Geometry with Trigonometry

Sec. 6.2 Algebraic note on linear equations 85

6.2 Algebraicnoteonlinearequations

6.2.1

It is convenient to note here some results on solutions of two simultaneous linear
equations in two unknowns.

(a) If

a 1 , 1 a 2 , 2 −a 1 , 2 a 2 , 1 = 0 , (6.2.1)

then the pair of simultaneous equations

a 1 , 1 x+a 1 , 2 y=k 1 ,

a 2 , 1 x+a 2 , 2 y=k 2 , (6.2.2)

has precisely one solution pair(x,y), and that is given by

(x,y)=

(

a 2 , 2 k 1 −a 1 , 2 k 2

a 1 , 1 a 2 , 2 −a 1 , 2 a 2 , 1

,

a 1 , 1 k 2 −a 2 , 1 k 1

a 1 , 1 a 2 , 2 −a 1 , 2 a 2 , 1

)

. (6.2.3)

(b) If

(a 1 , 1 ,a 1 , 2 )=( 0 , 0 ) and (a 2 , 1 ,a 2 , 2 )=( 0 , 0 ), (6.2.4)

and

a 1 , 1 a 2 , 2 −a 1 , 2 a 2 , 1 = 0 , (6.2.5)

then there is somej=0 such that

a 2 , 1 =ja 1 , 1 ,a 2 , 2 =ja 1 , 2. (6.2.6)

(c) If (6.2.4) holds, then for the system (6.2.2) of simultaneous equations to have

either no, or more than one, solution pair(x,y)it is necessary and sufficient

that (6.2.5) hold.

Note in particular that when (6.2.4) holds, for the pair of homogeneous linear

equations

a 1 , 1 x+a 1 , 2 y= 0 ,

a 2 , 1 x+a 2 , 2 y= 0 , (6.2.7)

to have a solution(x,y)other than the obvious one (0,0), it is necessary and

sufficient that (6.2.5) hold.