x = 3, y = 2; x = 2, y = 3; x = § ± 1 √− 151, y = § F } √. — 151. Check each set of real roots. The check for the imaginary roots is as follows: (1) 625 125 151 - 11325 755 — 151 +22801 + 625 16 135 √ 15111825 755 √ 151 +22801 = 97, 1582 = 97, 97 = 97. (2) § = 5 = 5. = 143. The equations 4+y4 = 97 and x + y 5 are called symmetrical equations, because they remain unaltered when x is written for y and y is written for x. An expression is symmetrical with respect to two or more letters, if it remains unaltered when the letters are interchanged. Every system of two equations that are symmetrical, or symmetrical except for the signs, can be solved by assuming xu + v and y = u — v. Instances of equations symmetrical except for the signs are xa — y1 — α, x − y = b. = II. A SYSTEM OF TWO EQUATIONS IN X AND Y, BOTH QUADRATIC 144. Any two equations of the form ax2+by2=c, or ax2+ bxy=c, or ax2 + bxy + cy2 : d, can be solved by assuming y = vx, and determining the constant v. = It is seen that in these equations all the terms which contain x and y are of the second degree in x and y. Some authors call these equations homogeneous. We avoid this term for the reason that the word homogeneous is more commonly used to designate equations like ax2+bxy+cy2=0, in which every term of the equation contains x and y, and is of the same degree in x and y. See § 8. To find the value of y care must be taken to use the value of v with that value of x which was obtained by substituting the value of v. When 4 was substituted for v in (5), yielding x = ±√3, the 4 must be multiplied by ±√3 (since y = vx), to obtain the corresponding value of y. Hence the values of x and y must be carefully paired as shown above. When and are used in the answers, the upper signs go together and the lower signs go together. 145. The method just explained possesses the great advantage of always yielding results. Very often, however, much shorter solutions can be given by special devices. Frequently a third equation can be derived from the two given equations which is simpler than one or both of those given. The solution is then obtained by the use of this simpler equation along with one of the original ones. For example, solve 2-3 xy = 143, y2+xy = 168. By adding (1) and (2) the simpler equation is obtained, Extract the square root of both sides, (3) is a linear equation. It gives Substitute in (2), x2 - 2xy + y2 = 25. x - y = ± 5. y2± 5y + y2 = 168, 2 y25y=168. (1) (2) (3) The sets of roots are x = 13, y = ± 8; x = ± 112, y = ± 21. POSSIBILITY OF SOLUTION BY QUADRATICS 146. It is readily seen that when, in a system of two equations, one equation is of the second degree and the other is of the first degree or linear, a solution may always be obtained by quadratics. If, however, both equations are quadratics, this is usually not the case. Only special types of quadratic equations, such as have been studied in this chapter, and others of similar character, admit of being solved by quadratics. The general case is far more complicated. Given and ax2 + bxy + cy2 + dx + ey + f = 0 α1x2 + b1xу + c ̧ÿ2 + d ̧x + e̟ ̧ÿ +fi = 0, to find x and y. If y is eliminated by substitution, the result is a quartic equation, which is of the fourth degree and cannot be solved by quadratics, except in special cases. It is shown in more advanced algebras that the algebraic solution of a quartic equation depends in general upon the solution of a certain cubic equation or equation of the third degree. The solution of cubic and quartic equations is not explained in this book. The given equations of the second degree in x and y can be solved by quadratics whenever the auxiliary cubic equation here mentioned possesses a rational root. This rational root can be found by the factor theorem (see § 78); the other two roots of the cubic can then be found by quadratics, as can also the four sets of roots of the given equations. (See F. Cajori, Theory of Equations, New York, 1914, pp. 71-73.) |