Elements of Algebra

318 ELEMENTS PART II. several divisors, we shall also have several values of $x$, as the following Table will shew: The divisors of 80 are $1 \quad 2 \quad 4 \quad 5 \quad 8 \quad 10 \quad 16 \quad 20 \quad 40 \quad 80$ therefore $x = 0 \quad 1 \quad 3 \quad 4 \quad 7 \quad 9 \quad 15 \quad 19 \quad 39 \quad 79$ and $y = 79 \quad 39 \quad 19 \quad 15 \quad 9 \quad 7 \quad 4 \quad 3 \quad 1 \quad 0$ But as the answers in the bottom line are the same as those in the first, inverted, we have, in reality, only the five following; viz. $x = 0, 1, 3, 4, 7$, and $y = 79, 39, 19, 15, 9$. 33. In the same manner, we may also resolve the general equation $xy + ax + by = c$; for we shall have $xy + by = c - ax$, and $y = \frac{c-ax}{x+b}$, or, dividing $c-ax$ by $x+b$, $y = -a + \frac{ab+c}{x+b}$; that is to say, $x+b$ must be a divisor of the known number $ab+c$; so that each divisor of this number gives a value of $x$. If we therefore make $ab+c = fg$, we have $y = \frac{fg}{x+b} - a$; and supposing $x+b=f$, or $x=f-b$, it is evident that $y=g-a$; and, consequently, that we have also two answers for every method of representing the number $ab+c$ by a product, such as $fg$. Of these two answers, one is $x=f-b$, and $y=g-a$; and the other is obtained by making $x+b=g$, in which case $x=g-b$, and $y=f-a$. If, therefore, the equation $xy + 2x + 3y = 42$ were proposed, we should have $a=2$, $b=3$, and $c=42$; consequently, $y = \frac{48}{x+3} - 2$. Now, the number 48 may be represented in several ways by two factors, as $fg$: and in each of those cases we shall always have either $x=f-3$, and $y=g-2$; or else $x=g-3$, and $y=f-2$. The analysis of this example is as follows: $\begin{array}{|l|rr|rr|rr|rr|rr|} \hline \text{Factors} & \multicolumn{2}{c|}{1 \times 48} & \multicolumn{2}{c|}{2 \times 24} & \multicolumn{2}{c|}{3 \times 16} & \multicolumn{2}{c|}{4 \times 12} & \multicolumn{2}{c|}{6 \times 8} \\ \hline & x & y & x & y & x & y & x & y & x & y \\ \hline \text{Numbers} & -2 & 46 & -1 & 22 & 0 & 14 & 1 & 10 & 3 & 6 \\ \text{or} & 45 & -1 & 21 & 0 & 13 & 1 & 9 & 2 & 5 & 4 \\ \hline \end{array}$ 34. The equation may be expressed still more generally, by writing $mxy = ax + by + c$; where $a, b, c$, and $m$, are