(2-3i)(z-1)+2i=5-8i

3 min read Jun 16, 2024

Solving the Complex Equation: (2-3i)(z-1)+2i=5-8i

This article will guide you through the process of solving the complex equation (2-3i)(z-1)+2i=5-8i, step by step. We'll break down each stage and explain the concepts involved.

1. Expanding the Equation

Firstly, we need to expand the equation by distributing the complex number (2-3i) to the term (z-1):

(2-3i)(z-1)+2i = 5-8i

2z - 2 - 3iz + 3i + 2i = 5 - 8i

2. Rearranging and Combining Terms

Now, let's rearrange the equation to group the real and imaginary terms:

2z - 3iz + 5i = 5 + 2

2z - 3iz = 7 - 5i

3. Expressing z in Standard Form

To solve for z, we need to express it in the standard form of a complex number, which is z = a + bi, where a and b are real numbers.

To do this, we can factor out z from the left-hand side of the equation:

z(2 - 3i) = 7 - 5i

4. Isolating z

Finally, to isolate z, we divide both sides of the equation by (2 - 3i):

z = (7 - 5i) / (2 - 3i)

5. Rationalizing the Denominator

To get z in the standard form, we need to rationalize the denominator. We can do this by multiplying both the numerator and denominator by the conjugate of the denominator, which is (2 + 3i):

z = (7 - 5i)(2 + 3i) / (2 - 3i)(2 + 3i)

6. Simplifying the Expression

Now, we can expand and simplify the expression:

z = (14 + 21i - 10i + 15) / (4 + 9)

z = (29 + 11i) / 13

7. Final Solution

Therefore, the solution to the equation (2-3i)(z-1)+2i=5-8i is:

z = 29/13 + 11i/13