Question

Find the value of $k$ for which each of the following systems of equations have infinitely many solution:
$2x+(k-2)y=k$
$6x+(2k-1)y=2k+5$

Answer 1

For the system to have infinitely many solutions, lines must coincide each other

$\Rightarrow$ratio of respective coefficients must be equal (x, y and constant coefficients)

$\Rightarrow \frac{2}{6}=\frac{k-2}{2k-1}=\frac{k}{2k+5}$

Equate first two equations ;

$\Rightarrow 2k-1=3k-6$

$\Rightarrow k=5$