ELIMINATION. In mathematics we often meet with instances where, given several state ments concerning several distinct quantities, we wish to discover precisely what is affirmed of a smaller group of these quantities. For ex ample, in the solution of simultaneous equa tions, such as Scia bly cl= 0 b,y c.-= 0, to obtain the value of x, we must derive from these two equations a single one not involving y. This process is called the elimination of y. In the case of linear simultaneous equations such as the above, the elimination may be performed by multiplying the first equation by b, and the second by b, and, subtracting or by solving the first equation for y and substituting this value in the second, or by solving both equations for y and equating the values thus obtained. All these methods give the result I (se, I0,6,1 all(see DETERMINANTS), and throughout all forms of elimination determinants are very convenient. Elimination between equations not linear is apt to be very complicated and difficult. ever, in the case of the elimination of a single unknown from two consistent algebraic equa tions, Sylvester's dialytic method forms an easy solution to the problem. This consists in obtaining from two equations in x of the mth and nth degrees, respectively, the n equa tions formed by multiplying the first equation by the powers of x from the 0th to the (n — 1)st and the m equations formed by multiplying the second equation by the powers of x from the 0th to the (m —1)st, and by eliminating from these the powers of x, considered as independ ent variables. We thus get m + n equations
in m + n-1 variables, and the condition (see DETERMINANTS) that these be consistent is that the determinant of the coefficients should vanish. For example, if our two equations are + asx' + °ix: + aux + ao -= 0 box' -I- bix + bo = 0 we obtain from these the equivalent family of equations a.x' + cox' + (he + /la + ao =0 aix' + (he + ale + aix' + aa =0 box' + bix + b,, .--- 0 bee ± boe + b. =0 + + thx* =0 boo + bie + bas = 0, which give the relation between the coefficients 0 a, a, a, a, ao a a, a, a, a2 0 0 0 0 b, b, b.
-= O.
0 0 b, b, b. 0 0 b, b, b. 0 0 b2 b, b. 0 0 0 In certain cases an analogous method may be applied to systems of three or more equa tions. A method of similar application to that of Sylvester had been discovered previously by Euler. (See ALGEBRA, ELEMENTARY; DETERMI