(x-yi)(2+3i)=(x-2i)/(1-i)

3 min read Jun 17, 2024

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).