(3-i)x(4+2i)y=2+6i

2 min read Jun 16, 2024

Solving the Equation: (3-i)x(4+2i)y = 2 + 6i

This equation involves complex numbers and requires a step-by-step approach to solve for the unknown variables x and y. Here's a breakdown of the process:

1. Expanding the Left-Hand Side

First, we need to expand the left-hand side of the equation by multiplying the complex numbers:

(3 - i)x(4 + 2i)y = (12 + 6i - 4i - 2i²)xy

Since i² = -1, we can simplify this expression further:

(12 + 6i - 4i + 2)xy = (14 + 2i)xy

2. Equating Real and Imaginary Components

Now we have the equation:

(14 + 2i)xy = 2 + 6i

To solve for x and y, we need to equate the real and imaginary components on both sides of the equation:

Real Components: 14xy = 2
Imaginary Components: 2xy = 6

3. Solving for x and y

We now have two equations with two unknowns. Let's solve for x and y:

From the real component equation: xy = 1/7
From the imaginary component equation: xy = 3

We see a contradiction here, as xy cannot be both 1/7 and 3. This means there is no solution for x and y that satisfies the original equation.

Conclusion

The equation (3-i)x(4+2i)y = 2 + 6i has no solution for x and y. This is because the real and imaginary components of the equation lead to conflicting values for the product xy.