How do I go about proving the integer linear program has an optimal solution, but that its linear program relaxation is unbounded? \begin{equation} \begin{array}{cl} {\max} & {x_1} \\ {\text{s.t.}} & {x_1 - \sqrt{5} x_2 = 0} \end{array} \end{equation}
I know that with the square root of 5 is irrational so I need to write down the LP relaxation and then prove how it is unbounded but I am confused how to go about that ..
Any help would be greatly appreciated.
To show that the LP relaxation is unbounded, let an arbitrary $M$ be given and construct a feasible solution $(x_1,x_2)$ for which the objective $x_1\ge M$. Explicitly, take $x_1=M$ and $x_2=M/\sqrt{5}$.