Elements of Algebra

212 ELEMENTS SECT. IV. A = $8x - 4s$, B = $8y - 2s$, and c = $8z - s$. Now, each having at the end of this game 24 guineas, we have three equations, the first of which immediately gives $x$, the second $y$, and the third $z$; farther, $s$ is known to be 72, since the three persons have in all 72 guineas at the end of the last game; but it is not necessary to attend to this at first; since we have 1st $8x - 4s = 24$, or $8x = 24 + 4s$, or $x = 3 + \frac{1}{2}s$; 2d $8y - 2s = 24$, or $8y = 24 + 2s$, or $y = 3 + \frac{1}{4}s$; 3d $8z - s = 24$, or $8z = 24 + s$, or $z = 3 + \frac{1}{8}s$; and adding these three values, we have $x + y + z = 9 + \frac{7}{8}s$. So that, since $x + y + z = s$, we have $s = 9 + \frac{7}{8}s$; and, consequently, $\frac{1}{8}s = 9$, and $s = 72$. If we now substitute this value of $s$ in the expressions which we have found for $x$, $y$, and $z$, we shall find that, before they began to play, A had 39 guineas, B 21, and C 12. This solution shews, that, by means of an expression for the sum of the three unknown quantities, we may overcome the difficulties which occur in the ordinary method. 617. Although the preceding question appears difficult at first, it may be resolved even without algebra, by proceeding inversely. For since the players, when they left off, had each 24 guineas, and, in the third game, A and B doubled their money, they must have had before that last game, as follows: A = 12, B = 12, and c = 48. In the second game, A and c doubled their money; so that before that game they had; A = 6, B = 42, and c = 24. Lastly, in the first game, B and c gained each as much money as they began with; so that at first the three persons had: A = 39, B = 21, c = 12. The same result as we obtained by the former solution. 618. Question 5. Two persons owe conjointly 29 pistoles; they have both money, but neither of them enough to enable him, singly, to discharge this common debt: the first debtor says therefore to the second, If you give me $\frac{2}{3}$ of your money, I can immediately pay the debt; and the second answers, that he also could discharge the debt, if the other would give him $\frac{3}{4}$ of his money. Required, how many pistoles each had?