Question

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

3x - 4y + 7 = 0

kx + 3y - 5 = 0

Answer 1

The given system of equation may be written as

3x - 4y + 7 = 0

kx + 3y - 5 = 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 = 3, b_1 = -4, c_1 = 7`

And `a_2 = k, b_2 = 3, c_2 = -5`

For a unique solution, we must have

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

We have,

`b_1/b_2 = (-4)/3`

and `c_1/c_2 = (-7)/5`

Clearly, `b_1/b_2 != c_1/c_2`

So, the given system will have no solution

`if a_1/a_2 = b_1/b_2 => 3/k = (-4)/3 => k = (-9)/4`

Clearly, for this value of k, we have `a_1/a_2 = b_1/b_2 != c_1/c_2`

Hence, the given system of equations has no solutions, when `k = (-9)/4`