Question

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

2x + 3y − 5 = 0
6x + ky − 15 = 0

Answer 1

The given system of equation is

2x + 3y − 5 = 0

6x + ky − 15 = 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 = -5`

And `a_2 = 6, b_2 = k,c_2 = -15`

For a unique solution, we must have

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

`=> 2/6 = 3/k`

`=> k = 18/2 = 9`

Hence, the given system of equations will have infinitely many solutions, if k = 9.