]

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

\[ x = \frac{5}{13}, \quad y = \frac{1}{13} \]