CHAPTER VII. 



SYSTEMS OF EQUATIONS. 



1. Definition. If a number of equations containing several 

 unknown quantities are supposed to be so related that they are 

 all satisfied simultaneously by the same set of values of the un- 

 known quantities, the equations are said to constitute a .Sy.y/d'w, 

 or a System of Simultaneous Equations. 



Thus the equations 



2X-\- y-\- 52"= 19 

 T,x+2y-i- 42-= 1 9 



are satisfied simultaneously by the set of values, 



-r=i,jj/=2, 2=s, 

 and are said to constitute a system. This set of values, or the 

 process of finding them, may be called the Solution of the system. 

 The reader is supposed to be already familiar with methods of 

 solution of a system of simple equations containing as many equa- 

 tions as different unknown quantities, such as the system given 

 above. The systems we propose to consider in this chapter are 

 those involving quadratics or equations of higher degrees. 



2 . The student should not suppose that every system of equa- 

 tions which may be proposed is capable of solution. It is one 

 requirement that the number of unknown quantities be just equal 

 to the number of equations in the system. But even this is not 

 all. Some of the equations in the system may contradict some 

 of the others, in which case a solution is impossible. For example, 

 take the system 



34-2J/=2.^- (i)\ 

 X- v= I (2) j 



From equation (2) 



x= I -j-y. 



Substitute this value of x in equation ( i ) and we obtain 



3 + 2V= 2 + 2J, 



or 1=0, 



