Solve for x and y

$x=\frac {120121-13y}{9}$

For $x$ to be maximum, $y$ has to be the minimum multiple of 13 such that $x$ is an integer. That occurs at $y=13×6=78$ .

Thus, $x_{max}=1327$

Or, you can solve the linear Diophantine equation to get

$x=36063+13n \ and \ y=-24042-9n$

and find the smallest $n$ , for which $y=-24042-9n$ is positive. that would be $n=-2672$ . So

$x=36063+13n \implies x=36063+13(-2672)=1327$