State True or False.The pair of linear equations 2x – 2y – 2 = 0, 4x – 4y – 5 = 0 is consistent.

Question

Accepted Answer

Given that pair of linear equations 2x – 2y – 2 = 0, 4x – 4y – 5 = 0 is consistent.We know that the pair of linear equations in two variables x and y is represented by the standard form,

a
      1
     
     x+
      b
      1
     
     y+
      c
      1
     
     =0

a
      2
     
     x+
      b
      2
     
     y+
      c
      2
     
     =0

Here

a
    1
   
   ,
    a
    2
   
   ,
    b
    1
   
   ,
    b
    2
   
   ,
    c
    1
   
   ,
    c
    2

are the coefficients of the equations.On comparing the ratios

a
      1

a
      2

,

b
      1

b
      2

,

c
      1

c
      2

, we can determine whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.For the given pair of linear equations to be consistent, the comparison of

a
      1

a
      2

,

b
      1

b
      2

and

c
      1

c
      2

must satisfy the condition

a
      1

a
      2

≠

b
      1

b
      2

or

a
      1

a
      2

=

b
      1

b
      2

=

c
      1

c
      2

and to be inconsistent, they must satisfy

a
      1

a
      2

=

b
      1

b
      2

≠

c
      1

c
      2

.
  Here,

a
      1
     
     =2,
      b
      1
     
     =−2,
      c
      1
     
     =−2,

a
      2
     
     =4,
      b
      2
     
     =−4,
      c
      2
     
     =−5.

Note down the coefficients and constants in the fraction form.

a
        1

a
        2

=
      2
      4

⇒

a
        1

a
        2

=
      1
      2

b
        1

b
        2

=
      
       −2
      
       −4

⇒

b
        1

b
        2

=
      1
      2

c
        1

c
        2

=
      
       −2
      
       −5

⇒

c
        1

c
        2

=
      2
      5

From the above ratios, we can observe that

1
      2
     
     =
      1
      2
     
     ≠
      2
      5

⇒

a
        1

a
        2

=

b
        1

b
        2

≠

c
        1

c
        2

So, the given lines are parallel and the equations have no solution. Hence, it is an inconsistent pair.Thus, the statement is false.So, the pair of linear equations 2x – 2y – 2 = 0, 4x – 4y – 5 = 0 are inconsistent.Therefore, option 2 is correct.

State True or False.

The pair of linear equations 2x – 2y – 2 = 0, 4x – 4y – 5 = 0 is consistent.

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