  We show that a complete hereditary cotorsion pair $(\C,\C^\bot)$ in an exact
category $\E$, together with a subcategory $\Z\subseteq\E$ containing
$\C^\bot$, determines a Waldhausen category structure on the exact category
$\C$, in which $\Z$ is the class of acyclic objects.
  This allows us to prove a new version of Quillen's Localization Theorem,
relating the $K$-theory of exact categories $\A\subseteq\B$ to that of a
cofiber. The novel idea in our approach is that, instead of looking for an
exact quotient category that serves as the cofiber, we produce a Waldhausen
category, constructed through a cotorsion pair. Notably, we do not require $\A$
to be a Serre subcategory, which produces new examples.
  Due to the algebraic nature of our Waldhausen categories, we are able to
recover a version of Quillen's Resolution Theorem, now in a more homotopical
setting that allows for weak equivalences.
