1. Problem: Solve the equation $$|z| + x - yi = 10 - 5i$$ where $$z = x + yi$$ and $$x, y$$ are real numbers. 2. Recall that $$|z| = \sqrt{x^2 + y^2}$$ is the magnitude of the complex number $$z$$. 3. The equation can be separated into real and imaginary parts: Real part: $$|z| + x = 10$$ Imaginary part: $$-y = -5$$ 4. From the imaginary part, solve for $$y$$: $$-y = -5 \implies y = 5$$ 5. Substitute $$y = 5$$ into the real part: $$\sqrt{x^2 + 5^2} + x = 10$$ $$\sqrt{x^2 + 25} + x = 10$$ 6. Isolate the square root: $$\sqrt{x^2 + 25} = 10 - x$$ 7. Since $$\sqrt{x^2 + 25} \geq 0$$, we require $$10 - x \geq 0 \implies x \leq 10$$. 8. Square both sides: $$x^2 + 25 = (10 - x)^2 = 100 - 20x + x^2$$ 9. Simplify: $$x^2 + 25 = 100 - 20x + x^2$$ Subtract $$x^2$$ from both sides: $$25 = 100 - 20x$$ 10. Solve for $$x$$: $$-20x = 25 - 100 = -75$$ $$x = \frac{75}{20} = \frac{15}{4} = 3.75$$ 11. Check the domain condition $$x \leq 10$$, which holds. 12. Final solution: $$x = 3.75, y = 5$$ So, $$z = 3.75 + 5i$$. 13. Verify: $$|z| = \sqrt{3.75^2 + 5^2} = \sqrt{14.0625 + 25} = \sqrt{39.0625} = 6.25$$ Left side real part: $$|z| + x = 6.25 + 3.75 = 10$$ Left side imaginary part: $$-y = -5$$ Matches right side $$10 - 5i$$. Answer: $$z = 3.75 + 5i$$