To convert the given polar equation $r - 5 \cos \theta = 3 \sin \theta$ into its rectangular form, we need to use the following conversions between polar and rectangular coordinates:

1. $x = r \cos \theta$
2. $y = r \sin \theta$
3. $r^2 = x^2 + y^2$

Let's rewrite and simplify the given equation step by step:

Step 1: Rewrite the equation

The polar equation is: $r - 5 \cos \theta = 3 \sin \theta$

Add $5 \cos \theta$ to both sides: $r = 5 \cos \theta + 3 \sin \theta$

Step 2: Substitute polar-to-rectangular conversions

We know $\cos \theta = \frac{x}{r}$ and $\sin \theta = \frac{y}{r}$. Substitute these into the equation: $r = 5 \frac{x}{r} + 3 \frac{y}{r}$

Multiply through by $r$ to eliminate the denominator: $r^2 = 5x + 3y$

Step 3: Replace $r^2$ with $x^2 + y^2$

Using $r^2 = x^2 + y^2$, we substitute: $x^2 + y^2 = 5x + 3y$

Step 4: Rearrange into standard rectangular form

Bring all terms to one side: $x^2 + y^2 - 5x - 3y = 0$

This is the rectangular representation of the given polar equation.

Would you like further clarification or additional examples?

Follow-up Questions

1. How do we identify the graph of $x^2 + y^2 - 5x - 3y = 0$?
2. Can we complete the square for $x$ and $y$ to find the center and radius if it represents a circle?
3. What are the general steps for converting any polar equation to rectangular form?
4. How do we determine the type of conic section represented by the rectangular equation?
5. What happens if the polar equation involves $r^2$ or higher powers of $\cos \theta$ and $\sin \theta$?

Tip:

Always simplify the equation as much as possible after substitution to make identifying the geometric shape easier.