14 i /.)//' i.lHtMETRY [Cu. 1. 



bers a definite valnr. thm at least one .l.-tiuitc and corre- 

 sponding value can be fomnl iu tin- oilier, so that, this pair 

 .iluea being substituted for the unknown numbers, i in- 

 equation will be satisfied. In this \\-.\\ an inlinitr nuiiil>cr of 

 pairs of values, that will satisfy the equation, may be I'miml. 



If, however, the equation is homogeneous in the two un- 

 knowns, i.e., of the form 



cu* + bxy + cy* = <>, 



thru th. : y may be regarded as a single number, and 



the equation has properties precisely like those discussed 

 Arts. 9, 10, and 11. 



To solve a system consisting of two or more independent 

 simultaneous equations, involving as many unknown ele- 

 ments, it is necessary to combine the equations so as to 

 eliminate all but one of the unknown elements, then to solve 

 the resulting equation for that one, and, by means of the 

 roots thus obtained, find the entire system of roo; 



EXERCISES 



1. Given the equation z 2 + 3 x - 4 + m (3 z 2 - 4)- 2 mx* = 0, find tlu 

 sura of the roots; the product of the roots; also the factors of the first 

 iber. 



2. Factor the following expressions : 



(1) r*-5z + 4; (3) mz 2 -3x + c; (5) 



r*+2z-8; (4) az*+tey + cfi (6) ll-27y- 



3. Without first solving the equation 



ar-3z-m(x + 2z*+4) =5** + 3 



find the sum, and the product, of its roots. For what value of m are its 

 root* equal? For what value of m does one root become infinitely 

 Urge? If all the terms are transposed to one member, what are the 

 factors of that member? 



-out first solving, determine the nature of the roots of the 

 equation (m - 2) (log or)* - (2m + 3) log x - 4m = 0. [Regard log x as 

 the unknown element.] 



