On comparing the ratios a1/a2 , b1/b2 and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.

On comparing the ratios a1/a2 , b1/b2  and c1/c2 find out whether the following pair of linear equations are consistent, or inconsistent.

(i) 3x + 2y = 5 ; 2x – 3y = 7

(ii) 2x – 3y = 8 ; 4x – 6y = 9

(iii) 3/2x + 5/3 y = 7 ; 9x – 10y = 14

(iv) 5x – 3y = 11 ; – 10x + 6y = –22

(v)4/3x + 2y =8  ; 2x + 3y = 12 

Solution 

(i) 3x + 2y = 5 ; 2x – 3y = 7

Convert the equation in form of  a1x + b1y + c1 = 0  and a2x+ b2y + c2 = 0

We get

3x + 2y - 5 =0  and  2x – 3y – 7  =0

Compare the equation with

                          

We get

a1 = 3,            b1       = 2,                 and c1 = -5

a2 =2              b2        =-3                  and c2 = -7

 

 

We get 

 

Hence both lines are Consistent

(ii) 2x – 3y = 8 ; 4x – 6y = 9

Convert the equation in form of  a1x + b1y + c1 = 0  and a2x+ b2y + c2 = 0

We get

2 x – 3 y - 8 = 0  and  4x – 6 y – 9  =0

Compare the equation with

                      

We get

a1 = 2,            b1       = -3,               and c1 = - 8

a2 = 4             b2        = - 6                and c2 = - 9 

 

So we get 

 

So both lines are Inconsistent

(iii)        3/2x + 5/3 y = 7 ; 9x – 10y = 14

Convert the equation in form of

 a1x + b1y + c1 = 0  and a2x+ b2y + c2 = 0

We get

3/2 x  + 5/3 y - 7 = 0  and  9x – 10 y - 14  =0

Compare the equation with

                          

We get 

a1 = 3/2,         b1       = 5/3,             and c1 = - 7

a2 = 9             b2        = - 10              and c2 = - 14

  So we get 

 

So both lines are Consistent

(iv)        5x – 3y = 11 ; – 10x + 6y = –22

Convert the equation in form of

 a1x + b1y + c1 = 0  and a2x+ b2y + c2 = 0

We get 

5 x  -3 y - 11 = 0  and  -10 x + 6 y + 22 =0

Compare the equation with

We get

a1 = 5             b1        = - 3,              and c1 = - 11

a2 = -10          b2        =  6                  and c2 =   22

 

So we get

                            

Hence 

So both lines are dependent and consistent  

(v)4/3x + 2y =8  ; 2x + 3y = 12

Convert the equation in form of

a1x + b1y + c1 = 0  and a2x+ b2y + c2 = 0

We get

4/3 x  + 2 y - 8 = 0  and  2x + 3 y - 12  =0

Compare the equation with

                                      

We get

a1 = 4/3,         b1       = 2,                 and c1 = -8

a2 = 2             b2        = 3                   and c2 = - 12 

So we get 

                     

So both lines are Dependent and consistent