1. **Problem:** Solve the equation $2z - 3 = \frac{-27 + 23i}{1 + i}$ where $z$ is the complex conjugate of $z$. Express $z$ in the form $x + yi$, where $x,y \in \mathbb{R}$. 2. **Formula and rules:** Recall that the complex conjugate of $z = x + yi$ is $\overline{z} = x - yi$. 3. **Step 1:** Simplify the right-hand side: $$\frac{-27 + 23i}{1 + i} = \frac{(-27 + 23i)(1 - i)}{(1 + i)(1 - i)} = \frac{-27 + 27i + 23i - 23i^2}{1 - i^2} = \frac{-27 + 50i + 23}{1 + 1} = \frac{-4 + 50i}{2} = -2 + 25i$$ 4. **Step 2:** Write $z = x - yi$ (since $z$ is conjugate of $z$), then: $$2(x - yi) - 3 = -2 + 25i$$ 5. **Step 3:** Expand and separate real and imaginary parts: $$2x - 2yi - 3 = -2 + 25i$$ Real: $2x - 3 = -2$ Imaginary: $-2y = 25$ 6. **Step 4:** Solve for $x$ and $y$: $$2x = 1 \Rightarrow x = \frac{1}{2}$$ $$-2y = 25 \Rightarrow y = -\frac{25}{2}$$ 7. **Answer:** $$z = \frac{1}{2} - \frac{25}{2}i$$ --- **Slug:** complex conjugate **Subject:** complex numbers **q_count:** 1 **desmos:** {"latex":"y=0","features":{"intercepts":true,"extrema":true}}