Question

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

kx - 2y + 6 = 0

4x + 3y + 9 = 0

Answer 1

The given system of equation is

kx - 2y + 6 = 0

4x + 3y + 9 = 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 = k, b_1 = -2,c_1 = 6`

And `a_2 = 4, b_2 = -3,c_2 = 9`

For a unique solution, we must have

`a_1/a_2 = b-1/b_2 = c-1/c_2`

`=>k/4 = (-2)/(-3) = 6/9`

Now

`k/4 = 6/9`

`=> k/4 = 2/3`

`=> k = (2xx4)/3`

`=> k = 8/3`

Hence, the given system of equations will have infinitely many solutions, if `k = 8/3`