91. A and B together can do a piece of work in 14 days, A and C in 14 days, and B and C in 23 days. How many days will it take each working alone? 92. A man buys two pieces of cloth, one of which contains 3 yards more than the other. For the larger piece he pays at the rate of $5 for 6 yards, and for the other at the rate of $7 for 5 yards. He sells the whole at the rate of 3 yards for $4, and makes $8 by the transaction. How many yards were there in each piece? 93. A gentleman distributing some money among beggars, found that in order to give them a cents each, he should need b cents more. He therefore gave them c cents each, and had d cents left. Required the number of beggars. 94. A man let a certain sum for 3 years at 5 per cent compound interest; that is, at the end of each year there was added to the sum due. At the end of the third year there was due him $2315.25. Required the sum let.. 95. A man starts in business with $4000, and adds to his capital annually one-fourth of it. At the end of each year he sets aside a fixed sum for expenses. At the end of three years, after deducting the fixed sum for expenses, his capital is reduced to $2475. What are his annual expenses? 96. A man invests one-third of his money in 35 per cent bonds, two-fifths in 4 per cent bonds, and the balance in 41 per cent bonds. His income from the investments is $595. What is the amount of his property? 97. At what time between 8 and 9 o'clock is the minutehand of a watch exactly 35 minutes in advance of the hourhand? 98. A fox is pursued by a greyhound, and has a start of 60 of her own leaps. The fox makes 3 leaps while the greyhound makes but 2; but the latter in 3 leaps goes as far as the former in 7. How many leaps does each make before the greyhound catches the fox? XIV. SIMPLE EQUATIONS. CONTAINING TWO UNKNOWN QUANTITIES. 180. If we have a simple equation containing two unknown quantities, as x + y = 12, it is impossible to determine the values of x and y definitely; because, if any value be assumed for one of the quantities, we can find a corresponding value for the other. Thus, if x = 9, then 9 + y = 12, or y 3; if x=8, then 8 + y = 12, or y = 4; etc. Hence, any of the pairs of values, x=9, y = 3 ; x =8, y = 4; etc., will satisfy the given equation. Similarly, the equation x - y = 4 is satisfied by any of the following pairs of values: x = 9, y = 5; x = 8, y = 4; etc. Equations of this kind are called indeterminate. But suppose we are required to find a pair of values which will satisfy both x + y = 12 and x - y = 4 at the same time. It is evident by inspection that the values x = 8, y = 4 satisfy both equations; and no other pair of values can be found which will satisfy both simultaneously. 181. Simultaneous Equations are such as are satisfied by the same values of their unknown quantities. Independent Equations are such as cannot be made to assume the same form. Thus, x + y = 9 and x - y = 1 are independent equations. But x+y=9 and 2x+2y=18 are not independent, since the first equation may be obtained from the second by dividing each term by 2. 182. It is evident from Art. 180 that two unknown quantities require for their determination two independent, simultaneous equations. Two such equations may be solved by combining them so as to form a single equation containing but one unknown quantity. This operation is called Elimination. 183. There are three principal methods of eliinination: 1. By Addition or Subtraction. 2. By Substitution. 3. By Comparison. Substituting the value of x in (1), 10 -3y = 19 -3y= 9 Whence, y=-3 Ans. x=2, y = - 3. This solution is an example of elimination by addition. 24 Substituting this value in (2), 10x - 14 = 10x10 x=- 1 Ans. x=- 1, y = 2. This solution is an example of elimination by subtraction. RULE. Multiply the given equations by such numbers as will make the coefficients of one of the unknown quantities equal. Add or subtract the resulting equations according as the equal coefficients have unlike or like signs. Note. If the coefficients which are to be made equal are prime to each other, each may be used as the multiplier for the other equation. If they are not prime, such multipliers should be used as will produce their lowest common multiple. Thus, in Ex. 1, to make the coefficients of y equal, we multiply (1) by 4, and (2) by 3. But in Ex. 2, to make the coefficients of x equal, since the L.C.M. of 15 and 10 is 30, we multiply (1) by 2, and (2) by 3. EXAMPLES. Solve the following by the method of addition or subtrac |