To eliminate the parameter $t$ from the given parametric equations and rewrite them as a Cartesian equation, we can use trigonometric identities. The given parametric equations are:

$x(t) = -3 \sin(t)$ $y(t) = 2 \cos(t)$

We start by isolating the trigonometric functions in terms of $x$ and $y$:

1. From $x(t) = -3 \sin(t)$: $\sin(t) = -\frac{x}{3}$

2. From $y(t) = 2 \cos(t)$: $\cos(t) = \frac{y}{2}$

Next, we use the Pythagorean identity for sine and cosine, which states: $\sin^2(t) + \cos^2(t) = 1$

Substituting the expressions for $\sin(t)$ and $\cos(t)$ into this identity, we get: $\left(-\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1$

Simplifying this, we obtain: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

This is the equation of an ellipse in Cartesian coordinates. Thus, the parametric equations $x(t) = -3 \sin(t)$ and $y(t) = 2 \cos(t)$ can be rewritten as the Cartesian equation: $\frac{x^2}{9} + \frac{y^2}{4} = 1$

If you have any questions or need further details, please let me know.

Here are 5 related questions for further practice:

1. What is the parametric form of the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$?
2. How would you convert the parametric equations $x(t) = 5 \cos(t)$ and $y(t) = 7 \sin(t)$ to Cartesian form?
3. What are the major and minor axes of the ellipse described by $\frac{x^2}{36} + \frac{y^2}{49} = 1$?
4. If the parametric equations are given by $x(t) = 4 \cos(t)$ and $y(t) = 3 \sin(t)$, what is the range of $t$ to complete one full cycle of the ellipse?
5. How can you determine the orientation of an ellipse given its parametric equations?

Tip: When converting parametric equations to Cartesian form, always look for trigonometric identities or algebraic relationships that can help eliminate the parameter.