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

3 min read Jun 16, 2024

Solving the Complex Equation: (1+3i)z - (2+5i) = (2+i)z

This article will guide you through the process of solving the complex equation (1+3i)z - (2+5i) = (2+i)z. We will break down each step and explain the reasoning behind them.

1. Rearranging the Equation

First, we need to rearrange the equation to isolate the variable 'z'. We can achieve this by moving all terms containing 'z' to one side of the equation and the constant terms to the other side.

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

2. Simplifying the Equation

Next, we can simplify the equation by combining the 'z' terms.

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

This simplifies to:

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

3. Isolating 'z'

To isolate 'z', we need to divide both sides of the equation by (-1 + 2i). However, division by a complex number requires a special technique. We need to multiply both the numerator and denominator of the right side by the conjugate of the denominator. The conjugate of (-1 + 2i) is (-1 - 2i).

z = (2+5i) * (-1 - 2i) / ((-1 + 2i) * (-1 - 2i))

4. Expanding and Simplifying

Now, we expand the numerator and denominator using the distributive property (FOIL method).

z = (-2 - 4i - 5i - 10i^2) / (1 + 2i - 2i - 4i^2)

Remember that i^2 = -1. Substitute this value into the equation:

z = (-2 - 4i - 5i + 10) / (1 + 4)

Simplify further:

z = (8 - 9i) / 5

5. Final Solution

Finally, we have isolated 'z' and obtained our solution:

z = (8/5) - (9/5)i

This is the solution to the complex equation (1+3i)z - (2+5i) = (2+i)z.