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 often eafier and rttorter methods to dedace an equation involving one unknown quantity only ; which will be belt learned by practice. Examp. VII. ("x4-)=« Supp. < 0+25=C x=a—y a—-y--z=b y-{-z=c a--0+2Z—b+c 2z=b+c—a f b--c-—a c+a—b  where the numerator is the difference of the produdfs of the oppofite coefficients in the orders in whichy is not found, and the denominator is the difference of the produdts of the oppofite coefficients taken from the orders that involve the two unknown quantities. For, from the firft equation, it is plain, that axzzc—by. . and x——~ f-—ey from the 2d, dx—f—ey. . and x—— therefore a d and cd—dbyzzaf—aey whence aey—dbyzzaf-—cd, J af—cd and, y— ae—db , ci—bf c r after the fame manner, x=——

It is obvious from the 3d and jth direftions, in what manner you are to work if there are four, or more, unknown quantities, and four, or more, equations given. By comparing the given equations, you may always at length difcover an equation involving only one unknown quantity; which, if it is a fimple equation, may always be refolved by the rules of the Taft chapter. We may conclude then, that “ When there are as many fimple Supp. equations given as quantities required, thefe quantities may be difcovered by the application of the preceding rulesi” , j’= 5 J 5x80—3x100= too=5-2then If indeed there are more quantities required than e5x8—3x7 19 > I? quations given, then the queftion is not limited to determinate quantities; but is capable of an infinite number of folutions. And, if there are more equations given than there are quantities required, it may be impoffible to THEOREM II. find the quantities that will anfwer the conditions of the queftion ; becaufe fome of thefe conditions may be Suppose now that there are three unknown quaninconfiftent with, others. tities and three equations, then call the unknown quantities x, y, and z. Thus, C ax--by-{-cz=:m Chap. XI. Ccntainingfome general UnzcyMMs <dx+ey+fzz=n for the exterminating unknown Quant ities (gx+hy+kr=p in given Equations. In the following Theorems, we call thofe coefficients Then ffiall z--aep—ahn--dhtn—dbp--gbn—gem ask—ahff-dhc—dbk--gbf-—gee ’ of the fame order that are prefixed to the fame unknown where the numerator confifts of all the different proquantities in the different equations. Thus in Theor. dudts that can be made of three oppofite coefficients 2d, a, d, g, are of the fame order, being the coefficients taken from the orders in which z is not found, and the of x; alfo b, e, h, are of the fame order, being the coef- denominator confifts of all the produdts can be made ficients ofy : and thofe are of the fame order that affedt of the three oppofite coefficients taken that from the orders no unknown quantity. But thofe are called oppojite coefficients that are taken that involve the three unknown quantities. each from a different equation, and from a different order of coefficients : As, a, e, and d, b, in the firft theo- Chap. XII. Of Quadratic E qu a t i o n s. rem ; and a, e, l, in the fecond; alfo, a, h. f; and d, b, k, See. In the folution of any queftion, where you have got an equation that involves one unknown quantity, but inVol. I. No. 4. Aa voltes