Find 'x' and 'y' such that 2x+3iy and 2+9i represent the same complex number.

Answer

To find the values of \( x \) and \( y \) such that the complex numbers \( 2x + 3iy \) and \( 2 + 9i \) are equal, we can follow these steps:

### Step 1: Set the complex numbers equal to each other
We start with the equation:
\[ 2x + 3iy = 2 + 9i \]

### Step 2: Equate the real parts
The real part of the left side is \( 2x \) and the real part of the right side is \( 2 \). Therefore, we can set up the equation:
\[ 2x = 2 \]

### Step 3: Solve for \( x \)
To find \( x \), we divide both sides of the equation by 2:
\[ x = \frac{2}{2} = 1 \]

### Step 4: Equate the imaginary parts
The imaginary part of the left side is \( 3y \) and the imaginary part of the right side is \( 9 \). Thus, we set up the equation:
\[ 3y = 9 \]

### Step 5: Solve for \( y \)
To find \( y \), we divide both sides of the equation by 3:
\[ y = \frac{9}{3} = 3 \]

### Conclusion
The values of \( x \) and \( y \) are:
\[ x = 1 \quad \text{and} \quad y = 3 \]

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