(2+3i)x+(2i-3)y=7i-4

4 min read Jun 16, 2024

Solving Complex Equations: A Step-by-Step Guide

This article explores how to solve equations involving complex numbers. We'll focus on the equation (2 + 3i)x + (2i - 3)y = 7i - 4, breaking down the solution process into clear steps.

Understanding the Equation

The equation presents a linear combination of two complex variables, x and y, multiplied by complex coefficients. Our goal is to find the values of x and y that satisfy this equation.

Step 1: Separating Real and Imaginary Parts

To solve for x and y, we need to separate the real and imaginary components of the equation. We can do this by expanding the equation:

(2 + 3i)x + (2i - 3)y = 7i - 4

2x + 3ix + 2iy - 3y = -4 + 7i

Now, group the terms with real coefficients and the terms with imaginary coefficients:

(2x - 3y) + (3x + 2y)i = -4 + 7i

Step 2: Equating Real and Imaginary Components

The equation states that the complex numbers on both sides are equal. This implies that the real parts and the imaginary parts must be equal.

Real part: 2x - 3y = -4 Imaginary part: 3x + 2y = 7

We now have two simultaneous equations with two unknowns.

Step 3: Solving the System of Equations

We can solve this system of equations using any method you prefer, such as substitution, elimination, or matrix methods.

Using elimination:

Multiply the first equation by 2 and the second equation by 3.
Add the resulting equations together.
Solve for x.
Substitute the value of x back into either of the original equations and solve for y.

Using substitution:

Solve one of the equations for one variable in terms of the other.
Substitute this expression into the other equation.
Solve for the remaining variable.
Substitute the value back into the equation you used for substitution to find the other variable.

Step 4: The Solution

By solving the system of equations, we will obtain values for x and y. These values represent the solution to the original complex equation.

Conclusion

Solving complex equations involves separating the real and imaginary parts, creating a system of equations, and then solving for the unknown variables. The process is similar to solving linear equations, but we need to account for the unique properties of complex numbers.