Farkas' lemma

Okay kiddo, let me tell you about Farkas' lemma. Basically, it's a way to figure out if there's a solution to a bunch of equations at the same time.

Imagine you have a bunch of toy blocks and you want to stack them together in a certain way. But you're not sure if it's possible. Farkas' lemma says that if you can't stack the blocks in the way you want, then there's a way to prove it using math.

You see, Farkas' lemma lets us look at a bunch of equations and figure out if they can all be true at the same time. We use something called linear programming to help us do this. It's like when you're trying to make a recipe and you need to use just the right amount of each ingredient.

So, Farkas' lemma tells us that if we can't find a solution that makes all the equations work, then there's another set of equations we can use to prove it. It's like having a backup plan in case things don't work out the way we wanted.

Overall, Farkas' lemma helps us solve problems involving a lot of equations all at once. It's pretty cool, huh?