  Consider the set $\{1,2,\dots,n\} = [n]$ and an equation $eq$. The rainbow
number of $[n]$ for $eq$, denoted $\operatorname{rb}([n],eq)$, is the smallest
number of colors such that for every exact $\operatorname{rb}([n],
eq)$-coloring of $[n]$, there exists a solution to $eq$ with every member of
the solution set assigned a distinct color. This paper focuses on linear
equations and, in particular, establishes the rainbow number for the equations
$\sum_{i=1}^{k-1} x_i = x_k$ for $k=3$ and $k=4$. The paper also establishes a
general lower bound for $k \ge 5$.
