Main theorem of elimination theory

Okay kiddo, imagine you have five colored marbles in a bag, and you want to know if you can get rid of any of them by combining them in different ways. This is called elimination theory, and it helps us solve problems like this one.

Now, imagine you also have a bunch of equations written down on a piece of paper. These equations represent relationships between the variables in the problem you're trying to solve. For example, you might have an equation that says "the sum of the first two marbles is equal to the third marble."

The main theorem of elimination theory tells us that we can use these equations to figure out which combinations of marbles can be eliminated, or written in terms of the other marbles. In other words, we can use the equations to simplify the problem and make it easier to solve.

To do this, we use a process called elimination. This involves adding or subtracting different combinations of the equations to get rid of certain variables and reduce the problem to a simpler form. For example, if we want to eliminate the third marble, we might add the equations "the sum of the first two marbles is equal to the third marble" and "the sum of the last two marbles is equal to the third marble," which would give us the equation "the sum of the first and last marbles is equal to twice the third marble."

By applying this process of elimination to all the variables in the problem, we can eventually solve for each of the marbles and determine which combinations can be eliminated. And that, my little friend, is the main theorem of elimination theory!