INDETERMINJTE PROBLEMS. 205 



By reducing the firft equation, x := —- — ; and by reducing 



the iecond, x zz : whence -^ — ^ , and con- 



mg ' e mg ' 



fequently,;'=:J:i:±t-^i±=^. But x = J:2=l±±.^ ^here- 

 fore alio x =: z • 



m'g—e 



Suppose I4^"+3it+24 = y"- ; then, taking 9 ir i^ from both 

 fides, I4x"+3i^+i5 — y"- — d^; but */(^^ — 4.ac) z=/(96i — 840) 

 = IT ; whence, if n = 2, the divifors, by Cafe III. will be 

 7X + 5 and 2,\:+3; wherefore, making 7?/.= 2, x =. 



'-^i^ = 5, and J = ^4+.o+ar-4. ^ ^^^ p^^ ,4.25+31.5+24 

 = 529 = (23)'- 



PROBLEM VIII. 



Lei c and d be known values of x and y in the exprejfion^ ax\-\-h 

 ■=. y, and from thefe let it be required to difcover others. 



Since ax^-\-b = j% and ac'-'rb — d\ fubtradling thefe equa- 

 tions, we {hall obtain ax^ — ac'^ :=z y'^ — d^, and by refolution, 

 [ax — ac){x-\-c) zz {y-\-d){y — d) ; whence ax — ac — my — md,. 



V+d 



and x-\-c =: -^—^ — . From the firft of thefe equations, x ■=. 



my — md+ac t r i ''+"' — """ i '"J — "'d+ac 



— ^ — , and from the iecond, x :=. : whence 



== 5 and y zz -, or — ^ — -^ . But x 



