1. **State the problem:** We are given the parametric equations $x = 3 \cos t$ and $y = 4 \sin t$ and need to find the Cartesian equation relating $x$ and $y$ without the parameter $t$. 2. **Express trigonometric functions:** From the parametric form, we have: $$x = 3 \cos t \implies \cos t = \frac{x}{3}$$ $$y = 4 \sin t \implies \sin t = \frac{y}{4}$$ 3. **Use the Pythagorean identity:** We know that for any angle $t$, $$\sin^2 t + \cos^2 t = 1$$ Substitute the expressions for $\sin t$ and $\cos t$: $$\left(\frac{y}{4}\right)^2 + \left(\frac{x}{3}\right)^2 = 1$$ 4. **Simplify the equation:** $$\frac{y^2}{16} + \frac{x^2}{9} = 1$$ 5. **Final Cartesian equation:** $$\frac{x^2}{9} + \frac{y^2}{16} = 1$$ This is the equation of an ellipse centered at the origin with semi-major axis 4 along the $y$-axis and semi-minor axis 3 along the $x$-axis.