An Elementary Treatise on Algebra

241. Finally, if there were more independent equations than unknown quantities, the problem would be more than determined, or over-limited; in fact, having calculated all the unknown quantities, by employing an equal number of equations; it is requisite that the values thus found, substituted in the remaining equations, reduce them to the form 0-0, which can only take place for certain relations between the known quantities. These relations are then the equations of condition necessary in order that the proposed question could be resolved; and if they are not satisfied, it shall be impossible.

242. In order to show the application of the formulæ of roots, (Art. 215), to the resolution of numerical equations, it is necessary to compare the proposed equations, term to term, with the general equations (Art. 207).

In order to resolve, for example, the three equations

7x+5y+2z=79,
8x+7y+92-122,
x+4y+5x=55,

we must compare, term to term, these equations with those of (Art. 207), which will give

a=7, b=5, c=2, d=79,

a'=8, b'=7, c'=9, d'=122,

a"=1, b′′=4, c"=5, d"=55.

Substituting these values in the general formulæ (c), (d), and (e), (Art. 215), and performing the operations indicated, we shall find

x=4, y=9, z=3.

243. It is important to remark that the same formulæ would still serve, when the proposed equations should not have all their terms affected with the sign +.

If we had for example,

3x-9y+8x=41;

-5x+4y+2z=-20;

11x-7y-6z=37.

The comparison of these with the general equations (Art. 207), by having regard to the signs, will give

a=+3, b——9, c=+8, d=+41, a'-5, b'+4, c'=+2, d'--20, a"+11, b"-7, c"-6, d"=+37. In substituting these values in the formulæ (c), (d), and (e); we must determine the sign which every term ought to have, according to the signs of the factors of which it is composed: it is thus that we would find, for instance, that the first term of the common denominator, which is ab'c", becoming +3x+4x-6, changes the sign, and the product is -72.

By observing the same with regard to the other terms, in the numerators, as well as in the common denominator, collecting into one sum those that are positive, and into another, those that are negative, we shall find

244. Having fully explained what concerns the elimination of unknown quantities in simple equations, and also illustrated the characters by which it may be known, if the proposed equations be determinate, indeterminate, or impossible; we may now proceed to the resolution of examples in determinate equations of the first degree: the practical rules that are necessary for this purpose, shall be pointed out in the two following sections.

§ II. RESOLUTION OF SIMPLE EQUATIONS,

Involving two unknown Quantities.

245. When there are two independent simple. equations, involving two unknown quantities, the value of each of them may be found by any of the following practical rules, which are easily deduced from the Articles in the preceding Section.

RULE I.

246. Multiply the first equation by the coefficient of one of the unknown quantities in the second equation, and the second equation by the coefficient of the same unknown quantity in the first. If the signs of the term involving the unknown quantity be alike in both, subtract one equation from the other; if unlike, add them together, and an equation arises in which only one unknown quantity is found.

Having obtained the value of the unknown quantity from this equation, the other may be determined by substituting in either equation the value of the quantity found, and thus reducing the equation to one which contains only the other unknown quantity.

Or. Multiply or divide the given equations by such numbers, or quantities, as will make the term that contains one of the unknown quantities the same in each equation, and then proceed as before.

Ex. 1. Given

of x and