Unification problems can be formulated and investigated in an algebraic setting, by identifying substitutions to modal algebra homomorphisms. This opens the door to applications of the notorious duality between modal algebras and descriptive frames. Through substantial use of this correspondence, we give a necessary and sufficient condition for modal formulas to be projective. Applying this result to a number of different logics, we then obtain concise and lightweight proofs of their projective-or non-projective-character. In particular, we prove that the projective extensions of K5 are exactly the extensions of K45. This resolves the open question of whether K5 is projective.