3 3 Similarly by taking v= 5 we get x=125=5. 5 3 of 3=5; or of 5=3, 5 Hence x=3 or 5 and y=3 i.e., x=3 or 5 and y=5 or 3. There are very few quadratic equations which do not admit of being solved in more ways than one, and many of the methods adopted require considerable ingenuity and skill. These methods cannot be fully explained here, but as an illustration of one of them we will work the last equation by another method. x3+y3=152 x2y+xy2 = 120 } Multiply equation (2) by 3, and add the result to equation (1). Then x+3y+3xy2+y3=512. .. (x+y)=512 and x+y=8. Now equation (2) may be written, and xy=15. We have now x+y=8 (3) and xy=15 (4) .. y2 — 8y = — 15 y2-8y+42=1 And x=8-y .. y=5 or 3 Solve the equations, x+y+xy=11; x2y+xy2 = 30. From (2) ay(x+y)=30. :. x+ÿ= (3) 30 xy Substitute this value of x+y in equation (1) 30 .. +ay-11 and a2y2-11xy=-30. xy = Completing the square, (xy)3 — 11xy + (11)2 = 121 – 30=1 4 4 ... x+y=5 or 6 from equation (3) Having these two equations we may find x and y by the same method as was employed in finishing the last equation. The result is x=3, 2 or 5, 1 y=2, 3 or 1, 5. Lastly solve the equations, We x2+y=85; xy=42. may proceed by the 1st method; but the following method is instructive: By addition x2+y2+2xy=169 By subtraction a2+y2-2xy= Hence (x+y)2=169 (x-y)2= 1 - 1 •. x+y=+13 x―y=+ 1. By adding these equations we eliminate y, and by subtracting them we eliminate a. Thus +7 or +6 and y = ±6, +7. 1 1 5 (9) x+y=21; xy=61. (10) x+y==+== (11) x2+xy+y2=7; x—y=1. 2* (12) x2+xy+y2=52; xy-x2=8. (18) 2x2-3xy+y2=24; 3x2-5xy+2y2=33. (20) 2x2+3xy—4y2 = 10; x2-2xy +3y2=3. (21) x2+y2=146; x+y=16. (22)+y=9=x2- y2. Xx- -y (23) 3x2+5x-8y=36; 2x2-3x — 4y=3. (26) 2 9 xy +y x-y 10 + = x-y x+y 3; x2+y2=25. (27) x2=4x+3y; y2=4y+3x. (28) x2+y2−1=2xy; xy(xy+1)=420. Problems leading to Quadratic Equations with two 1. The sum of two numbers is 14 and their product 45. Find the numbers. 2. The difference of two numbers is 3 and the sum of their squares 65. Find the numbers. 3. Find two numbers, such that their sum divided by their difference may be 7, and the difference of their squares may be 144. 4. The sum of two numbers is 4 and the sum of their cubes is 28. Find the numbers. 5. The product of two numbers increased by their difference is 13, and their difference taken from the sum of their squares is 26. Find the numbers. 6. Find two numbers such that their product is equal to 3 times their difference, and their difference added to the sum of their squares 29 is 18. 7. Find two numbers such that their difference added to the difference of their squares is, and their sum added to the sum of their squares is 36. 43 8. Divide 24 into two parts whose product shall be to the sum of their squares as 3 to 10. 9. Divide 30 into two parts such that the sum of their squares divided by the difference of their squares may be 5 10. The hypotenuse of a right-angled triangled multiplied by one of the sides is 80 feet, but if multiplied by the other is only 60 feet. Find the three sides of the triangle. 11. The fore wheel of a coach makes 176 revolutions per mile more than the hinder one and the sum of their circumferences is 25 feet. Find the circumference of each. 12. There is a certain number of 2 digits, and if the number be divided by the product of the digits the quotient will be 3; and if 18 be added to the number the digits will be reversed. Find the number. 13. A person bought sorts of cloth, giving 14 shillings for the one sort and £4 19s. for the other. For every yard of the better sort he gave as many shillings as he had yards in all, and for every yard of the worse as many shillings as there were yards of the better sort more than the worse. How many yards of each sort did he buy? x2+xy=99 xy-y2=14 14. A vintner sold 7 dozen of sherry and 12 dozen of claret for £50, and he sold 3 dozen more of sherry for £10 than he did of claret for £6. Required the price of each. 15. A gentlemen left £210 to be divided among three servants in continued proportion, so that the first shall have £90 more than the last. Find the share of each. 16. Two men start at the same time from two towns A and B and each travels towards the other. When they meet it is found that one has travelled 84 miles more than the other, and that by continuing to travel at the same rate they will complete the journey in 9 and 16 days respectively. Find their rates of travelling. 16x-9y=84 16x 9y y х 17. A passenger and luggage train one travelling at 10 miles an hour less speed than the other set out at the same time, the one from London and the other from Edinburgh, a distance of 400 miles apart, and pass each other at a certain station on the road. The passenger train sets out to return from Edinburgh 2 hours after the luggage train starts from London and they pass each other at the same station. At what rates do they travel and how far is the station from London ? ANSWERS. EXERCISE II. Page 5. (1) 14x+6y+19z. (3) 9a+196+13z. (5) 20x+30y+24z. (2) 26x+24y+20z. (4) 8xy+16yz+18xz. (7) 3x+xy+2xz+4y+2yz+3z. (8) 4xy+5xy+6xz+9yz. (9) 12x+xy+2xz+10y+3yz+10z+2. (10) 17x+xy+2xz+27y+3yz+23z. Numerical results. (5) 152. (6) 156. (7) 40. (8) 112. (9) 90. (10) 166 (4) 2a-b+10c-12. (5) -6ab-5ac+8bc+6. (4) 2c. (7) -18x+8y +41. (8) xy-11xz+9yz. (9) 15x+12y+19z+20. (10) 2xz+2yz-28. (11) 0. (12) 3xy-5xz+15yz-101. Numerical results. (1) -24. (2) 16. (3) 47. (4) 18. (5) 27. (8) 8. (9) 98. (10) -28. (11) 0. EXERCISE VI. Page 12. (12) -77. (1) 4α-x. (2) 3c-a. (3) −5a. (4) a−3b+c+d. (5) 2x-4a+46. (6) 2a. (7) 4x. (8) 46. (9) 1+5x (10) 5-8x. (11) 3a-2b. (12) x. |