Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically:

(i)              x + y = 5, 2x + 2y = 10

Convert the equation in form of 

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

We get

               x + y  -5 = 0, 2 x + 2 y  -10 = 0

Compare the equation with

              

We get

a1 = 1,            b1       = 1,                 and c1 = -5

a2 = 2             b2        = 2                   and c2 = - 10

                                    

 So we get 

                                  

Dependent and consistent  so it will have many solutions .

(ii)           x – y = 8, 3x – 3y = 16

Convert the equation in form of

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

We get

   x – y -8 = , 3x – 3y -16 = 0

Compare the equation with

                         

We get

a1 = 1,  b1     = - 1,              and c1 = - 8

a2 = 3  b2      =  - 3               and c2 = - 16

 

So we get 

  

So both lines are Inconsistent

(iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0

Compare the equation with

              

We get

a1 = 2,            b1       =  1,                and c1 = - 6

a2 = 4             b2        =  - 2               and c2 = - 4

 

So we get 

So both lines are consistent

Solve the equations graphically

2x + y – 6 = 0,

Subtract 2x and add 6 both side we get

 y  = 6 – 2x

plug x = 1,2 and 3 we get

y         =  6 - 2 * 1      = 4

 y         = 6 -  2 * 2      = 2

y          =  6 -  2 * 3     = 0

X	1	2	3
y	4	2	0

 4 x – 2 y – 4 = 0

Divide by 2 we get

2 x – y - 2                  = 0 

Add y both side we get

2 x – 2            = y

or

y          = 2x – 2

plug x  = 1.2 and 3 we get

y          = 2 * 1 – 2      = 0

y          = 2 * 2 – 2      = 2

y          = 2 * 3 – 2      =  4

X	1	2	3
y	0	2	4

Answer x = 2 and y = 2

(iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

Compare the equation with

         

We get

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

a2 = 2             b2        =  - 2               and c2 = - 5

So both lines are Inconsistent