1. **State the problem:** We are given a complex number $z = x + iy$ that satisfies the equation $$\frac{z}{z+2} = 2 - i.$$ We want to find the values of $x$ and $y$. 2. **Write the equation explicitly:** Substitute $z = x + iy$ into the equation: $$\frac{x + iy}{(x + iy) + 2} = 2 - i.$$ 3. **Simplify the denominator:** $$(x + iy) + 2 = (x + 2) + iy.$$ 4. **Set up the equation:** $$\frac{x + iy}{(x + 2) + iy} = 2 - i.$$ 5. **Multiply both sides by $((x + 2) + iy)$ to clear the denominator:** $$x + iy = (2 - i)((x + 2) + iy).$$ 6. **Expand the right side using distributive property:** $$(2 - i)((x + 2) + iy) = 2(x+2) + 2iy - i(x+2) - i(iy).$$ 7. **Calculate each term:** $2(x+2) = 2x + 4$ $2iy = 2iy$ $-i(x+2) = -ix - 2i$ $-i(iy) = -i^2y = y$ (because $i^2 = -1$) 8. **Combine the terms:** $$(2x + 4) + 2iy - ix - 2i + y = (2x + 4 + y) + (2y - x - 2)i.$$ 9. **Equate the real and imaginary parts from both sides:** Real parts: $x = 2x + 4 + y$ Imaginary parts: $y = 2y - x - 2$ 10. **Solve the real part equation:** $$x = 2x + 4 + y \implies x - 2x - y = 4 \implies -x - y = 4 \implies x + y = -4.$$ 11. **Solve the imaginary part equation:** $$y = 2y - x - 2 \implies y - 2y = - x - 2 \implies -y = - x - 2 \implies y = x + 2.$$ 12. **Substitute $y = x + 2$ into $x + y = -4$: ** $$x + (x + 2) = -4$$ $$2x + 2 = -4$$ $$2x = -6$$ $$x = -3.$$ 13. **Find $y$ using $y = x + 2$: ** $$y = -3 + 2 = -1.$$ **Final answer:** $x = -3$ and $y = -1$.