Elements of Algebra

188 ELEMENTS SECT. IV. Then the women will be $20-x$. Now, the $x$ men having spent 24 shillings, the share of each man is $\frac{24}{x}$. The $20-x$ women having also spent 24 shillings, the share of each woman is $\frac{24}{20-x}$. But we know that the share of each woman is one shilling less than that of each man; if, therefore, we subtract 1 from the share of a man, we must obtain that of a woman; and consequently $\frac{24}{x}-1 = \frac{24}{20-x}$. This, therefore, is the equation, from which we are to deduce the value of $x$. This value is not found with the same ease as in the preceding question; but we shall afterwards see that $x=8$, which value answers to the equation; for $\frac{24}{8}-1 = \frac{24}{12}$ includes the equality $2=2$. 569. It is evident therefore how essential it is, in all problems, to consider the circumstances of the question attentively, in order to deduce from it an equation that shall express by letters the numbers sought, or unknown. After that, the whole art consists in resolving those equations, or deriving from them the values of the unknown numbers; and this shall be the subject of the present section. 570. We must remark, in the first place, the diversity which subsists among the questions themselves. In some, we seek only for one unknown quantity; in others, we have to find two, or more; and, it is to be observed, with regard to this last case, that, in order to determine them all, we must deduce from the circumstances, or the conditions of the problem, as many equations as there are unknown quantities. 571. It must have already been perceived, that an equation consists of two parts separated by the sign of equality, =, to shew that those two quantities are equal to one another; and we are often obliged to perform a great number of transformations on those two parts, in order to deduce from them the value of the unknown quantity: but these transformations must be all founded on the following principles; namely, That two equal quantities remain equal, whether we add to them, or subtract from them, equal quantities; whether we multiply them, or divide them, by the same number; whether we raise them both to the same power, or extract their roots of the same degree; or lastly,