Elements of Algebra

CHAP. IV. OF ALGEBRA. 215 $y = \frac{abcdn-abdn+abn-bn}{abcd-1} = \frac{n \cdot (abcd-abd+ab-b)}{abcd-1}$ $z = \frac{abcdn-abcn+bcn-cn}{abcd-1} = \frac{n \cdot (abcd-abc+bc-c)}{abcd-1}$ $u = \frac{abcdn-bcdn+cdn-dn}{abcd-1} = \frac{n \cdot (abcd-bcd+cd-d)}{abcd-1}$ 622. Question 7. A captain has three companies, one of Swiss, another of Swabians, and a third of Saxons. He wishes to storm with part of these troops, and he promises a reward of 901 crowns, on the following condition; namely, that each soldier of the company, which assaults, shall receive 1 crown, and that the rest of the money shall be equally distributed among the two other companies. Now, it is found, that if the Swiss make the assault, each soldier of the other companies will receive half-a-crown; that, if the Swabians assault, each of the others will receive $\frac{1}{3}$ of a crown; and, lastly, if the Saxons make the assault, each of the others will receive $\frac{1}{4}$ of a crown. Required the number of men in each company? Let us suppose the number of Swiss to be $x$, that of Swabians $y$, and that of Saxons $z$. And let us also make $x + y + z = s$, because it is easy to see, that, by this, we abridge the calculation considerably. If, therefore, the Swiss make the assault, their number being $x$, that of the other will be $s - x$: now, the former receive 1 crown, and the latter half-a-crown; so that we shall have, $x + \frac{1}{2}s - \frac{1}{2}x = 901$. In the same manner, if the Swabians make the assault, we have $y + \frac{1}{3}s - \frac{1}{3}y = 901$. And, lastly, if the Saxons make the assault, we have $z + \frac{1}{4}s - \frac{1}{4}z = 901$. Each of these three equations will enable us to determine one of the unknown quantities, $x$, $y$, and $z$; For the first gives $x = 1802 - s$, the second $2y = 2703 - s$, the third $3z = 3604 - s$. And if we now take the values of $6x$, $6y$, and $6z$, and write those values one above the other, we shall have $6x = 10812 - 6s$, $6y = 8109 - 3s$, $6z = 7208 - 2s$, and, by addition, $6s = 26129 - 11s$; or $17s = 26129$;