It is clear that we will not be buying many apples. For "large" problems, we would need general theory, or a suitable computer program. But this is a small problem, and algebra need not come into play.
You will be buying at most $5$ big apples.
Try buying $0$ big apples, and the rest small. It is easy to see that we can't get to exactly $1000$ using small apples, since $120$ does not divide $1000$ exactly.
Try buying $1$ big apple, and the rest small. Then we need to get to $800$ grams using only small apples. Can't be done, since $120$ is not a factor of $1000$.
Try buying $2$ big apples, and the rest small. That leaves $600$ grams to be made with small apples. This can be done, since $\frac{600}{120}=5$.
Try buying $3$ big apples, the rest small. Can't be done. Try buying $4$. Can't be done.
Try buying $5$ big apples. Works fine.
With $2$ big, we had a total of $7$ apples, with $5$ big we had a total of $5$ apples. So we get the largest number of apples by buying $2$ big, $5$ small, a total of $7$.
It might have been easier to see what's going on by knocking off a $0$, and using $12$, $20$, $100$. It might have been still easier to further divide by $4$, and using $3$, $5$, $25$.
Remark: Note that there are no $x$'s and/or $y$'s in the analysis.
For me, writing $120x+200y=1000$, and then perhaps $3x+4y=25$, would be automatic. But the rest would have been exactly like in the above solution: a scan of the possibilities, leading to a quick answer. The equation would only be used as an organizing device, easier (for me) than visualizing small yellow apples and big red apples. Remember that this is a concrete problem. Symbols can be very useful, but it is even more useful to experiment, in order to find out what's really going on.