1. **State the problem:** We are given a complex number $Z = x + yi$ in standard form, and the equation $$\frac{\overline{Z} + \sqrt{2}xi - 25}{\left(\cos\left(\frac{7\pi}{2}\right) - i \sin\left(\frac{7\pi}{2}\right)\right)^{-1}} = \sqrt{18}.$$ We need to find the real numbers $x$ and $y$. 2. **Recall important formulas and rules:** - The conjugate of $Z = x + yi$ is $\overline{Z} = x - yi$. - Euler's formula relates cosine and sine to complex exponentials: $\cos \theta - i \sin \theta = e^{-i\theta}$. - The reciprocal of $\cos \theta - i \sin \theta$ is $\left(\cos \theta - i \sin \theta\right)^{-1} = \cos(-\theta) - i \sin(-\theta) = \cos \theta + i \sin \theta$. 3. **Evaluate the denominator:** Calculate $\cos\left(\frac{7\pi}{2}\right)$ and $\sin\left(\frac{7\pi}{2}\right)$. Since $\frac{7\pi}{2} = 3\pi + \frac{\pi}{2}$, and cosine and sine have period $2\pi$: $$\cos\left(\frac{7\pi}{2}\right) = \cos\left(3\pi + \frac{\pi}{2}\right) = \cos\left(\pi + \frac{\pi}{2}\right) = \cos\left(\frac{3\pi}{2}\right) = 0,$$ $$\sin\left(\frac{7\pi}{2}\right) = \sin\left(3\pi + \frac{\pi}{2}\right) = \sin\left(\pi + \frac{\pi}{2}\right) = \sin\left(\frac{3\pi}{2}\right) = -1.$$ So the denominator inside the inverse is: $$\cos\left(\frac{7\pi}{2}\right) - i \sin\left(\frac{7\pi}{2}\right) = 0 - i(-1) = i.$$ 4. **Find the inverse:** $$\left(i\right)^{-1} = \frac{1}{i} = -i.$$ 5. **Rewrite the original equation:** $$\frac{\overline{Z} + \sqrt{2}xi - 25}{-i} = \sqrt{18}.$$ Multiply both sides by $-i$: $$\overline{Z} + \sqrt{2}xi - 25 = -i \sqrt{18}.$$ 6. **Substitute $\overline{Z} = x - yi$ and simplify:** $$x - yi + \sqrt{2}xi - 25 = -i \sqrt{18}.$$ Group real and imaginary parts: Real part: $x - 25$ Imaginary part: $-y i + \sqrt{2} x i = i(\sqrt{2} x - y)$ So the left side is: $$ (x - 25) + i(\sqrt{2} x - y).$$ 7. **Equate real and imaginary parts to the right side:** Right side is $-i \sqrt{18} = 0 - i \sqrt{18}$, so real part is 0 and imaginary part is $-\sqrt{18}$. Set real parts equal: $$x - 25 = 0 \implies x = 25.$$ Set imaginary parts equal: $$\sqrt{2} x - y = -\sqrt{18}.$$ Substitute $x=25$: $$\sqrt{2} \times 25 - y = -\sqrt{18}.$$ 8. **Solve for $y$:** $$25 \sqrt{2} - y = -3 \sqrt{2}$$ (since $\sqrt{18} = 3 \sqrt{2}$) Rearranged: $$-y = -3 \sqrt{2} - 25 \sqrt{2} = -28 \sqrt{2}$$ $$y = 28 \sqrt{2}.$$ **Final answer:** $$x = 25, \quad y = 28 \sqrt{2}.$$