Solving Complex Equation: (x  yi)(2 + 3i) = (x  2i) / (1  i)
This article will guide you through the process of solving the complex equation: (x  yi)(2 + 3i) = (x  2i) / (1  i). We will utilize the properties of complex numbers and algebraic manipulation to find the values of x and y.
Expanding and Simplifying

Expand the left side of the equation: (x  yi)(2 + 3i) = 2x + 3xi  2yi  3yi² = (2x + 3y) + (3x  2y)i
Remember that i² = 1. 
Rationalize the denominator on the right side: (x  2i) / (1  i) = [(x  2i)(1 + i)] / [(1  i)(1 + i)] = (x + xi  2i  2i²) / (1  i²) = (x + 2) + (x  2)i
Equating Real and Imaginary Parts
Now we have the equation: (2x + 3y) + (3x  2y)i = (x + 2) + (x  2)i. For two complex numbers to be equal, their real and imaginary parts must be equal. This gives us two equations:
 Real Part: 2x + 3y = x + 2
 Imaginary Part: 3x  2y = x  2
Solving the System of Equations
We now have a system of two linear equations with two unknowns. We can solve for x and y using various methods such as substitution or elimination.
Using Elimination:

Rearrange the equations:
 x  3y = 2
 2x  2y = 2

Multiply the first equation by 2:
 2x + 6y = 4
 2x  2y = 2

Add the two equations together:
 4y = 6

Solve for y:
 y = 3/2

Substitute the value of y back into either original equation to solve for x:
 x  3(3/2) = 2
 x + 9/2 = 2
 x = 5/2
Solution
Therefore, the solution to the complex equation (x  yi)(2 + 3i) = (x  2i) / (1  i) is x = 5/2 and y = 3/2. This means the complex number is (5/2 + 3/2i).