## Solving Complex Equations: A Step-by-Step Guide

This article will guide you through the process of solving the complex equation **(x + 2y) + (2x - 3y)i + 4i = 5**.

### Understanding Complex Numbers

A complex number is a number that can be expressed in the form **a + bi**, where **a** and **b** are real numbers, and **i** is the imaginary unit, defined as the square root of -1.

In our equation, the real part is **(x + 2y)** and the imaginary part is **(2x - 3y)i + 4i**.

### Isolating Real and Imaginary Components

To solve the equation, we need to equate the real and imaginary parts on both sides of the equation.

**Real Part:**
x + 2y = 5

**Imaginary Part:**
(2x - 3y)i + 4i = 0

Simplifying the imaginary part, we get: (2x - 3y + 4)i = 0

Since the imaginary unit 'i' cannot be zero, the coefficient of 'i' must be zero: 2x - 3y + 4 = 0

### Solving the System of Equations

Now we have two equations with two unknowns:

**x + 2y = 5****2x - 3y + 4 = 0**

We can solve this system using various methods, such as substitution or elimination.

**Using Elimination:**

Multiply the first equation by 3 and the second equation by 2:

**3x + 6y = 15****4x - 6y + 8 = 0**

Adding the two equations together, we get:
**7x + 8 = 15**

Solving for x:
**7x = 7**
**x = 1**

Substitute the value of x back into either of the original equations to solve for y. Let's use the first equation:

**1 + 2y = 5**
**2y = 4**
**y = 2**

### Solution

Therefore, the solution to the complex equation **(x + 2y) + (2x - 3y)i + 4i = 5** is **x = 1** and **y = 2**.