Replace the polar equation \( r=\frac{9}{\sin \theta-4 \cos \theta} \) with an equivalent Cartesian equation. Then identify the graph. The equivalent Cartestian equation is \( y=\square \)

To convert the polar equation \( r = \frac{9}{\sin \theta - 4 \cos \theta} \) into Cartesian coordinates, we can use the relationships between polar and Cartesian coordinates: \[ x = r \cos \theta, \quad y = r \sin \theta, \quad r = \sqrt{x^2 + y^2}, \quad \sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r} \] Starting with the given polar equation: \[ r = \frac{9}{\sin \theta - 4 \cos \theta} \] We can multiply both sides by \( \sin \theta - 4 \cos \theta \): \[ r(\sin \theta - 4 \cos \theta) = 9 \] Substituting \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \): \[ r\left(\frac{y}{r} - 4 \frac{x}{r}\right) = 9 \] This simplifies to: \[ y - 4x = 9 \] Now, we can rearrange this equation to express \( y \) in terms of \( x \): \[ y = 4x + 9 \] Thus, the equivalent Cartesian equation is: \[ y = 4x + 9 \] This is the equation of a straight line with a slope of 4 and a y-intercept of 9. ### Identification of the Graph The graph of the equation \( y = 4x + 9 \) is a straight line. In summary, the equivalent Cartesian equation is: \[ y = 4x + 9 \]