Question

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

8x + 5y = 9

kx + 10y = 18

Answer 1

The given system of equation is

8x + 5y - 9 = 0

kx + 10y - 18 = 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 = 8, b_1 = 5, c_1 = -9`'

And `a_2 = k, b_2 = 10, c_2 = -18`

For a unique solution, we must have

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

`=>8/k = 5/10 = (-9)/(-18)`

Now

`8/k = 5/10`

`=> 8 xx 10 = 5 xx k`

`=> (8xx10)/5 = k`

`=> k = 8 xx 2 = 16`

Hence, the given system of equations will have infinitely many solutions, if k = 16