  Given a linear equation $\mathcal{E}$, the $k$-color Rado number
$R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of
$\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The
degree of regularity of $\mathcal E$, denoted $dor(\mathcal E)$, is the largest
value $k$ such that $R_k(\mathcal E)$ is finite. In this article we present new
theoretical and computational results about the Rado numbers $R_3(\mathcal{E})$
and the degree of regularity of three-variable equations $\mathcal{E}$.
  % We use SAT solvers to compute many new values of the three-color Rado
numbers $R_3(ax+by+cz = 0)$ for fixed integers $a,b,$ and $c$. We also give a
SAT-based method to compute infinite families of these numbers. In particular,
we show that the value of $R_3(x-y = (m-2) z)$ is equal to $m^3-m^2-m-1$ for
$m\ge 3$. This resolves a conjecture of Myers and implies the conjecture that
the generalized Schur numbers $S(m,3) = R_3(x_1+x_2 + \dots x_{m-1} = x_m)$
equal $m^3-m^2-m-1$ for $m\ge 3$. Our SAT solver computations, combined with
our new combinatorial results, give improved bounds on $dor(ax+by = cz)$ and
exact values for $1\le a,b,c\le 5 $. We also give counterexamples to a
conjecture of Golowich.
