47. A box contains 550 coins of silver and gold. Each silver coin is worth as many cents as there are gold coins, and each gold coin is worth as many cents as there are silver coins, and the whole is worth $500. How many coins are there of each kind? 48. There is a number consisting of two digits whose sum is 12 and the sum of whose squares is 90. What is the number? 49. A and B start at the same time to walk 20 miles. A travels a mile an hour faster than B and arrives an hour earlier. At what rate did each travel? 50. Find three consecutive numbers whose sum is equal to of the product of the last two. 51. The cost of lunch for a party was $15. If each paid 20 cents more than there were persons, how many were in the party? 52. A party had a dinner that cost $50. If there had been 5 persons more, the share of each would have been 50 cents less. How many persons were in the party? 53. Find two consecutive numbers, the sum of whose reciprocals is . 54. A man worked a certain number of days for $ 45. Had he received a dollar a day less than he did, he would have had to work 16 days longer for the same money. How many days did he work? 55. What number is greater than its reciprocal? 56. The frame of a picture 18 inches by 12 inches is of uniform width, and its area is equal to that of the picture. How wide is the frame? 57. If there were 2 fewer telegraph poles in a mile, the distance between the poles would be increased by 24 feet. If the poles are placed at equal intervals, find the number of poles in a mile. 58. A certain number is expressed by two digits whose product is 35, and if 18 is subtracted from the number, the order of digits will be reversed. Find the number, 59. By traveling 4 kilometers an hour faster than usual a train made a run of 675 kilometers in one hour less than its regular time. How many kilometers an hour did it travel? 60. A garden 56 yards long and 16 yards wide is surrounded by a walk of uniform width, without the garden. If the walk contains 640 square yards, what is its width? 61. The difference between two numbers is 9, and the square of the larger minus twice the square of the smaller is 158. Find the numbers. 62. If a bar of iron weighing 40 pounds be drawn out 2 feet longer, it will weigh one pound less per linear foot. What is its length? The larger 63. A man bought two farms for $2800 each. contained 10 acres more than the smaller, but he paid $5 more an acre for the smaller than for the larger. How many acres were in each ? 64. A man bought a certain quantity of sugar for $66. If sugar were to rise 1 cent a kilo, he would obtain 50 kilos less for the same money. How many kilos of sugar did he buy? 65. A rectangular lot 20 rods by 15 rods is surrounded by a fence within which is a drive occupying as much area as the rest of the lot. Find the width of the drive. 66. A rectangular lawn 20.5 meters long and 8.5 meters wide is surrounded by a path of uniform width, wholly without the lawn. If the area of the path is 62 square meters, what is its width? 67. If the circumference of the fore wheel of a buggy is 1 foot less than that of the hind wheel, and the former makes 48 more revolutions than the latter in going a mile, what is the circumference of each wheel? 68. A cistern is fitted with two pipes, it 3 minutes sooner than B can empty it. the cistern will be filled in 18 minutes. be filled if A is open and B is closed? A and B. A can fill SIMULTANEOUS QUADRATIC EQUATIONS 362. In a system of quadratic equations, as in a linear system, there must be as many equations given as there are unknown numbers, otherwise there can be no definite solution obtained. The given equations must be consistent and independent (Arts. 221-223). In simultaneous quadratic equations, as in simple equations, a solution requires the elimination of all but one of the unknown numbers; and, generally, by elimination, two simultaneous quadratic equations containing two unknown numbers will produce an equation of the fourth degree, which is usually insolv able by means of quadratics. However, a solution may be effected by the methods for solving quadratics in the following special cases: 1. When one equation is linear. 2. When each equation is quadratic and one is homogeneous, or when both are homogeneous (Art. 106). 3. When the equations are symmetric with respect to the unknown numbers when the unknown numbers can be interchanged without affecting the equations (Art. 107). Certain simultaneous equations that do not fall under the preceding classes may be readily solved by special devices. Often equations that belong to one or more of these classes can be solved more expeditiously by means of such devices. 363. It will tend to simplicity to consider first a particular problem in which each of the equations contains only the squares of the unknown numbers. Obviously in this case one of the unknown numbers may be eliminated by addition or subtraction, and the value of the other be found by substitution. Substituting either of these values of x in (1), we find y = ± 2. is, x and y each has two values. When x 3, y=2; when x y=2. Hence the following pairs of values satisfy both equations: = That - 3, Verify Are the given equations homogeneous with respect to x and y? these results by substituting the pairs of values in the given equation. 364. One Equation Linear. In this case, one of the unknown numbers can be found in terms of the other from the linear equation, and that value substituted in the quadratic. The resulting equation will be a quadratic in one unknown number, which may always be solved. Certain special examples in which one equation is of the third degree and the other of the first may be solved in a similar manner. Note that the equations of the given system are simultaneous, but they are not simultaneous quadratics, since one of the equations is of the first degree. In this case we should substitute in the linear equation the values found for one unknown number in order to obtain the values of the other. This is important because the quadratic equation and the equation obtained by substituting in it the value of one unknown number derived from the linear equation are not necessarily equivalent to the given system. In this example, the system (1), (3) is not necessarily equivalent to the given system (1), (2). How many pairs of roots result from this solution? How many pairs should be expected when both equations are quadratic? |