Question

Find the value of $\lambda$ if the following equations are consistent
$x+y-3=0$
$(1+\lambda )x+(2+\lambda )y-8=0$
$x-(1+\lambda )y+(2+\lambda )=0$

Answer 1

For the equations to be consistent, $D=0.$
$\therefore D=\left|\begin{array}{ccc}1& 1& -3\\ 1+\lambda & 2+\lambda & -8\\ 1& -1-\lambda & 2+\lambda \end{array}\right|=0$
Here the equations are in two variables x and y. If they are consistent then the values of x and y obtained from first two should satisfy the third and hence D = 0
Apply $C_2-C_1,C_3+3C_1$
$D=\left|\begin{array}{ccc}1& 0& 0\\ 1+\lambda & 1& -5+3\lambda \\ 1& -2-\lambda & 5+\lambda \end{array}\right|=0$
or $(5+\lambda )+(2+\lambda )(-5+3\lambda )=0$
or $5+\lambda +(-10+\lambda +3{\lambda }^2)=0$

or $3{\lambda }^2+2\lambda -5=0$ or $(\lambda -1)(3\lambda +5)=0$

$\therefore \lambda =1,-5/3.$