From the second 96-3x+6y=6x+4y, 197. If there be three independent simple equations, and three unknown quantities, reduce two of the equations to one, containing only two of the unknown quantities, by the preceding rules ; then reduce the third equation and either of the former to one, containing the same two unknown quantities; and from the two equations thus obtained the unknown quantities which they involve may be found. The third quantity may be found by substituting their values in any of the proposed equations. From the 1st two equns. 6a+ 9y+ 12% 48,] From the 1st and 3rd 10x + 15y + 20% = 801 by subtr. 27y+ 14% = 68) hence 135y + 70% = 340 The same method may be applied to any number of independent simple equations, in which the number of unknown quantities is the same as the number of equations. Another method. a1x+b1y+c1z = d1* . . . . . . (1)] ...... а ̧x+by+С ̧2 = d2 .(2)}; to find x, y, and z. ...... Multiply (2) by m, (3) by n, and to the resulting equations add (1); then we have (a,+ ma2+na ̧)x + (b ̧ +mb ̧ + nb2)y + (c, + mc, + nc ̧) z = d1 + md +nd ̧• Now to find x, let the arbitrary multipliers m and n be such that the coefficients of y and z in this last equation are separately equal to 0 ; that is, * The small figures here give no particular values to the quantities to which they are annexed, a, and a being as different as a and b; but it is often convenient to use the same letter thus slightly varied to mark some common meaning of such letters, and thereby assist the memory. Thus in this instance, a1, ɑg, aз, have this common property, viz. that all are coefficients of x, a, in the 1st, a, in the 2nd, and a, in the 3rd, equation. Similarly for the coefficients of y and z. Similarly, by making the coefficients of x and x, or of x and y, separately equal to 0, the value of y, or of z, may be found. Hence, the following Rule may be deduced, and will be found easy of application: To find x, multiply the 1st equation by bc-b3c2, the 2nd by bac, b, c, and the 3rd by bc-b,c,; then add together the resulting equations, and a simple equation will be obtained in which y and z do not appear. A similar rule may be stated for finding either y or ; or having found the value of x, the equations are reduced to simple equations of two unknown quantities y and z, so that y and z may be found by any of the methods of Art. 196. 198. That the unknown quantities may have definite values, there must be as many independent equations as unknown quanti ties. When there are more equations than unknown quantities, the value of any one of these quantities may be determined from different equations; and should the values thus found differ, the equations are incongruous; should they be the same, one or more of the equations are unnecessary. When there are fewer equations than unknown quantities, one of these quantities cannot be found, but in terms which involve some of the rest, whose values may be assumed at pleasure; and in such cases the number of answers is indefinite. Thus, if x + y = a, then a = a -y; and assuming y at pleasure, we obtain a value of a such, that a + y = a. These equations must also be independent, that is, not deducible one from another. Let x + y = a, and 2x + 2y = 2a; these are not independent equations, since the latter equation being deducible from the former, it involves no different suppositions, nor requires any thing more for its truth, than that x + y = a should be a just equation. It is sometimes, however, not easy to discover at once whether proposed equations be independent or not. Thus in the equations it is not obvious at first sight that the third equation is derived from the other two. But by multiplying the first equation by 4, and subtracting the second, the result is the third equation; and accordingly the usual process, being applied to find x, y, z, would certainly fail. = As examples of incongruous equations, the following may be instanced: x+y=7, and 3x + 3y 30, from which we get 7 = 10; or, again x + y = 7, 3x − y = 1, and x + 2y = 10, from which we get 10 = 12. PROBLEMS WHICH PRODUCE SIMPLE EQUATIONS. 199. From certain quantities which are known to investigate others which have a given relation to them, is the business of Algebra. When a question is proposed to be resolved, we must first consider fully its meaning and conditions. Then substituting one or more of the symbols x, y, ≈ &c. for such unknown quantities as appear most convenient, we must proceed as if they were already determined, and we wished to try whether they answer all the pro |