Question

Find the value of k for which each of the following system of equations have no solution :

2x - ky + 3 = 0

3x + 2y - 1 = 0

Answer 1

The given system of the equation may be written as

2x - ky + 3 = 0

3x + 2y - 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 = -k, c_1 = 3`

And

`c_2 = 3, b_2 = 2, c_2 = -1`

For a unique solution, we must have

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

We have

`a_1/a_2 = 2/3`

and `c_1/c_2 = 3/(-1)`

Clearly, `a_1/a_2 != c_1/c_2`

So, the given system will have no solution. If

`a_1/a_2 = b_1/b_2 i.e 2/k = (-k)/2 => k = (-4)/3`

Hence, the given system of equations has no solutions `k = (-4)/3`