Farkas's lemma

Imagine you have a bunch of toys and you want to divide them equally among your friends. Farkas's Lemma is like a rule that helps you figure out if you can do that or not.

Farkas's Lemma says that you can divide the toys equally among your friends if and only if you can find some weights (numbers) that you can multiply with the number of toys in each group so that the total of these weighted numbers is equal to zero.

Let's say you have 3 friends and 12 toys. You want to divide the toys equally among them. You can represent this situation as an equation:

3x + 3y + 3z = 12

where x, y, and z are the number of toys each friend gets.

To find out if this equation can be solved (i.e., if you can divide the toys equally), you can use Farkas's Lemma. You can assign some weights (numbers) to x, y, and z. Let's say you assign weights of 1, -1, and 0 respectively.

Then, you multiply these weights with the equation:

1(3x) + (-1)(3y) + 0(3z) = 12

Simplifying this equation, you get:

3x - 3y = 12

Now, you can see that this equation can never be equal to zero (which is what you need to divide the toys equally), no matter what values you assign to x, y, and z. Therefore, you cannot divide the toys equally among your friends.

Farkas's Lemma helps mathematicians solve many different kinds of problems, not just ones involving toys and friends. But the basic idea is the same - it helps you figure out if you can find a solution to a problem, based on some rules or conditions.