(x+2y)+(2x-3y)i+4i=5

3 min read Jun 16, 2024

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.