Question

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

4x + 5y = 3

kx + 15y = 9

Answer 1

The given system of equation is

4x + 5y -3 = 0

ks + 15y - 9 = 0

The system of equation is of the for

`a_1x + b_1y + c_1= 0`

`a_2x + b_2y + c_2 = 0`

Where `a_1 = 4, b_1 = 5, c_1 = -3`

And `a_2 = k, b_2 = 15, c_2 = -9`

For a unique solution, we must have

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

`=> 4/k = 5/15 = (-3)/(-9)`

Now

`4/k = 5/15`

`=> 4/k = 1/3`

`=> k = 12`

Hence, the given system of equations will have infinitely many solutions, if k = 12