CHAP, ii] SYSTEM OF LINEAR EQUATIONS. 81 



four variables is treated; also 



3x + 5y + 7z = 560, 9z + 25y + 49z = 2920. 



E. Be"zout 195 solved x + y + z = 41, 24z + 19y + 10z = 741 by elimin- 

 ating x and showing that the integral solutions of 5y + 142 = 243 are 

 z = 5u 3, y = 57 14w. 



Abbe Bossut 196 solved by eliminating z 



x + y + z = 22, 24x + 12y + 6z = 36. 



A. G. Kastner 197 treated the problem of Pratorius 190 and its generaliza- 

 tion : Three peasants have a, b, c eggs, respectively, where a, b, c are distinct 

 numbers. They sold x, y, z eggs respectively at the price m per egg and 

 the remainder at n. Each received the same total amount of money. 

 Find x, y, z, mfn. We have 



mx + n(a x) = my + n(b y) = mz + n(c z), 

 where x, a x, etc., are to be positive integers. We get 



m b a c b (b c}x + (c a)y 



+ 1 = - - T lj Z = - -r r - . 



n x y y z (b a) 



Give successive values to x and solve the equation in y, z. 

 A. G. Kastner 198 discussed the " Regel Coci." From (1), 



b ah (/ h}x 



y -- ^h - - 



whence 



b ah 



% = ~^ T~ 



f - h 



Also, ag + (/ g}x = 6, so that we have limits for x. 



J. D. Gergonne 199 considered n equations in m > n variables, 



anXi + + aimXm = ki (i = 1, , n), 



with integral coefficients, and stated a priori that 



xj = T 3 + Aja + B y /3 + . - - 



where a, ft, are parameters in number m n at least. Substitute these 

 expressions for the x's into the given equations and equate the coefficients 

 of a, of |8, etc. Some of the resulting conditions show that TI, T 2 , 

 is a set of solutions of the given equations. The remaining conditions 

 show that the A's, the B's, are sets of solutions of 



+ + dimX m = (i = 1, , ri), 



195 Cours de Math., 2, 1770, 94-6. 



196 Cours de Math., II, 1773; ed. 3, I, Paris, 1781, 414. 



197 Leipziger Magazin fur reine u. angew. Math., 1788, 215-227. 



198 Math. Anfangsgrunde, I, 2 (Fortsetzung der Rechenkunst, ed. 2, 1801, 530). 



199 Annales de Math, (ed., Gergonne), 3, 1812-13, 147-158. 

 7 



