We are tasked with solving the equation:

$\frac{1+z}{1-z} = 2 - i$

Let's solve this step by step.

Step 1: Multiply both sides of the equation by $1 - z$ to eliminate the denominator on the left-hand side:

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

Step 2: Expand the right-hand side of the equation:

$1 + z = (2 - i)(1 - z)$ $= 2(1 - z) - i(1 - z)$ $= 2 - 2z - i + iz$ So the equation becomes: $1 + z = 2 - 2z - i + iz$

Step 3: Rearrange the terms so that all terms involving $z$ are on one side and constant terms are on the other side:

$1 + z = 2 - i - 2z + iz$ Now move all the terms involving $z$ to one side: $1 + z + 2z - iz = 2 - i$ Combine like terms: $1 + 3z - iz = 2 - i$ Now, move the constants to the other side: $3z - iz = 2 - i - 1$ Simplify the right-hand side: $3z - iz = 1 - i$

Step 4: Factor out $z$ on the left-hand side:

$z(3 - i) = 1 - i$

Step 5: Solve for $z$:

$z = \frac{1 - i}{3 - i}$

To simplify this complex fraction, we multiply both the numerator and denominator by the complex conjugate of the denominator, which is $3 + i$:

$z = \frac{(1 - i)(3 + i)}{(3 - i)(3 + i)}$

Step 6: Simplify the denominator:

The denominator is a difference of squares: $(3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10$

Step 7: Simplify the numerator:

Expand $(1 - i)(3 + i)$: $(1 - i)(3 + i) = 1(3 + i) - i(3 + i)$ $= 3 + i - 3i - i^2 = 3 + i - 3i + 1 = 4 - 2i$

Thus, we have: $z = \frac{4 - 2i}{10}$

Step 8: Final result:

$z = \frac{4}{10} - \frac{2i}{10} = \frac{2}{5} - \frac{i}{5}$

Therefore, the solution to the equation is: $z = \frac{2}{5} - \frac{i}{5}$

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Would you like a more detailed explanation of any step, or do you have additional questions on complex numbers?

Here are some related questions to explore further:

1. How do you simplify a complex fraction like $\frac{1 - i}{3 - i}$?
2. What are the real and imaginary parts of a complex number?
3. How do you multiply and divide complex numbers?
4. What is the geometric interpretation of a complex number?
5. How do you convert a complex number to polar form?

Tip: When dividing complex numbers, multiplying by the conjugate of the denominator is a key technique to rationalize the denominator.