Question

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y – 7 = 0

(a – 1)x + (a + 1)y = (3a – 1)

Answer 1

The given system of equations is

2x + 3y – 7 = 0

(a – 1)x + (a + 1)y – (3a – 1) = 0

It is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 2, b_1 = 3, c_1 = -7`

And `a_2 = a - 1, b_2 = a + 1, c_2 = -(3a - 1)`

The given system of equations will be have infinite number of solutions, if

`a_1/a_2 = b_1/b_2 = c_1/c_2`

`=> 2/(a - b) = 3/(a + 1) = (-7)/(-(3a - 1))`

`=> 2/(a - 1) = 3/(a + 1) = (-7)/(-(3a - 1))`

`=> 2/(a - 1) = 3/(a + 1) = (-7)/(3a -1)`

`=> 3/(a - 1) = 3/(a + 1) and 3/(a + 1) = 7/(3a - 1)`

`=> 2(a + 1) = 3(a - 1) and 3(3a - 1) = 7(a + 1)`

`=>2a + 2 = 3a - 3 and 9a - 3 = 7a + 7`

`=> 2a - 3a = -3 and 9a - 3 = 7a + 7`

`=> -a = -5 and 2a  = 10`

`=> a = 5 and a = 10/2 = 5`

`=> a = 5`

Hence, the given system of equations will have infinitely many solutions, if a = 5.