7.5. Petya has 4 Soviet copper coins - one each of 1, 2, 3, and 5 kopecks. He learned from the internet the following fact: these coins should weigh exactly as many grams as their denomination. Petya wants to check this fact using a balance scale. Will he be able to do this if he has only one 9-gram weight?
Answer: It will. Solution: Let $x$ be the weight of a 1-kopeck coin, $y$ be the weight of a 2-kopeck coin. If the fact being checked is true, then $(x+y)$ is the weight of a 3-kopeck coin, and ($x+2y$) is the weight of a 5-kopeck coin. To verify this, we will perform two weighings: 1) 1 kopeck + 2 kopecks = 3 kopecks (?); 2) 2 kopecks + 3 kopecks = 5 kopecks (?). If at least one of these equalities does not hold (the scales are not in balance), then the fact being checked is false. If the equalities hold, we will verify by performing two more weighings the following equalities: 1 kopeck + 3 kopecks + 5 kopecks = 9 grams (?), and 2 kopecks + 3 kopecks + 5 kopecks = 1 kopeck + 9 grams (?). If at least one of these equalities does not hold, then the fact being checked is false. If both equalities hold, we have two equations: $x+x+y+x+2y=9$ and $y+x+y+x+2y=9+x$, from which we get $x=1, y=2$, i.e., the fact being checked is true.