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which, being identical, or both the same, furnish no determinate answer. And in fact, if the three equations be properly examined, it will be found, that the third is merely the difference of the first and second, and, consequently, involves no condition but what is contained in the other two.
By comparing in like manner any numerical equations with the general equations (Art. 207), if the formulæ of roots give = 0, y=0, and z=" : the three equations are contradictory, that is to say, there are no finite values of x, y, z, 'which would satisfy the three equations at once, and the reciprocal takes place, as we shall presently see.
237. When the equations with three unknown quantities are incongruous, each of the values of the unknown quantities becomes equal to infinity. Let for example,
a"=ma', 6"=mb, c'=mc', d"und', conditions under which the third equation is in opposition to the second. The substitution of these values will reduce the common denominator to zero, as can be easily verified; but the numerators shall always have finite and real values; therefore
1 the roots shall have the form
0 cally, if the roots are infinite, we may conclude that the equations are contradictory.
In fact, if we make the denominator equal to zero, we shall have
ab'c" -ac'6" -ca'b" - bu'c" a":
substituting this value in the third equation, we
0 since the roots are infinite, and not of the form
; the third equation is therefore incompatible with the two others; since that, in combining it with these and the supposed relations of the coefficients, we are conducted to a result contrary to the hypothesis.
238. We can extend the considerations which have been just explained, to any number of equations whatever; and we shall arrive at this general conclusion.
If the roots of equations of the first degree have the form, or the question which we propose to resolvi
0 by these equations is indeterminate or impossible, and reciprocally.
In order to explain this analytically, it may be remarked, that according to the rules relative to the resolution of equations, it is necessary that this resolution makes known all the values of the unknown quantities proper to verify the proposed cquations.
Now when a problem is indeterminate, it is impossible that the final equation which contains the last unknown quantity in the first degree only, gives all the different values of which this unknown quantity is susceptible; besides, it is absurd to suppose that it will give one of these values rather than any other; it is necessary therefore that the value of the unknown quantity may be such as not to imply a contradiction with the enunciation, and it is what happens in fact when the calculation gives for the result • ,
Moreover, we cannot obtain such an expression only in this case, as the demonstration of the reciprocal proposition completely proves.
239. When the equations express contradictory conditions, no finite value, whatever it may be, can verify them; thus algebra, by giving then infinite
l values for the unknown quantities, indicates clearly that there exist no numbers which, substituted at once in the equations, in place of the unknown quantities, could satisfy them ; for infinity is a limit which surpasses every assignable quantity.
240. If we had less equations than unknown quantities, the application of the preceding methods will lead to a final equation containing many unknown quantities, and therefore the value of any one of them cannot be ascertained except in terms of the others; and by assuming values of these others, we may obtain an infinite number of corresponding values of the former quantities, which will satisfy the conditions proposed; the problem is therefore indeterminate, (Art. 230). This would also be the case, as has been already proved (Art. 237), when there are as many equations as unknown quantities ; but not independent of each other.
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 +-22=79,
x + 4y + 5z=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"=1, =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
a=4, y=9, 2=3.
243. It is important to remark that the same forinulæ would still serve, when the proposed equations should not have all their terms affected with the sign + If we had for example,
11x--7--62=37. The comparison of these with the general equaLions (Art. 207), by having regard to the signs, will give
a=t-3, b=-9, c=+8, d=+-41,
b a"=+11,6"=-7, ("=-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 2774- 2834
=+2. 592-622 -30
3022-2932 +-90 y =
3. 592-622 -30 3859-3889 -30
=+1. 592-622 -30