press nothing more than the first two, since that we have only tripled two equal results; we should have but one translation, and consequently a single equation. It can therefore happen that we may have less equations than unknown quantities, and then the question is said to be indeterminate; because the number of conditions would be insufficient for the determination of the unknown quantities, as we shall see clearly illustrated in the following section.

1. ELIMINATION OF UNKNOWN QUANTITIES FROM

ANY NUMBER OF SIMPLE EQUATIONS.

203. Elimination is the method of exterminating all the unknown quantities, except one, from two, three, or more given equations, in order to reduce them to a single, or final equation, which shall contain only the remaining unknown, and certain known quantities.

204. In order to simplify the calculations, by avoiding fractions, we shall here make use of literal equations, which will modify the process of elimination: And also, to avoid the inconvenience arising from the multitude of letters which must be employed in order to represent the given quantities, when the number of equations involving as many unknown quantities surpasses two, we shall represent by the same letter all the coefficients of the same unknown quantity; but we shall affect them with one or more accents, in order to distinguish them, according to the number of equations.

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'x+b'y=c'.

(A), (B).

are

The coefficients of the unknown quantity 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).quations By transposition, these quantities become

5x+3y-y+2x=7+5,
9x-2y-x+7y=16-3;

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,

By

mx+6y-7=px-2y+3,
rx-9y+6=3y-3x+12,

be reduced to the form of equations (A) and (B).
transposition, these equations become
mx+by-px+2y=3+7,.
rx-9y-3y+3x=12—6;

by reduction, we shall have

(m—p)x4-8y=10,

(r+3)x-12y=6;

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 &, c 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
a'x+by+cz=d'
a'x+by+c"z=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 quan

tities.

(1):

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'x+b'y = c'

(A); (B).

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,

d'axa'bya'c;

aa'x+ab'y=ac'.

Subtracting the first of these from the second, the unknown x 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'—a'b) x=b'c- `bc' ;

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 x 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.