If we assume that x = vy, we may, by substituting vy for x in the given equations, obtain the equivalent system To obtain the solutions of System II., we may proceed as follows: Since the given equations are simultaneous, these "expressed" values for y must be equal. The solutions of the v-eliminant (9) of the derived System II. are found to be v = } and v = 7. (10) Since neither of these values could have been introduced by the multipliers 2+2 and v — 1, when deriving (9) from (8) they must also be roots of equation (8). The system composed of the v-eliminant (9), the assumed equation ry and either equation (6) or equation (7), constitutes a system of equations which is equivalent to System II. :7, in equa x = xy. Substituting the solutions of equation (9), v = 3/2 and v = tion (7), we obtain: Substituting 3/2 for v, y= ± 2. Substituting 7 for v, To find the corresponding values of x, we may either substitute these values of y in the given equations (1) or (2), or we may substitute corresponding values of v and y in x = vy. The four solutions of the given system are thus found to be These four sets of values are found to satisfy the equations of the given system. By extracting the square root of 3, we may obtain from (iii.) and (iv.) approximate values of x and y, correct to any required number of significant figures. Ex. 2. Solve the system of homogeneous equations Since this is a 2-2 system, we may expect to find four solutions. By assigning the value zero to y, it may be seen that the corresponding values of x in the two equations are not equal. Accordingly, it follows that y = 0 is no part of any solution of the given system. Hence we may assume that x = vy, and substituting vy for x in the given equations, we obtain the equivalent derived system The values of x may be found by substituting corresponding values of v and y in the assumed equation x = Substituting {=47. v $, We find x = ±6 Accordingly the solutions of the given system of equations are: x = + 62, } (i.) y = +7. I=- 6, x = + & √ = 5, } (iv.) y = 4√ 5. S By substitution these values are all found to satisfy the given equations. Since we have found four sets of values, it appears that, in deriving the v-eliminant (7), no solutions were lost. By referring to Fig. 4, it will be seen that the graphs of equations (1) and (2) intersect in but two points, the coördinates of which are x = = 6, y = 7, and x = - - 6, y = −7. We must accordingly interpret the imaginary values (iii.) and (iv.) of z and y as indicating that the graphs have no points of intersection the coördinates of which are these solutions. IV. Reduction of Systems of Equations by Division 26. Representing by A, B, C, and D expressions which are integral with reference to two unknowns, x and y, it may be seen that if the members AC and BD of one equation (1) of a System I., composed of two equations, contain as factors the corresponding members A and B of the remaining equation (2) of the system, then the given System I. is equivalent to the derived double system (i.) and (ii.). That is, A CBD, A = B, (1) Given System. It should be observed that the derived equation CD (3) is obtained by dividing the members of equation (1) by the corresponding members of (2), while the remaining equation (2) of the given System I. is carried over unchanged into the derived system (i.). The remaining system (ii.) is composed of the equations formed by equating to zero separately the factors A and B which are common to the corresponding members of equations (1) and (2) of the given system. The Principle may be established as follows: Substituting for B in (1) the equal value A, we obtain the equivalent - equation Or, Factoring, A CA. D. Accordingly, from System I. we may obtain the equivalent system A(CD) = 0, (7) II. Equivalent A-B=0. (8) 83 Derived System. By the principle of § 16, this single system is equivalent to the derived Ex. 1. Solve the system of equations 2 x2 — x = y2 – 1, (1)I. Given System. x = y + 1. Since the members of equation (2) are contained as factors in the corresponding members of equation (1), the given single System I. is equivalent to the derived double system Equation (3) is formed by dividing the members of equation (1) by the corresponding members of equation (2). Equations (4) and (5) are obtained by equating separately to zero the members of equation (2) which are contained as factors in the corresponding members of equation (1). The solutions of the derived systems (i.) and (ii.) are The number of sets of values thus obtained is equal to the order, 2, of the given system. By substitution these sets of values will be found to satisfy both of the given equations. 27. Whenever one of the expressions represented by A or B (see § 26) is a known number, it follows that the derived system (i.) A=0, (4), B = 0, (5), will have no finite solutions. Ex. 2. Solve the system of equations x3 — y3 = 26, (1) 8} I. Given System. x y = 2. Dividing the members of equation (1) by the corresponding members of equation (2), we may derive the equivalent system x2 + xy + y2 = 13, (3)II. Equivalent. Derived System. y and 2, Observe that by separately equating to zero the factors x which are common to the corresponding members of equations (1) and (2), we would have as one of the expected conditional equations a known number equal to zero, that is, 2 = 0. Hence the expected derived "double" system reduces to a single System II., equivalent to the one given. Accordingly, although the given system of equations is of the third order, the number of finite solutions does not exceed the order of the derived equivalent System II., that is, there will be but two sets of values. |