Elements of Algebra

CHAP. I. OF ALGEBRA. 187 certain conditions, or properties, which were to agree with the quantity sought. 566. In Algebra, when we have a question to resolve, we represent the number sought by one of the last letters of the alphabet, and then consider in what manner the given conditions can form an equality between two quantities. This equality is represented by a kind of formula, called an equation, which enables us finally to determine the value of the number sought, and consequently to resolve the question. Sometimes several numbers are sought; but they are found in the same manner by equations. 567. Let us endeavour to explain this farther by an example. Suppose the following question, or problem, was proposed: Twenty persons, men and women, dine at a tavern; the share of the reckoning for one man is 8 shillings, for one woman 7 shillings, and the whole reckoning amounts to 7l. 5s. Required the number of men and women separately? In order to resolve this question, let us suppose that the number of men is = $x$; and, considering this number as known, we shall proceed in the same manner as if we wished to try whether it corresponded with the conditions of the question. Now, the number of men being = $x$, and the men and women making together twenty persons, it is easy to determine the number of the women, having only to subtract that of the men from 20, that is to say, the number of women must be $20-x$. But each man spends 8 shillings; therefore $x$ number of men must spend $8x$ shillings. And since each woman spends 7 shillings, $20-x$ women must spend $140-7x$ shillings. So that adding together $8x$ and $140-7x$, we see that the whole 20 persons must spend $140+x$ shillings. Now, we know already how much they have spent; namely, 7l. 5s. or 145s; there must be an equality, therefore, between $140+x$ and 145; that is to say, we have the equation $140+x=145$, and thence we easily deduce $x=5$, and consequently $20-x=20-5=15$; so that the company consisted of 5 men, and 15 women. 568. Again, Suppose twenty persons, men and women, go to a tavern; the men spend 24 shillings, and the women as much: but it is found that the men have spent 1 shilling each more than the women. Required the number of men and women separately? Let the number of men be represented by $x$.