Algebraify

In many problems we can make great progress by encoding the problem as an algebraic equation or system of algebraic equations.

Example 1

Problem: Peter, Emma, and Kyler played chess with each other. Peter won 4 games and lost two games. Emma won 3 games and lost 3 games. If Kyler lost 3 games, how many games did he win?

Let $x$ be the number of games that Kyler won and let $n$ be the total number of games that Peter, Emma, and Kyler played together. If we sum all the wins and all the losses, then we will count the number of games played exactly twice! Therefore

4 + 2 + 3 + 3 + 3 + x = 2 n .

Thus $15 + x = n$ . Moreover, if we count the total number of losses, then we will count each game exactly once. Therefore

2 + 3 + 3 = n

Thus $n = 8$ total games were played. Putting this into the previous equation, we see that $15 + x = 16$ . Hence $x = 1$ , meaning that Kyler won one game.