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Therefore, when a problem of the first degree with two unknown quantities is possible, impossible, or indeterminate; we are conducted to the values of x and y, finite, infinite, or of the form that is to say, indeter
0 minate'; and the reciprocal takes place.
234. Let us extend this discussion to the roots (c), (d), and (c), (Art. 215) of the three equations (C), (D), and (E).
The problem can be indeterminate under a very great number of hypotheses upon the values of the coefficients. We can suppose, between the coefficients, any one whatever of these relations, I st. a=a'=u", b='=6", d=d=d"; 2nd. a=a',b"=b', c"=c', d"=d'; 3d. a" =aa---a, 6"=ab-b, c"=ac' --C, d"=ad-d;
By introducing any one whatever of these hypotheses into the formulæ of the roots, we shall al
the first system of hypotheses says that these three equations make but one, the second expresses that the third equation is but the first, and the third announces that the last equation is a combination of the other two.
Let us examine in particular the case where we have d=0, d=0, d"=0; the three proposed equations become then, dividing them by x, and putting
The first two are sufficient to determine p and
9, and the substitution of these values in the third, shall give an equation of condition, that is, a relation between the given quantities, a, b, c, a', b', c', a", b", c", necessary in order that the three equations may be satisfied otherwise than by the values, x=0, y=0, 2=0, which verify them.
, =. If the equation of condition take place, the calculation, which shall only find finite values for the ratios =p,- = of the three unknown quantities,
y leaves one of them entirely arbitrary, so that the question is susceptible of an indefinite number of solutions.
In order to prove then, that the values of the unknown- quantities which reduce to zero, by making in the nrimerators of the general formulæ, d=0, d=0, d"=0, are really of the form must eliminate pand 9 from the above equations ; the first two give
ch'-be these values substituted in the third, change it into this :
ab'c"--adb"+ca'b"-ba'c" +-bc'a" -cb'a"=0; therefore
0 0 0
o' This is still what we would have found for the values of the unknown quantities, if one of the equations, the third, for example, were comprised in the two others, and we would fall again into this case by the hypotheses a"=na + ma', b"=nb tomb', c"=nce mc', d"=nd+md.
235. Reciprocally, if the values of the unknown
0 quantities present themselves under the form may conclude with certainty that one of the equations is comprised in the two others, and that consequently the problem is indeterminate. The calculation necessary for arriving at this conclusion, being very tedious, the principal parts will be only indicated; however, the student would do well, in order to exercise bimself, to effect the whole of it.
Let us first demonstrate that any one whatever of the equations which we obtain, by making the numerators and the common denominator of the roots equal to zero, is comprised in the other three.
If the numerators, for instance, be equal to zero,
b'c" — c'b=a', cb-bc"=bl, hc' ---cb=c?;
a'6" --b'a"=u', bah-ab"=63, ab-bd=c3
a'd+d+-c3d"=0. If we take d, d', d", for the anknown quantities,
0 they shall have
for their values, since they are
0 not nought, and the denominator shall be albc3-acb3 +ca3-bac3 +67ca3-c?bas =0.
If we substitute here, for a,b,c', a’, ba, c? &c. their values above, it will become, after proper reductions, a quantity precisely equal to the square of
ab'c"--ac'6" + ca'b"-ba'c"#bca"-cb'a" ;
therefore this common denominator is equal to nothing, in consequence of the numerators; the same demonstration might be applied to other cases.
This being premised, if in the numerators, equated to zero, of the values of x, y, z,. we take the values of a", 1", c", and substitute them in the third equation,
a"x+by+c":=d", we shall find another which will satisfy itself in consequence of the first two.
In fact, the numerators of w and y being equal to zero, will give
db'c" —bd'c" +-bc'd" —cb'd"
dc cd By substituting these values in the numerator of s, we find an equation of condition which satisfies itself, since it becomes 0=0. By putting for a' and 6" their values in the equation a"+by+o": =d" it becomes, after taking away the denominator and making proper reductions, de" (a'x+by+cz)---d'c”(ax+by+cz)=0;
db"d-dcd=0, after having replaced a'x+by+c'z by d', and axt by+cz by d. It is necessary to observe here that the preceding values of a" and b", which reduce to zero the numerator of the value of z, render also the common denominator of the formulæ of roats equal to nothing And in fact, if it were not so, we should have
d ďł d x=0, y=0, and z=
that is to say,
so that z would admit of three values, when every other unknown quantity which depend on z, should admit bot of one.
0 Thus the three roots becoming
one of the three equations is comprised in the other two.
236. The preceding considerations will still appear more evident from the resolution of numerical equations :
Let, for example, the three equations to be resolved, be
x+3y-2z=2. By comparing these with the general equations (Art. 207,) and regarding the signs, we will have
a=1, b=-2, c=1, d=5,
a"=1, b=3, c= -2, d"=3; substituting these values in the general formula of roots, (c), (d), and (e), (Art. 215), we shall haye 2-21+30+8-14--540-40 +
0 -2+3+-6--87-2-1 11-11
-- -14+2+4+20-5-7 26-26 Y=
- 2+3+6-87-2-1 11-11 0 -10+15+21-28 +4-2 40-400
--2+37-6-8+2-1 11-11 0 Therefore the values of x, y, and
z, are indetersignate, in the proposed equations, which will also
; appear obvious; since, by eliminating x from each of them, and then equating the results, we shall have these two equations,