Elements of Algebra

CHAP. II. OF ALGEBRA. 313 $20 - p$, which evidently is also less than 30. Now, as we may, in this equation, assume all numbers for $p$ which do not exceed 10, we shall have the following eleven answers: the number of men $p$, of women $q$, and of children $r$, being as follow: $p = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10;$ $q = 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0;$ $r = 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20;$ and, if we omit the first and the last, there will remain 9. 26. Question 2. A certain person buys hogs, goats, and sheep, to the number of 100, for 100 crowns; the hogs cost him $3\frac{1}{2}$ crowns apiece; the goats, $1\frac{1}{3}$ crown; and the sheep, $\frac{1}{2}$ a crown. How many had he of each? Let the number of hogs be $p$, that of the goats $q$, and of the sheep $r$, then we shall have the two following equations: 1. $p + q + r = 100$, 2. $3\frac{1}{2}p + 1\frac{1}{3}q + \frac{1}{2}r = 100$; the latter of which being multiplied by 6, in order to remove the fractions, becomes, $21p + 8q + 3r = 600$. Now, the first gives $r = 100 - p - q$; and if we substitute this value of $r$ in the second, we have $18p + 5q = 300$, or $5q = 300 - 18p$, and $q = 60 - \frac{18p}{5}$: consequently, $18p$ must be divisible by 5, and therefore, as 18 is not divisible by 5, $p$ must contain 5 as a factor. If we therefore make $p = 5s$, we obtain $q = 60 - 18s$, and $r = 13s + 40$; in which we may assume for the value of $s$ any integer number whatever, provided it be such, that $q$ does not become negative: but this condition limits the value of $s$ to 3; so that if we also exclude 0, there can only be three answers to the question; which are as follow: When $s = 1, 2, 3,$ We have $\left\{ \begin{array}{l} p = 5, 10, 15, \\ q = 42, 24, 6, \\ r = 53, 66, 79. \end{array} \right.$ 27. In forming such examples for practice, we must take particular care that they may be possible; in order to which, we must observe the following particulars: Let us represent the two equations, to which we were just now brought, by 1. $x + y + z = a$, and 2. $fx + gy + hz = b$, in which $f, g$, and $h$, as well as $a$ and $b$, are given numbers.