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

Jasonbradley

4 min read Jun 16, 2024

Solving Complex Equations: A Step-by-Step Guide

This article will guide you through solving the complex equation:

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

Let's break it down step by step:

1. Simplify the Fractions

First, we need to simplify the fractions by multiplying the numerator and denominator by the complex conjugate of the denominator.

For (1+i)x - 2i / (3+i):

The conjugate of (3+i) is (3-i).
Multiply both numerator and denominator by (3-i):
- [(1+i)x - 2i] * (3-i) / (3+i)*(3-i)

For (2-3i)y + i / (3-i):

The conjugate of (3-i) is (3+i).
Multiply both numerator and denominator by (3+i):
- [(2-3i)y + i] * (3+i) / (3-i)*(3+i)

2. Expand and Simplify

Now, expand the products and simplify the expressions:

(1+i)x - 2i / (3+i):

[(1+i)x - 2i] * (3-i) / (3+i)*(3-i) = [(3x - xi + 3ix + x) - (6i + 2)] / (9 + 1)
Simplifying further: (4x + 2ix - 2 - 6i) / 10

(2-3i)y + i / (3-i):

[(2-3i)y + i] * (3+i) / (3-i)*(3+i) = [(6y + 2iy - 9iy + 3y) + (3i - y)] / (9 + 1)
Simplifying further: (9y - 7iy - y + 3i) / 10

3. Substitute and Combine Like Terms

Substitute the simplified fractions back into the original equation:

(4x + 2ix - 2 - 6i) / 10 + (9y - 7iy - y + 3i) / 10 = i

Combine the real and imaginary terms separately:

Real terms: (4x - 2 + 9y - y) / 10 = 0
Imaginary terms: (2x - 6 - 7y + 3) / 10 = 1

4. Solve the System of Equations

Now we have two equations with two unknowns (x and y).

Equation 1: 4x + 8y - 2 = 0
Equation 2: 2x - 7y - 3 = 1

Solve this system of equations using any method you prefer (substitution, elimination, etc.). The solution for x and y will be the solution for the original complex equation.

Conclusion

Solving complex equations involving fractions requires careful manipulation and simplification. By following the steps outlined above, you can systematically solve such equations and find the values of the unknown variables. Remember to always focus on separating real and imaginary terms and then solving the resulting system of equations.