Solve In :
If we rearrange the equation we find
$\displaystyle y = \frac{1000 - 19x}{13}$.
You need to substitute an integer $\displaystyle x$ in order to get an integer $\displaystyle y$.
For this to happen, $\displaystyle 1000 - 19x$ needs to be a multiple of 13.
Have a go from here...
$\displaystyle y = \frac{1001-1 - 13x-6x}{13}$
$\displaystyle y = 77-x-\frac{1+ 6x}{13}$.
The smallest value of x that will satisfy is 2. Since gcd(6,13)=1, the next next value will be 2+13=15, the next 28 and so on.
The largest value of x is less than or equal to $\displaystyle [\frac{1000}{19}]$(Why?)