Question

For which value of a and b does the following pair of linear equations have an infinite number of solutions ?

2x+3y = 7
(a-b)x+(a+b)=3a+b-2​

Answer 1

Answer:

hey mate, here is your answer

Step-by-step explanation:

Given Equation :

                     1 . 2 x + 3 y = 7 ⇒ (1)

                     2 . ( a -b) x + ( a + b ) = 3 a + b -2 ⇒ ( 2)

→ 2 x + 3y = 7

→ 2x+ 3y -7 = 0

→  comparing with a 1 X  + b 1 y + + c 1 = 0

→ ∴ a 1 = 2  , b 1 = 3, c 1 = -7

now , ( a - b) x + ( a + b)  y = 3a + b -2

     → ( a - b) x + ( a + b) y - ( 3 a + b -2 ) = 0

    →   comparing with a 2 + b2 y + c 2 = 0

  →   ∴ a 2 = ( a -b) , b 2 y + c 2 =  0

  →  ∴ a 2 = ( a - b) , b 2 = ( a + b) ,

 →  c 2 = -( 3 a + b -2)

so , a 1 = 2 , b1 = 3, c1 = -7  

and  a2 = ( a -b) , b2 = ( a + b) ,c2 =( -3 a +b -2)

it is given that the equation has infinite numbers of solutions

so, a1/a2 = b1/b2 =c1/c2

putting in values

2 ( a- b) = 3/( a+ b) = -7 / -( 3a + b-2)

2 / ( a-b) = 3/( a+ b)  =  -7/ -( 3a + b-2)

= solving 2/( a-b) = 3/( a + b)

           →   2 ( a +b) = 3 ( a-b)

          →     2a + 2b = 3a -3b

           →   2b + 3b =3a - 2a

          → 5b = a

         →  a = 5 b      (3) equation

solving  2 / ( a - b)  = 7 / ( 3a + b-2)

            = 2 ( 3a + b-2) = 7( a -b)

           = 6a + 2 b -4 = 7a - 7 b

          = 2 b - 4 + 7b = 7 a - 6 a

         = 9b - 4 = a      ( equation = 4)

= comparing( 3 )and (4)

= 5 b =9b - 4

= 4 = 9b - 5b

= 4 =4b

= 4b = 4

= b = 4/4 = 1

now, putting b = 1 in (3) equation

                    a = 5b

                    a = 5 (1)

                     a =5

therefore ,for a = 5 ,b =  1 the given set of equations have infinitely many solutions.

thank you ,good night