205. In the first place, any two simple equations, each involving the same two unknown quantities, may, in general, be written thus: ax+by=c (A), (B). The coefficients of the unknown quantity x are represented both by a; those of y by b; but the accent, by which the letters of the second equation are affected, shows that we do not regard them as having the same value as their correspondents in the first. Thus a' is a quantity different from a,« b' a quantity different from b. 206. We can readily see, by a few examples, how any two simple equations, each involving the same two unknown quantities, may be reduced to the above form. Ex. 1. Let the two simple equations, 5x+3y-5=y-2x+7, 9x-2y+3=x-7y+16, be reduced to the form of equations (A) and (B). 5x+3y-y+2x=7+5, by reduction, we shall have 7x+2y=12, 8x+5y=13; equations which are reduced to the form of (A) and (B), and which may be expressed under the form of the same literal equations, by substituting a, b, and c, for 7, 2, and 12; and a', b', and c', for 8, 5, and 13. Ex. 2. Let the two simple equations," mx+6y-7-px-2y+3, rx-9y+6=3y-3x+12, be reduced to the form of equations (A) and (B). By transposition, these equations become mx+6y-px+2y=3+7, by reduction, we shall have (mp)x+3y=10, which are reduced to the form required, and which may be expressed under the form of the same literal equations, by substituting a for m-p; b for 3. formc for 10, a' for r+3, b' for -12, and c for 6. In like manner any two simple equations may be reduced to the form of equations (A) and (B); hence we may conclude that a, b, c, a', b', and c', may be any given numbers or quantities whatever, positive or negative, integral or fractional. It is to be always understood, that when we make use of the same letters, marked with different accents, they express different quantities. Thus, in the following equations, a, a', a", are three different quantities; and the same of others. 207. Any three simple equations, each involving the same three unknown quantities, may be expressed thus; ax+by+cz=d (C), (D), (E); where a, b, c, d, a', b', c', d', a", b", c", d", are known quantities; and x, y, z, unknown quantities whose values may be found in terms of the known quantities. And so on for five, or more simple equations. 208. Analysts make use of various methods of eliminating unknown quantities from any number of equations, so as to have a final equation containing only one of the unknown quantities; some of which are only applicable in particular cases; but the most general methods of exterminating unknown quantities in simple equations, are the following. FIRST METHOD. 209. Let us consider, in the first place, the equations, ax+by=c (A); It is evident that if one of the unknown quantities, x, for example, had the same coefficient in the two equations, it would be sufficient to subtract one from the other, in order to exterminate this unknown: Let, for example, the equations be 10x+11y=27, 10x+9y=15; if the second be subtracted from the first, we shall have 11y-9y-27-15, or 2y=12. It is very plain, that we can immediately render the coefficients of x equal, in the equations (A) and (B); By multiplying the two members of the first by a', the coefficient of x in the second; and the two members of the second by a, the coefficient of x in.. the first; we shall thus obtain, a'ax+aby=a'c; Subtracting the first of these from the second, the unknown will disappear, we shall have only (ab'-a'b)y=ac-a'c, an equation which contains no more than the unknown quantity y, and we will deduce from it By eliminating in the same manner the unknown .quantity y, from the proposed equations; we would arrive at the equation (ab'-ab)x b'c-be'; from which we will deduce 210. The process which we have just employed, may be applied to all simple equations, for exterminating any number whatever of unknown quantities. If we apply this process to three equations, involving x, y, and z, we will at first eliminate a between the first and second; then between the second and third; and we shall thus arrive at two equations, which involve only y and z, and between which we will afterward eliminate y, as in the preceding article. If we effect the equation in z, at which we will arrive, we shall have a factor too much in all its terms; and consequently it will not be the most simple which might be obtained. SECOND METHOD. 211. Let us resume again the equations, (A) ax+by=c; a'x+b'y=c'. . (B): If we find the value of xin terms of y and the known quantities in each of these equations, we shall have : the equality of the second members, furnishes the equation c_by_c-b'y α = a' which, by making proper reductions, gives ac' a'c y=ab-ab of by substituting this value for y, in one of the values of *, we shall, after the reductions, have . These values of x and y are the same as before. Now, it is evident, that by proceeding in the same manner, with three equations containing x, y, and z, we will find the value of x in each of them, then by comparing these values, we shall arrive at two equations, involving only y and z, from which we can eliminate y, as in equations (A) and (B). And, we can proceed, in a similar manner, when there are four equations with four unknown quantities; and so on, for five, or more equations. |