(1+i)x-(1-2i)y=1+4i

4 min read Jun 16, 2024

Solving Complex Equations: (1+i)x - (1-2i)y = 1+4i

This article will guide you through the process of solving a complex equation involving variables x and y. The equation is:

(1+i)x - (1-2i)y = 1+4i

Where i represents the imaginary unit, defined as the square root of -1.

Understanding Complex Numbers

Before we delve into the solution, let's briefly recap the properties of complex numbers:

Standard Form: Complex numbers are typically expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
Equality: Two complex numbers are equal if and only if their real and imaginary parts are equal.

Solving the Equation

Expand the equation: First, distribute the complex numbers:

x + ix - y + 2iy = 1 + 4i
Group real and imaginary terms: Rearrange the terms:

(x - y) + (x + 2y)i = 1 + 4i
Apply the equality property: For the equation to hold true, the real and imaginary parts on both sides must be equal:

x - y = 1 x + 2y = 4
Solve the system of equations: Now we have a system of two linear equations in two variables. We can solve this using various methods, such as elimination or substitution.

Method of Elimination:
- Multiply the first equation by 2: 2x - 2y = 2
- Subtract the second equation from the modified first equation: x - 3y = -2
- Solve for x: x = 3y - 2
- Substitute this value of x into either of the original equations and solve for y: (3y - 2) - y = 1 2y = 3 y = 3/2
- Substitute the value of y back into the expression for x: x = 3(3/2) - 2 x = 5/2
Solution: The solution to the complex equation is:
- x = 5/2
- y = 3/2

Verification

To ensure our solution is correct, we can substitute the values of x and y back into the original equation:

(1 + i)(5/2) - (1 - 2i)(3/2) = 1 + 4i

Simplifying the left-hand side confirms that it equals the right-hand side, verifying our solution.

Conclusion

By understanding the properties of complex numbers and applying algebraic methods, we have successfully solved the complex equation (1+i)x - (1-2i)y = 1+4i. The solution provides the values of x and y that satisfy the given equation.