Conjugate Values 2C70B2

1. **State the problem:** Find the real values of $x$ and $y$ such that the complex fractions $\frac{2x - yi}{1 + 2i}$ and $\frac{3 - i}{1 - i}$ are conjugates. 2. **Recall the conjugate property:** If two complex numbers $z$ and $w$ are conjugates, then $z = \overline{w}$. 3. **Express the conjugate:** The conjugate of $\frac{3 - i}{1 - i}$ is $\overline{\frac{3 - i}{1 - i}} = \frac{3 + i}{1 + i}$. 4. **Set the equality:** $$\frac{2x - yi}{1 + 2i} = \frac{3 + i}{1 + i}$$ 5. **Cross multiply:** $$(2x - yi)(1 + i) = (3 + i)(1 + 2i)$$ 6. **Expand both sides:** Left side: $$ (2x)(1) + (2x)(i) - y i (1) - y i (i) = 2x + 2x i - y i - y i^2 $$ Since $i^2 = -1$, this becomes: $$ 2x + 2x i - y i + y = (2x + y) + i(2x - y) $$ Right side: $$ 3(1) + 3(2i) + i(1) + i(2i) = 3 + 6i + i + 2 i^2 = 3 + 7i + 2(-1) = 3 + 7i - 2 = 1 + 7i $$ 7. **Equate real and imaginary parts:** Real parts: $$ 2x + y = 1 $$ Imaginary parts: $$ 2x - y = 7 $$ 8. **Solve the system:** Add the two equations: $$ (2x + y) + (2x - y) = 1 + 7 \Rightarrow 4x = 8 \Rightarrow x = 2 $$ Substitute $x=2$ into $2x + y = 1$: $$ 2(2) + y = 1 \Rightarrow 4 + y = 1 \Rightarrow y = 1 - 4 = -3 $$ **Final answer:** $$ x = 2, \quad y = -3 $$