Question

Find the value of k for which each of the following system of equations has infinitely many solutions :

x + (k + 1)y =4

(k + 1)x + 9y - (5k + 2)

Answer 1

The given system of the equation may be written as

x + (k + 1)y - 4 = 0

(k + 1)x + 9y - (5k + 2) = 0

The system of equation is of the form

`a_1x + b_1y + c_1 = 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 1,b_1 = k + 1, c_1 = -4`

And `a_2 = k +1, b_2 = 9, c_2 = -(5k + 2)`

For a unique solution, we must have

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

`=> 1/(k +1) = (k +1)/9 = (-4)/(-(5k + 2))`

`=> 1/(k +1) = (k +1)/9 and (k+1)/9 = 4/(5k +2)`

`=> 9 = (k +1)^2 and (k +1)(5k + 2) = 36`

`=> 9 = k^2 + 1 + 2k and 5k^2 + 2k + 5k + 2 = 36`

`=> k^2 + 2k + 1 - 9 = 0 and 5k^2 + 7k + 2 - 36 = 0`

`=> k^2 + 2k - 8 = 0 and 5k^2 + 7k - 34 = 0`

`=> k^2 + 4k -2k - 8 = 0  and 5k^2 + 17k - 10k - 34 = 0`

 => k(k + 4) - 2(k + 4) = 0 and (5k + 17) - 2(5k + 17) = 0

=> (k + 4)(k -2) = 0 and (5k + 1) (k -2) = 0

`=> (k = -4 or k = 2) and (k = (-17)/5  or k = 2)`

k = 2 satisfies both the conditions

Hence, the given system of equations will have infinitely many solutions, if k = 2