Then the difference will be 3xy + 3xy2=a3-b, Having now found x+y and xy, the values of x and y may be found as in the second Example. Raise x+y to the 4th power, and x*+4x3y+6x2y2+4xy3+y1=a* ; Subtract from this the 2d equation, x* 4x3y + 6x2y3 + 4xy3 = aa — b. Let now a-b=c (A) xy Divide the remainder by ry, and we have 4x2+6xy+4y2=· Now the first equation squared is.... ..x2 + 2xy + y2=a2; 4x2+8xy+4y2=4a3 (B) Therefore, From (B) the greater take (A) the less, then 2.ry=4a2——; Clear the remainder of the fraction, and we get 2xy=4a2xy—c. By transposition and division, x2y2-2a2xy=—c. Hence we have xy-a2= ±√a*—c, Wherefore x and y may now be found as in Example second. (85.) 7. Let x+y=a, and x+y=b. Then the first equation raised to the 5th power, x3+5x3y+10x3y2+10x2y°+5x¥*+y3=a* ; The second ditto is by the question,} +y=b. Therefore the difference, 5x y +10x3y2+10x2y3+5xy"—a—b=c, C Hence, by subtraction,.. aay+xy2=a3— 5xy' From which equation the value of xy may be found And then x and y may be found as in Example 2. (86.) 8. Given x+y=s, and xy=p; to find the value of x2+yo, ±3 + y3; x*+y* ; &c. in terms of (s) and (p). First, the square of the 1st equation, x+2xy + y2=so ; Therefore, by transposition, x+y2=s2—2xy=s3—2p. And, thirdly, .... $3-2ps, x2+ps+y3 = s3-2ps; = s3—3ps. (x3+y3) × (x+y) (-3ps)s, Or, which is the same, x+xy x (x2+y2)+y* = = s*-3ps2, 9. There are two numbers, the sum of which is 19, and the sur of their squares is 193; required those numbers. Ans. 12 and 7 10. The sum of the squares of two numbers is 261, and their product is 90; required those two numbers. Ans. 15 and 6 11. The sum of two numbers is 9, and the sum of their cubes is 351; required those two numbers. Ans. 7 and 2 12. The sum of two numbers is 7, and the sum of their 4th Ans. 5 and 2. powers is 641; what are those numbers? 13. The sum of two numbers is 6, and the sum of their 5th Ans. 2 and 4. powers is 1056; what are those numbers? 14. The sum of two numbers is 13, and their product is 30; it is required to find the sum of their 4th powers. Ans. 10081. (87.) The THIRD METHOD.-Equations of the form Sax2 + bxy=c in which the sum of the indices of the unknown quantities is the same in every term of both equations, are managed by substituting for one unknown quantity the product of the other and a third unknown quantity. Examples. (98) 1. Let bxy=a; and xy+dy2=c. Assume x=vy. Then, by substitution, we have vy—bvy2=a (A), And.. From the equation (A) ... vy2 + dy2 =e (B). Take the equation (B) y2= v+d' But having the value v, the value of y may be found from the e equation y2= v+d; and, consequently, the value of x (= vy) will be known. (89.) 2. What are those two numbers, the sum of which multiplied by the greater is 154; and whose difference multiplied by the less is equal to 24? Solu.-Let x the greater, and y = the less number; Then will (x+y) x=x2+xy=77, And (x-y) y=xy-y2=12. .. Therefore, v = or 1. 7 Now either of these values of v answers the conditions of the From which it appears that the numbers are 4 and 7. 3. It is required to find two numbers, such, that the square of the greater minus the square of the less may be 56, and the square of the less plus one-third of their product may be 40. Ans. 9 and 5. 4. There are two numbers, such, that thrice the square of the greater plus twice the square of the less is 110; and half their product plus the square of the less is 4; it is required to find these two Ans. 6 and 1. numbers. Miscellaneous Examples for practice. 1. What two numbers are those whose sum is 29, and their product 100? Ans. 25 and 4. 2. What two numbers are those wnose sum is 20, and their pro Auct 36? Ans. 2 and 18 3. To divide the number 33 into two such parts that their product shall be 162. Ans. 27 and 6. 4. To divide the number 60 into two such parts, that their product may be to the sum of their squares in the ratio of 2 to 5. Ans. 20 and 40. 5. The difference of two numbers is 5, and a fourth part of the. product is 26; required those two numbers. Ans. 13 and 8. 6. The difference of two numbers is 3, and the difference of their cubes is 117; what are those numbers? Ans. 2 and 5 7. The difference of two numbers is 6; and if 47 be added to twice the square of the less, it will be equal to the square of the greater; what are the numbers ? Ans. 11 and 17. 8. There are two numbers whose sum is 30; and one-third of their product plus 18 is equal to the square of the less number; it is required to find those numbers. Ans. 9 and 21. 9. What number is that which, when divided by the product of its two digits, the quotient is 3; and if 18 be added to it, the digits will be inverted : Ans. 24 10. There are two numbers whose product is 120: if 2 be added to the less, and 3 subtracted from the greater, the product of the sum and remainder will also be 120; what are these numbers ? Ans. 15 and 8. 11. What two numbers are those whose sum multiplied by the greater is equal to 77; and whose difference multiplied by the lesser is equal to 12? Ans. 4 and 7. 12. To resolve the number 6 into two such factors, that the sum of their cubes shall be 35. Ans. 3 and 2. 13. To divide the number 20 into two such parts, that their product shall be equal to 105 ? Ans. 10-5, and 10-√−5. 14. A grazier bought as many sheep as cost him 601. and, after reserving fifteen, he sold the remainder for 541. and gained 2s. a head by them; how many sheep did he buy? Ans. 75. 15. To find a number such, that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3. 16. The sum of two numbers is 8, and the sum of their cubes is 152; what are the numbers? Ans. 3 and 5. 17. Two partners, A and B, gained 1401. by trade; A.'s money was three months in trade, and his gain was 60l. less than his stock; and B.'s money, which was 501. more than A.'s, was in trade five months; what was A.'s stock? Ans. 1001. 18. To divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36. 19. A and B distribute 12001. each among a certain number of persons; A relieves 40 persons more than his friend B, and B gives to each person 51. a piece more than A. Now, how many persons were relieved by A and B respectively? Ans. 120 by A, and 80 by B. 20. The sum of two numbers is 6, and the sum of their 5th power s 1056; what are the numbers ? Ans. 2 and 4. 21. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. Ans. 10 and 14. 22. A and B set off at the same time to a place distant 300 miles : A travels at the rate of one mile an hour quicker than B, and gets to the end of his journey 10 hours before him. How many miles per hour did each person travel? Ans. A travelled 6, and B 5 miles an hour. 23. The sum of two numbers is 7, and the sum of their 4th power is 641; what are the numbers ? Ans. 2 and 5. 24. A company at a tavern had §1. 15s. to pay for their reckoning; but, before the bill was settled, two of them left the room, and they who remained had 10s a piece more to pay than before: how many were there in company? Ans. 7. |