(1+2i)z=4-3i+2z

Jasonbradley

3 min read

·

Jun 16, 2024

15

Share

Solving Complex Equation: (1+2i)z = 4-3i+2z

This article will guide you through the process of solving the complex equation (1+2i)z = 4-3i+2z. We will utilize basic algebraic manipulations and the properties of complex numbers to arrive at the solution.

Understanding the Equation

The equation (1+2i)z = 4-3i+2z involves a complex variable 'z' and complex constants. Our aim is to isolate 'z' and find its value.

Solving for 'z'

1. Rearrange the terms:

   Subtract 2z from both sides: (1+2i)z - 2z = 4-3i

2. Factor out 'z':

   z(1+2i-2) = 4-3i

3. Simplify:

   z(-1+2i) = 4-3i

4. Isolate 'z':

   Divide both sides by (-1+2i): z = (4-3i) / (-1+2i)

5. Rationalize the denominator:

   Multiply both numerator and denominator by the complex conjugate of the denominator, which is (-1-2i): z = (4-3i)(-1-2i) / ((-1+2i)(-1-2i))

6. Expand and simplify:

   z = (-4-8i +3i +6i^2) / (1 + 4) z = (-4 - 5i - 6) / 5 (Since i^2 = -1) z = (-10 - 5i) / 5 z = -2 - i

Solution

Therefore, the solution to the equation (1+2i)z = 4-3i+2z is z = -2 - i.

Verification

To verify our solution, substitute z = -2 - i back into the original equation:

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

As both sides of the equation are equal, our solution z = -2 - i is verified.