Question

IF the equations 2x - 3y = 7 and (a+b)x - (a+b-3)y =4a+b have infinitely many solutions then find a and b​

Answer 1

Equations:

Comparing each of equations with the standard form of linear equation (i.e., ax + by + c = 0 ), we get

1) 2x - 3y = 7

{ a1 = 2, b1 = -3, c1 = 7 }

2) (a + b)x - (a + b - 3)y = 4a + b

{ a2 = (a + b), b2 = -(a + b - 3), c2 = 4a + b }

Now,

For infinitely many Solutions:

==> a1/a2 = b1/b2 = c1/c2

==> 2/(a + b) = -3/-(a + b - 3) = 7/(4a + b)

Case -1 :

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

==> 2a + 2b - 6 = 3a + 3b

==> a + b = -6 ...(1)

Case-2:

==> -3/-(a + b - 3) = 7/(4a + b)

==> 3(4a + b) = 7(a + b - 3)

==> 12a + 3b = 7a + 7b - 21

==> 5a - 4b = -21 ...(2)

Multiply (1) by 5

==> 5a + 5b = -30 ...(3)

Subtract (2) from (3)

==> 5a + 5b - 5a + 4b = -30 - (- 21)

==> 9b = -9

==> b = -1

Putting b = -1 in (1)

==> a + b = -6

==> a - 1 = -6

==> a = -5

Hence, For infinitely many solutions, value of a is -5 and value of b is -1