Examples to be solved by Elimination by Substitution. as you retain ;" B says to C, "give me $ 1400, and I shall have thrice as much as you have remaining;" C says to A, "give me $420, and then I shall have 5 times as much as you retain." How much has each? Ans. A $980, B $ 1540, C $ 2380. 7. Three soldiers, in a battle, make $96 booty, which they wish to share equally. In order to do this, A, who made most, gives B and C as much as they already had; in the same manner, B then divided with A and C; and after this, C with A and B. If, by these means, the intended equal division is effected, how much booty did each soldier make? Ans. A $52, B $ 28, C $ 16. 8. A, B, C, D, E play together on this condition, that he who loses shall give to all the rest as much as they already have. First A loses, then B, then C, then D, and at last also E. All lose in turn, and yet at the end of the 5th game they all have the same sum, viz. each $32. How much had each when they began to play? Ans. A $81, B $ 41, C $21, D $11, E $6. 153. Second Method of solving the Problem of art. 142, called that of Elimination by Comparison. Find the value of either of the unknown quantities in all the equations in which it is contained; place either of the values thus obtained equal to each of the others, and the equations thus formed will be one less in number than those from which they are obtained, and will contain one unknown quantity less. By continuing this process on these new equations, the number of equations will finally be reduced to one. Examples to be solved by Elimination by Comparison. 154. EXAMPLES. 1. To solve any two equations of the first degree with two unknown quantities. Solution. These equations may, as in art. 146, be reduced to the forms Ax+By+ M = 0, A'x+By+M' = 0. The values of x, obtained from these equations, are which, being placed equal to each other, give being the same values as those obtained in art. 146. Examples to be solved by Elimination by Comparison. Solution. The values of x, obtained from these equations, are x= 12 y z the first of which being placed equal to each of the others gives, by reduction, %= 4, y = 3; whence we get, from either value of x, by substitution, 7 A person has two horses, and two saddles, one of which cost $50, the other $2. If he places the best saddle upon the first horse, and the worst upon the second, then the latter is worth $8 less than the other; but if he puts the worst saddle upon the first horse, and the best upon the other, then the latter is worth 32 times as much as the first. What is the value of each horse? Ans. The first $30, the second $70. 8. What fraction is that, whose numerator being doubled, and denominator increased by 7, the value becomes ; but the denominator being doubled, and the numerator increased by 2, the value becomes ? Ans. t. 9. A wine merchant has two kinds of wine. If he mix 3 gallons of the worst with 5 of the best, the mixture is worth $1 per gallon; but, if he mix 3 gallons of the worst with 8 gallons of the best, the mixture is worth $1,033 per gallon. What does each wine cost per gallon? Ans. The best $1,12, the worst $ 0,80. 10. A wine merchant has two kinds of wine. If he mix a gallons of the first with b gallons of the second, the mixture is worth c dollars per gallon; but, if he mix a' gallons of the first with b' gallons of the second, the mixture is worth c' dollars per gallon. What does each wine cost per gallon? 11. Three masons, A, B, C, are to build a wall. A and B, jointly, could build this wall in 12 days; B and C could accomplish it in 20 days; A and C would do it in 15 days. What time would each take to do it alone? Ans. A requires 20, B 30, C 60 days. 12. Three laborers are employed in a certain work. A and B would, together, complete it in a days; A and C require b days; B and C require c days. In what time would each accomplish it singly? 13. A cistern may be filled by three pipes, A, B, C. By the pipes A and B, it could be filled in 70 minutes; by the pipes A and C, in 84 minutes; and by the pipes B and C, in 140 minutes. In what time would each pipe fill it? Ans. A in 105, B in 210, C in 420 minutes. 14. A, B, C play faro. In the first game A has the bank, B and C stake the third part of their money, and win. In the second game B has the bank, A and C stake the third part of their money and also win. Then C takes the bank, A and B stake the third part of their money and also win. After this third game they count their money, and find that they have all the same sum of 64 ducats. How much had each when they began to play? Ans. A had 75, B 63, C 54. |