For no particularly good reason, note that our numbers are all divisible by $13$. So effectively it is the same problem if the hats cost $23$, the ties cost $16$, and the total fortune is $59500$.
There is some ambiguity, as it is not clear whether all the money has to be spent. Assume it must. Then we are looking for non-negative integer solutions of $23x+16y=59500$, with the side condition that $x\ge 2y$.
There is a general procedure for finding one integer solution to such an equation: use the Extended Euclidean Algorithm to find a solution of $23s+16t=1$ in integers, then multiply by $59500$.
Or else we can get lucky, and note that $(3)(23)+(1)(16)=85$ and $59500=(85)(700)$. So a solution in integers is $(x_0,y_0)$, where $x_0=2100$ and $y_0=700$.
Thus all integer solutions have the shape
$$x=2100- 16t, \qquad y=700+23t,$$
where $t$ is an integer, positive, zero, or negative.
Finally, we need to make sure that $x\ge 2y$ (and that $x$, $y$ are non-negative). This will give us three linear inequalities in $t$, one of them superfluous.