§ 4. SYSTEMS OF FRACTIONAL EQUATIONS. 1. If some or all of the equations of a system be fractional, and lead, when cleared of fractions, to linear equations, the solution of the system can be obtained by the preceding methods. Any solution of the linear system which is derived by clearing of fractions, is a solution of the given system, unless it is a solution of the L. C. D. (equated to 0) of one or more of the fractional equations. (See Ch. X., Art. 4.) Transferring and uniting terms in (3), and dividing by 2, 33 x 26 y = 27. The solution of (1) and (4) is 15, 18. (4) In clearing (2) of fractions, we multiplied by 11 (4 y +5). Since x 15, y = 18 is not a solution of 4y+5=0, it is a = Observe that the given equations are neither linear nor fractional. Yet they can be transformed so that they will contain only the reciprocals of x, y, and z. Dividing (1) by xy, (2) by xz, (3) by yz, we have: Consequently, a solution of the given system is 12, 12, — 12. It is important to notice that we cannot assume that the system (II.) is equivalent to the system (I.), since the equations of (II.) are derived from the equations of (I.) by dividing by expressions which contain the unknown numbers. But if any solution of (I.) be lost by this transformation, it must be a solution of the expressions (equated to 0) by which the equations of (I.) were divided; that is, of xy=0, xz = 0, yz = 0. (III.) The system (III.) has the solution 0, 0, 0, and this solution. evidently satisfies the system (I.). We therefore conclude that the given system has the two solutions 12, 12, -12, and 0, 0, 0. This system can be readily solved by making the following Solving equations (4), (5), and (6), we obtain u = 1, v = 1, w=-1. Substituting these values in the system (I.), we have Whence, x+y+2=6, 2xy=4, y-3x=-1. z −1. x=3, y=2, z=1. 2. As in a system of integral equations, so in a system of fractional equations, the equations must be consistent and independent. These values constitute a solution of equations (1) and (3), but not of (1) and (2). For they form a solution of the L. C. D. (equated to 0) of the fractions in (2); that is, of We conclude, therefore, that equations (1) and (2) do not have any solution. It can, in fact, be shown that they are inconsistent. For (1) is equivalent to Dividing both members of the last equation by (x − 1) (y − 2), we obtain Equation (5) is evidently inconsistent with (2). It should be noticed that in clearing (2) of fractions no unnecessary factor was used. The explanation of the apparent contradiction of the principle proved in Ch. X., Art. 3, is that this principle holds only when the fractional equation contains but one unknown number. |