For infinitely many solutions, the coefficient determinant and every replaced determinant must vanish.

$D=\left|\begin{array}{c}111\\ 123\\ 13\lambda \end{array}\right|=(2\lambda -9)-(\lambda -3)+(3-2)=\lambda -5$.

$D=0\Rightarrow \lambda =5$.

For consistency with infinite solutions, applying the same condition with the constants column ($5,9,\mu$) gives $\mu =13$.

(Quick check: the third equation must be a linear combination of the first two. Subtracting the first from the second: $y+2z=4$. Twice that plus the first: $x+3y+5z=13$, matching $\lambda =5,\mu =13$.)

$\therefore \lambda +\mu =5+13=18$.