1. The problem is to eliminate the parameter $0$ from the parametric equations: $$x = 3 \cos 0 - 5 \sin 0$$ $$y = 4 \sin 0 + 5 \cos 0$$ 2. Note that $0$ here represents the parameter (angle), so we can rename it as $\theta$ for clarity: $$x = 3 \cos \theta - 5 \sin \theta$$ $$y = 4 \sin \theta + 5 \cos \theta$$ 3. The goal is to express $y$ solely in terms of $x$ by eliminating $\theta$. 4. Write the system: $$x = 3 \cos \theta - 5 \sin \theta$$ $$y = 4 \sin \theta + 5 \cos \theta$$ 5. Solve for $\cos \theta$ and $\sin \theta$ from these equations. Multiply the first equation by 4 and the second by 3: $$4x = 12 \cos \theta - 20 \sin \theta$$ $$3y = 12 \sin \theta + 15 \cos \theta$$ 6. Rearrange to isolate $\cos \theta$ and $\sin \theta$: Multiply original equations: $$x = 3 \cos \theta - 5 \sin \theta$$ $$y = 4 \sin \theta + 5 \cos \theta$$ We can use a system approach or try to express $\cos \theta$ and $\sin \theta$ as: From the first equation: $$3 \cos \theta = x + 5 \sin \theta$$ From the second equation: $$5 \cos \theta = y - 4 \sin \theta$$ Multiply the first equation by 5: $$15 \cos \ theta = 5x + 25 \sin \theta$$ Multiply the second equation by 3: $$15 \cos \theta = 3y - 12 \sin \theta$$ Set equal: $$5x + 25 \sin \theta = 3y - 12 \sin \theta$$ Combine $\sin \theta$: $$25 \sin \theta + 12 \sin \theta = 3y - 5x$$ $$37 \sin \theta = 3y - 5x$$ So: $$\sin \theta = \frac{3y - 5x}{37}$$ Similarly substitute back into $3 \cos \theta = x + 5 \sin \theta$: $$3 \cos \theta = x + 5 \times \frac{3y - 5x}{37} = x + \frac{15y - 25x}{37} = \frac{37x + 15y - 25x}{37} = \frac{12x + 15y}{37}$$ Therefore: $$\cos \theta = \frac{12x + 15y}{111}$$ 7. Using identity $\sin^2 \theta + \cos^2 \theta = 1$: $$\left(\frac{3y - 5x}{37}\right)^2 + \left(\frac{12x + 15y}{111}\right)^2 = 1$$ Multiply both sides by $111^2$ (since $111 = 3 \times 37$): $$ (3y - 5x)^2 + \left(\frac{12x + 15y}{3}\right)^2 = 111^2$$ Square terms and simplify carefully gives the Cartesian equation relating $x$ and $y$ without $\theta$. **Final Cartesian relation:** $$\left(\frac{3y - 5x}{37}\right)^2 + \left(\frac{12x + 15y}{111}\right)^2 = 1$$