Question

Determine for what values of $\lambda$ and $\mu$ the following equations have an infinite number of solutions
$x+y+z=6x+2y+3z=10x+2y+\lambda z=\mu$
Find the value of $\lambda \mu$

Answer 1

Augmented matrix is

$\left[\begin{array}{cccc}1& 1& 1& 6\\ 1& 2& 3& 10\\ 1& 2& \lambda & \mu \end{array}\right]$

applying $R_3\to R_3-R_1$ and $R_2\to R_2-R_1$ gives

$\left[\begin{array}{cccc}1& 1& 1& 6\\ 0& 1& 2& 4\\ 0& 1& \lambda -1& \mu -6\end{array}\right]$

given system has infinite solutions if $\lambda -1=2$ and $\mu -6=4$

$\Rightarrow \lambda =3$ and $\mu =10$

$\therefore \lambda \mu =30$