Formulating equation for two mutual exclusive integers I have a problem with formulating non-binary linear equation with mutual exclusivity
\begin{cases}
x_1 \gt 0, \quad \text{OR} &x_2 \gt 0 \\[2ex]
x_1 + x_2 = 300
\end{cases}
As the result I want the equation to have only two possible solutions $x_1 = 300$ or $x_2 = 300$.

No way to solve this as an LP. But we can solve this as a MIP (Mixed Integer Programming) problem:
$$
\begin{array}{l}
x_1 \le 300 \delta \\
x_2 \le 300 (1-\delta) \\
x_1+x_2 = 300\\
\delta \in \{0,1\}\\
x_1,x_2 \ge 0
\end{array}
$$ 
To be complete: some MIP solvers also allow socalled SOS1 variables which do a similar thing.
