Chapter 3, Simultaneous Problems, part 1

It often happens that two or three problems are so entangled up together that it seems impossible to solve any one of them until the others have been solved. For instance, we might get out three answers of this kind:

x equals half of y; y equals twice x; z equals x multiplied by y.

The value of each depends on the value of the others.

When we get into a predicament of this kind, three courses are open to us. We can begin to make slapdash guesses, and each argue to prove that his guess is the right one; and go on quarrelling; and so on; as I described people doing about arithmetic before Algebra was invented. Or we might write down something of this kind: The values cannot be known. There is no answer to our problem. We might write: and accept those as answers and give them forth to the world with all the authority which is given by big print, wide margins, a handsome binding, and a publisher in a large way of business; and so make a great many foolish people believe we are very wise. Some people call this way of settling things Philosophy; others call it arrogant conceit. Whatever it is, it is not Algebra. The Algebra way of managing is this: We say: Suppose that x were Unity (1); what would become of y and z? Then we write out our problem as before; only that, wherever there was x, we now write 1.