How to express $r$ in $\phi$ with equation $r \cos(\phi) + r^2\sin(\phi) = 2$?

Question

If we convert an equation in Cartesian coordinates to polar coordinates, can we always represent one variable by the rest?

Answer 1

$x=r \cos\phi;\;y=r\sin\phi$

$r=\sqrt{x^2+y^2}$

The equations can be written as

$r\cos\phi+ r(r\sin\phi)=2$ and then

$x+y\sqrt{x^2+y^2}=2$

Answer 2

Hint:

This is a quadratic equation in $r$, waiting for you to solve it.