Question

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

2x + 3y = 7

(k + 1)x + (2k - 1)y - (4k + 1)

Answer 1

The given system of equation may be written as

2x + 3y - 7 = 0

(k + 1)x + (2k - 1)y - (4k + 1) = 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 = 2, b_1 = 3, c_1 = -7`

And `a_2 = k + 1, b_2 = 2k - 1, c_2 = -(4k + 1)`

For a unique solution, we must have

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

`=> 2/(k +1) = 3/(2k - 1) = (-7)/(-(4k + 1))`

`=> 2/(k + 1) = 3/(2k - 1) and 3/(2k - 1) = 7/(4k + 1)`

`=> 2(2k - 1) = 3(k + 1) and 3(4k + 1) = 7(2k - 1)`

=> 4k - 2 = 3k + 3 and 12k + 3 = 14k -7

=> 4k - 3k =- 3 + 2 and 12k - 14k = -7-3

=> k = 5 and -2k = -10

`=> k = 5 and  k = 10/2 = 5`

=> k = 5 satisfies both the conditions

Hence, the given system of equations will have infinitely many s if k = 5