Question

Prove that there is a value of $c(\ne 0)$ for which the system
$6x+3y=c-3$
$12x+cy=c$
has infinitely many solutions.

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)

$\Rightarrow \frac{6}{12}=\frac{3}{c}=\frac{c-3}{c}$

Equate first two equations ;

$\Rightarrow c=6$