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Rado numbers are closely related to Ramsey numbers, but pertaining to equations and integers instead of cliques within graphs. For every integer \( m \geq 3 \) and every integer \( c \), let the 2-color Rado number \( r(m,c) \) be the least integer, if it exists, such that for every 2-coloring of the set \( \{1,2,\ldots,r(m,c)\} \) there exists a monochromatic solution to the equation
\[
\sum_{i=1}^{m-1} x_i + c = x_m
\]
The values of \( r(m,c) \) have been determined previously for nonnegative values of \( c \), as well as all values of \( m \) and \( c \) such that \( -m+2 < c < 0 \) and \( c < -(m-1)(m-2) \). In this paper, we find \( r(m,c) \) for the remaining values of \( m \) and \( c \).