The equation given in the image is:

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

We need to solve for $z$, where $i$ represents the imaginary unit ($i^2 = -1$).

Step-by-step Solution:

1. Expand the left-hand side: Use the distributive property to expand $(z + i)(1 - i)$:

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

   Simplify each term:

   $z(1 - i) = z - zi$ $i(1 - i) = i - i^2$ Since $i^2 = -1$, the second term simplifies to:

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

   Now combine both parts:

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

   Rearranging:

   $= z - zi + i + 1$

2. Equating with the right-hand side: Now, equate the expanded form to the right-hand side $2 + 3i$:

   $z - zi + i + 1 = 2 + 3i$

3. Separate real and imaginary parts: Collect the real and imaginary terms on both sides of the equation:

   • Real part: $z + 1 = 2$
   • Imaginary part: $-zi + i = 3i$

4. Solve for $z$:

   • From the real part: $z + 1 = 2$

     Solving for $z$:

     $z = 2 - 1 = 1$

   • From the imaginary part: $-zi + i = 3i$

     Factor out $i$:

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

     Equate the coefficients of $i$:

     $-z + 1 = 3$

     Solving for $z$:

     $-z = 3 - 1 = 2$ $z = -2$

Conclusion:

There is a contradiction here, as we obtained two different values for $z$: $z = 1$ and $z = -2$. Upon reviewing the steps, it seems the equation is inconsistent, suggesting a possible error or typo in the problem. Let me know if you'd like further clarification or if you'd like to recheck any part of this solution.

Do you want to explore these concepts further, or do you have any other questions?

Related Questions:

1. How can we solve a system of equations with real and imaginary parts?
2. What are the rules for multiplying complex numbers?
3. How can you factor and expand expressions involving complex numbers?
4. What are common strategies for solving complex number equations?
5. How do we represent complex numbers on the Argand diagram?

Tip:

When solving equations involving complex numbers, always separate real and imaginary parts and solve them independently.