State whether the following statement is true or false.If the equations

3x−y
=14
and

(a+b)x−2by
=5a+2b+1
are a pair of dependent linear equations, then the value of

a
and

b
are 5 and 1 respectively.

Question

State whether the following statement is true or false.If the equations

3x−y
   =14
   and

(a+b)x−2by
   =5a+2b+1
   are a pair of dependent linear equations, then the value of

a
   and

b
   are 5 and 1 respectively.

Accepted Answer

The given equations are,

3x−y
   =14

(a+b)x−2by
   =5a+2b+1
  We know that if two lines are coincident lines, then the pair of the linear equation is called dependent. A pair of linear equations

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

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

is coincident or dependent if ( and only if)

a
      1

a
      2

=

b
      1

b
      2

=

c
      1

c
      2

Since we have,

3x−y
   =14

(a+b)x−2by
   =5a+2b+1
  As these equations are dependent, therefore we have,

3
    
     a+b
   
   =
    
     −1
    
     −2b
   
   =
    
     14
    
     5a+2b+1

⇒
      3
      
       a+b
     
     =
      
       −1
      
       −2b

⇒6b=a+b

⇒a=5b

and

−1
      
       −2b
     
     =
      
       14
      
       5a+2b+1

⇒5a+2b+1=28b

⇒25b+2b+1=28b

⇒b=1

Thus if the equations

3x−y
   =14
   and

(a+b)x−2by
   =5a+2b+1
   are a pair of dependent linear equations, then the value of

a
   and

b
   are 5 and 1 respectively.Therefore the given statement is true.Hence option (1) is correct.

State whether the following statement is true or false.

If the equations $3 x − y = 14$ and $(a + b) x − 2 b y = 5 a + 2 b + 1$ are a pair of dependent linear equations, then the value of $a$ and $b$ are 5 and 1 respectively.

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State whether the following statement is true or false.If the equations 3x−y =14 and (a+b)x−2by =5a+2b+1 are a pair of dependent linear equations, then the value of a and b are 5 and 1 respectively.