We give a precise algebraic characterization of the power of Sherali--Adams relaxations for solvability of valued constraint satisfaction problems (CSPs) to optimality. The condition is that of bounded width, which has already been shown to capture the power of local consistency methods for decision CSPs and the power of semidefinite programming for robust approximation of CSPs. Our characterization has several algorithmic and complexity consequences. On the algorithmic side, we show that several novel and well-known valued constraint languages are tractable via the third level of the Sherali--Adams relaxation. For the known languages, this is a significantly simpler algorithm than those previously obtained. On the complexity side, we obtain a dichotomy theorem for valued constraint languages that can express an injective unary function. This implies a simple proof of the dichotomy theorem for conservative valued constraint languages established by Kolmogorov and Živný [J. ACM, 60 (2013), 10], and also a dichotomy theorem for the exact solvability of minimum-solution problems. These are generalizations of minimum-ones problems to arbitrary finite domains. Our result improves on several previous classifications by Khanna et al. [SIAM J. Comput., 30 (2001), pp. 1863--1920], Jonsson, Kuivinen, and Nordh [SIAM J. Comput., 38 (2008), pp. 329--365], and Uppman [Proceedings of ICALP'13, Springer, Berlin, 2013, pp. 804--815].