If you have a bit of grounding in vector calculus, the next section provides a simple proof. Since every game has its own unique and separate equation, what we now have is a simultaneous equation model. So long as the number of games played exceed the number of teams, the system is saturated and a solution exists. X is a matrix indicating who won and who lost. R is a column vector of theoretical power ratings for each team.
E is a column vector containing errors.
Setting the derivative to 0 satisfies the first-order condition for a minimum. The proof is a little tedious, but it can be found in most mathematical statistics books. Ah, the beauty of mathematics. The first method tries to explain the entire winning margin in all game played. It favors teams that win and win big.