(/r. Ar^rvi^ 



Systems of Equations. 



lOI 



The student may show that the method set forth above will 

 solve the general system and hence any possible example under it. 



16- Case II. To show that any system of two S3mimetrical 

 quadratics can be solved, we will start with the general case, 

 which is evidently, 



xf'+ax-^-dxy+ay-^y^^c \ . > 



x'-\-dx-hexy-\-dy-\-y'=/ j ' ^ 



If x^ and y^ appeared in either of these equations with coeffic- 

 ients the system could be reduced to the given form by dividing 

 the equation through by that coefficient. 



Through the given system substitute 21-^21' for x and 2c— w for 

 J/, where 21 and w are two new unknown quantities. Then (a) 

 becomes 



22l^-{-22(f-^2a2l-\-d2l^ — dz£f=c} , i. 



2ir-\-2'uf-\-2d2i-\-e2i^ — e7jif=f\ ^ ^ 



Subtracting the second of these equations from the first we obtain 



2(a—d)ii-\-(b—e)ie—(b—e)iir=c—f, 

 or 2(a—d)2i-\-(b—e)(2i'—'i£f)==c—f. 



Whence 21= — =^ — ^ — 7 — —-, 



2(a—d) 



Now if the right-hand side of this equation be substituted for 

 21 in the terms 2aic and 2dii of system (b), that system will con- 

 tain no powers of the unknown quantities but the second and will 



