.1 \ \LTTIi QBOMXTB1 [Cn. v. 



. let it he ivqiliivd In \\rilr tin- relation of a lllir that is 



tllel t<> tlu line 



Tin- slope <f this line is -\, lu-nn- any othi-r lim- \\hosr 

 IS 8 is parallel to the given linr, 



i.e., y = 3* + /s . . . 



is, for all values of l>, parallel to lino (1). 



If it he required that the line (^) shall also pass through 

 a given point, (1, 5) for example, it is only necessary to 

 determine rightly the value of b. This is <h>ne by remem- 

 bering that if the line (2) passes through the point (1, 5), 

 th.-n these coordinates must satisfy equation (2), 



i.e., 5 = 3.1+6, whence b = '2. 



Therefore the line y = 3ar-f 2 is not only parallel to t In- 

 line y = 3 a; + 7, but also passes through the point (1, 5). 



Similarly y = Jz + 6, whatever the value of b, is per- 

 pendicular to y = 3 x + 7. 



Again, the line 3a; + 5y-|-A: = 0, whatever the value of k, 

 is parallel to the line 3a;-f5y 15 = 0; and the lim- 

 5:r 3y-f& = is perpendicular to 3 x + 5 y 15 = 0. 

 !!] aijain the arbitrary constant k maybe so determined 

 that this line shall pass through any given point. So also 

 the lines A^x + B$ + C l = and A^x + B$ 4- C 9 = are 

 llel, while A^x + B$ 4- C l = and B^x - A$ -f (7 a = 

 are perpendicular to each other. 



This condition for parallelism and for perpendicularity 

 of two lines may also be stated thus : two lines are parallel 

 if their equations differ (or may be made to differ) only in 

 constant terms ; two lines are perpendicular if the coeffi- 

 cients of x and y in the one are equal (or can b" made equal), 

 respectively, to the coefficients of y and x in the other. 



