:.: H i in: -//.M /<,/// i i.vg 91 



Because of this fact, such an equu <ft?n spoken of 



m linear equ.ui.n. 



IK. In the equation Ax + fly + C = 0. there are apparently three 

 OQsUnta; in rrality. there are but two independent constant*, vit. the 

 ratio* of the ooaOoienta (cf . Art. S8). This corresponds to the (act that 

 a straight line ia determined geometrically by two conditions. 



58 Reduction of the general equation Ax 4- By +C = O to 

 the standard forms. Determination of a, ft, m, />. and a in 



terms of .1. II, and < 



the standard form ? +? 1 (ymm. 



. 



That i lie equation 



Ax+By + C-0 . 



represent* fonw > line baa just been shown (Art. 



again, since multi|li< -ation by a constant, and 

 do not change the locus (Art. 88), therefore 



-j^-a-lt . (2) 



A B 



represents the same line. But equation (2) is in the re- 

 quired form (A; :M<1 its mi. reepts are : 



the standard form y = mx + b (tlopt 

 form). 



the resulting equation aewrta [tee Art. 20. (1)] that the area of the triangle 

 formed by the potato /*,, /',, and /*, is tero; /. . theee three polnu lie on a 

 straight line ; but they are any three points on the locus of Ax + Hy + C = 0. 

 hence that locus Is a straight line. 



These reductions constitute a second proof of the theorem of Art. 57. 

 if C = 0, the line represented by (1) goes through the origin, and the 

 symmetric form of the equation is inapplicable (Art. M); but, in that case, 

 the above reduction also fails, since it is not permissible to divide the mem- 



K*-r> OJ .v:. "j .-i:; D 1 > IO, 



