Systems of Equations. 99 



— , =4 



\^x .x'—y 



13. Solution of Systems of two Quadratics. If we have 

 a system of two quadratic equations containing two unknown 

 quantities and attempt to eliminate one of the unknown quantities 

 it will be found in general that the resulting equation is of the 

 fourth degree. Thus take the system 



rH-xr=io. J 

 We find from the first equation that 



Substitute this ^'alue for y in the second equation, and it be- 

 comes 



or, expanding and collecting terms, 



.1 * -f .r-^ — 5 jt '* — 5.r — 25=10. 

 Now, since we are not yet familiar with the solution of equa- 

 tions of a degree higher than the second, the treatment of sys- 

 tems of two quadratics in general cannot be taken up at this 

 place. But there are two important special cases of systems of 

 two quadratics whose treatment will involve no knowledge be- 

 yond the solution of quadratic equations, and the.se we will now 

 consider. The cases referred to are \ 



I. Where the terms in each equation containing the un- 

 known quantities constitute a homogeneous expression with respect 

 to the unknown quantities. 



II. Where the equations are symmetrical.* 



14. Case I. We will illustrate the first case, and also the 

 method of elimination which may be applied to any example of 

 it, by the following solution : 



o 1 4-1 ♦ ^ -^"— 2.1)'= 5 (i) 



Solve the system -. - .^ ^' \A 



(3.V-— ioj''=35. (2) 



♦For the deflnitione of homogeneous and syrametrioal see I, Arts. 7 and 8. 



