23. 2 x2-3 y2-12x-12 y = 23, 24. 4x2+9 y2-8x-18 y = 12, 25. 5 x2 + 12 y2 = 32, 26. x2 - 3 xy + 2 y2 = 3, 4x-7y = 6. 27. 3 x2 + y2 = 49, 12x + y = 49. 28. 2x2 - y2 = 46, 5x y = 23. 29. x2 + y2 = 82, 9x+y=82. 30. (x-1)2+(y—2)2=10, 3x+y=15. 31. 4 x2 — 9 y2 = 108, 4x-3y=18. 32. x2+4y252, = x + 2y = 10. *Eliminate x2 and y2 and get a simple equation. 45. 31 x2+31 y2 + 67 x + 25 y = 432, 2x2+2y2+3x-5 y = 12. 46. 10 x2 -2xy - y2 + 5 = 0, 15 x2-11 xy + 2 y2 = 0. 47. 10 x2-21 xy + 170 = 0, 20 x2 - 33 xy + 10 y2 = 0. 48. 41 x2 - 46 xy + 13 y2 = 4, 10 x2 - 11 xy + 3 y2 = 0. 49. 18 x2-11 xy — 2 y2 — — 12, = 5 x2-20 xy + 13 y2 = 17. 50. 3 xy-7 x2 = 2, 13 xy-25x2-y2 = 5. 51. 6 x2-9 xy + 4 y2 = 34, 25x239 xy + 15 y2 = 85. 52. 8 y2-9 xy = 20, 4x2-17 xy + 12 y2 = 24 169. Symmetric equations. An expression is symmetric with respect to x and y if it is unaltered by interchanging x and y. Examples, x2 + xy + y2, x2y + xy2, 203+ y3. General Method. If x + y = 8 and xy=p, then x2 + y2, x3 + y3, x2 + y2, etc., can be expressed in terms of 8 and p. x2 + y2 = (x + y)2 — 2 xy = s2 — 2p. x3 + y3 = (x + y)3 — 3 xy(x + y) = 88 — 3 ps. x2 + y2 = (x2 + y2)2 — 2 x2y2 = (82 — 2 p)2 — 2 p2 Example 1. Solve x3 + y3 = 351, x2y + xy2 = 126. Example 2. Solve x + x2y2 + y2 = 481, x2 + xy + y2 = 37. (1) (2) Solving (4) and (2) by method of § 168 or § 169, The required solutions are (1, 2), (2, 1), (4, − 1), (− 1, 4). CHAPTER XVI SURDS. FRACTIONAL EXPONENTS. RADICAL EQUATIONS, ETC. 170. If n is a positive integer and a is a positive rational number which is not a perfect nth power, then the /a is called a surd of the nth order. Thus, √6 is a surd of the second order, √1.44 is not a surd, for (1.2)2=1.44. III. a="/a= Va. (Chapter XII.) 172. A surd is in its simplest form when the radicand is an integer and is as small as possible. Simplify (1) √63; (2) 5 √8; (3) √16. 1. √63 = √9 × 7 = √9 × √7 = 3√7. 5√2 5√2 5√2 5 or √2. √8.√2 √16 4 5 2. The process in (2) is called rationalizing the denominator. |