Question

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

\[kx + 3y = 2k + 1\]
\[2\left( k + 1 \right)x + 9y = 7k + 1\]

Answer 1

The given system of the equation may be written as

\[kx + 3y = 2k + 1\]
\[2\left( k + 1 \right)x + 9y = 7k + 1\]

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 = k, b_1 = 3, c_1 = -(2k + 1)`

And `a_2 = 2(k +1), b_2 = 9, c_2 = -(7k + 1)`

For a unique solution, we must have

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

`=> 1/(2(k +1)) = 3/9 = (-(2k + 1))/(-(7k + 1))`

`=> k/(2(k +1) = 3/9 and 3/9 = (2k + 1)/(7k + 1)`

`=> 9k = 6(k +1) and 21k +3 = 18 k +9`

`=> 9k - 6k = 6 and 21k - 18k = 9 - 3`

`=> 3k = 6 and 3k = 6`

`=> k = 6/3 and k = 6/3`

=> k = 2 and  k = 2

=> k = 2 satisfies both the conditions

Hence, the given system of equations will have infinitely many solutions, if k = 2.