We are tasked with converting the polar equation

$r = \frac{7}{\sin(\theta) + 4\cos(\theta)}$

into a rectangular equation.

Step 1: Relate polar coordinates to rectangular coordinates:

The key relationships between polar and rectangular coordinates are:

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

Additionally, $\cos(\theta) = \frac{x}{r}$ and $\sin(\theta) = \frac{y}{r}$.

Step 2: Rewrite the equation using $x$ and $y$:

The given equation is:

$r = \frac{7}{\sin(\theta) + 4\cos(\theta)}.$

Replace $\sin(\theta)$ and $\cos(\theta)$ using their polar forms:

$\sin(\theta) = \frac{y}{r}, \quad \cos(\theta) = \frac{x}{r}.$

Substitute into the denominator:

$r = \frac{7}{\frac{y}{r} + 4\frac{x}{r}}.$

Step 3: Simplify the denominator:

Factor out $\frac{1}{r}$ from the denominator:

$r = \frac{7}{\frac{1}{r}(y + 4x)}.$

Simplify further by multiplying numerator and denominator by $r$:

$r = \frac{7r}{y + 4x}.$

Step 4: Eliminate $r$:

To eliminate $r$, multiply through by $y + 4x$ (assuming $y + 4x \neq 0$):

$r(y + 4x) = 7r.$

Divide through by $r$ (again assuming $r \neq 0$):

$y + 4x = 7.$

Final Answer:

The rectangular equation is:

$y = -4x + 7.$

Verification:

1. The original polar equation simplifies cleanly into a linear rectangular equation.
2. The final result $y = -4x + 7$ matches the requirement of writing $y$ as a function of $x$.

Let me know if you'd like more clarification or details! Here are related questions to explore:

1. How do you convert other polar equations to rectangular form?
2. What are some key differences between polar and rectangular coordinate systems?
3. How do you derive the general form of conic sections in polar form?
4. How can you graph $r = \frac{7}{\sin(\theta) + 4\cos(\theta)}$?
5. What steps are necessary to convert rectangular equations back to polar form?

Tip: Always start with the polar-to-rectangular relationships when converting between systems.