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

4 min read Jun 16, 2024

Solving Complex Equations: A Step-by-Step Guide for (2-i)x + (3-2i)y = -1 + 2i

This article will guide you through solving the complex equation (2-i)x + (3-2i)y = -1 + 2i. We'll break down the process into manageable steps and explore the key concepts involved.

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

Key Properties:

Addition/Subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i

Solving the Equation

Separate Real and Imaginary Parts:

We begin by separating the real and imaginary parts of the equation:

(2x + 3y) + (-x - 2y)i = -1 + 2i
Equate Corresponding Parts:

For the equation to hold true, the real parts on both sides must be equal, and the imaginary parts must also be equal. This gives us two separate equations:
- Real Part: 2x + 3y = -1
- Imaginary Part: -x - 2y = 2
Solve the System of Equations:

Now we have a system of two linear equations with two unknowns. We can use various methods to solve for x and y. One common approach is elimination:
- Multiply the second equation by 2: -2x - 4y = 4
- Add the first equation to the modified second equation: y = 3
- Substitute y = 3 into either original equation to find x: 2x + 3(3) = -1, which gives x = -5
Solution:

Therefore, the solution to the complex equation is:
- x = -5
- y = 3

Verification

We can verify our solution by plugging the values of x and y back into the original equation:

(2 - i)(-5) + (3 - 2i)(3) = -1 + 2i

Simplifying the left side, we get:

-10 + 5i + 9 - 6i = -1 + 2i

This simplifies to:

-1 + 2i = -1 + 2i

Since both sides are equal, our solution is correct.

Conclusion

Solving complex equations involves separating real and imaginary parts, creating a system of equations, and solving for the unknowns. By applying these steps, we can efficiently find solutions to complex equations and verify their accuracy. This process is crucial in various areas of mathematics, physics, and engineering.