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:
\[
z = \frac{2}{5} - \frac{i}{5}
\]

---

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.

The query submitted by the user:$$ \frac{1+z}{1-z}=2-\imaginaryI $$

Learn how to solve the complex equation $ \frac{1+z}{1-z} = 2 - i $ using algebraic techniques and complex number operations. This solution covers key concepts in complex number manipulation, including multiplying by the conjugate, simplifying fractions, and solving for the unknown variable. Suitable for advanced students (Grades 11-12).

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