Systems of Equations. 97 



Thus take the system 



and derive the system 





The only set of values which will satisfy (a) is x=4, y=i. 

 This set satisfies (b) and no solution is lost. 



The system (a) is equivalent to the system (b) and to two 

 other systems (see Art. 7), but the other two systems are incom- 

 patible. 



As an example of the case in which solutions may be lost, 

 consider the system 



^'=9— _y' 3 '^^ 



which is satisfied by either of the two sets x=o, jK=3 and x=2i, 

 j=o. If we divide the second equation by the first, member by 

 member, we pass to the system 



x=s-j 



which is satisfied only by the values ^=3, y=o. 



II. Solution of a Linkar-QuadrxITic System. We now 

 propose to take up the solution of those systems involving two 

 unknowns which consist of one linear and one quadratic equation. 

 It is convenient to call this a linear-quadratic system. We will 

 proceed by first working the following particular example : 



x-—2y'^—\ (2) j 



(") 



From equation ( 1 ) the value of x in terms of j is easily seen to be 

 Substituting this value of .r in equation (2) we obtain 



('5-j/-2;'==i (a) 



or 25— loj'+y— 2)/^=i (s) 



Unitipg and transposing terms 



y+iq)/=24, . (6) 



whence, solving this quadratic, 



_y=2 or —12, 

 and from equation (i) ' 



.r=3ori7. 



A— 12 



