One of the salespeople is wrong... which one?
We tried out a bunch of different substitution approaches but I can't figure this one out. If you can explain it to me, I can explain it to him.
On
Statement 1 is the same as "6 trinkets minus 5 gewgaws has the same value as 7 baubles", which is clearly inconsistent with statement 3. So one of these two sales people is wrong.
Similarly, statement 2 is the same as "5 gewgaws minus 4 baubles has the same value as 9 trinkets", which is inconsistent with statement 1. So one of these two sales people is wrong.
Thus sales person 1 is wrong. Assuming that all three items have a positive value.
Technically, none of the salespeople are wrong if trinkets, gewgaws, and baubles are worthless.
If this case is to be excluded, assign some arbitrary worth to one of the three items. For example, let trinkets be worth 1. Then we have three equations:
$$7b + 5g = 6$$ $$4b - 5g = -9$$ $$3g - 4b = -6$$
If all three salesmen valued baubles, gewgaws, and trinkets equally, then these three equations would represent lines that intersected at a single point: the agreed upon value for baubles and gewgaws when trinkets are assumed to be worth 1.
However, this is not the case. If salesman 1 and salesman 2 are correct, then baubles are worth -0.273 and gewgaws are worth 1.582. If salesman 1 and salesman 3 are correct, then baubles are worth 1.171 and gewgaws are worth -0.439. If salesman 2 and salesman 3 are correct, then baubles are worth 7.125 and gewgaws are worth 7.5.
If we further assume that no item is allowed to have negative value, then only salesman 2 and salesman 3 can be correct, but technically we still would have to prove this holds no matter what value we assign $t$.
However, value is subjective. I see no reason why three different salesmen can't have conflicting views about the values of objects. The fact that salesman one disagrees with two other salesmen about an exchange rate is insufficient proof that he is 'wrong' in my book.