Convert the following Cartesian equation into a polar equation. $y=4 x+7$

Step 1 ：We are given the Cartesian equation in the form of y = mx + c, where m is the slope and c is the y-intercept. The equation is y = 4x + 7.

Step 2 ：To convert this into polar coordinates, we use the relationships x = rcos(θ) and y = rsin(θ), where r is the distance from the origin to the point (x, y) and θ is the angle from the positive x-axis to the line connecting the origin and the point (x, y).

Step 3 ：Substitute x = rcos(θ) and y = rsin(θ) into the given equation, we get rsin(θ) = 4rcos(θ) + 7.

Step 4 ：Rearrange the equation, we get -rsin(θ) + 4rcos(θ) + 7 = 0.

Step 5 ：Solve for r, we get r = 7/(sin(θ) - 4cos(θ)).

Step 6 ：Thus, the polar equation is \(\boxed{r = \frac{7}{\sin(\theta) - 4\cos(\theta)}}\).