Pam owns an inventory of unopened packages of corn and rice

This question is based on the concept of "integral solutions". Since in such real world examples, x and y cannot be negative or fractional, usually these equations have a finite number of solutions.
After you get one solution, you will quickly know how many solutions the equation has. But getting the first set of values which satisfy the equation requires a little bit of brute force.
But the good thing here is that this is a DS question. You don't need to find the actual solution. The only thing you "need" is to establish that there is a single solution only. (Obviously, there has to be a solution since she does own $282 worth of packages.)

That can be done relatively easily.
First, check out this post for a conceptual discussion on this question type (case 2):
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/06 ... -of-thumb/

Once you understand this, the following will make sense to you.

17x + 13y = 282
If x = 0, y is 21.something (not an integer)
If x = 1, y = 20.something
If x = 2, y = 19.something
If x = 3, y = 17.something

Now you know that there will be only one set of values satisfying this equation. Why? Because y will be less than 17 in the first set of values which satisfy this equation. So if you want to get the next set that satisfies, you will need to subtract it by 17 which will make y negative. So in any case, there will be a unique solution to this equation.
Hence statement 1 is sufficient.

Answer (A)