If the system of equations $$x + y + z = 5$$, $$x + 2y + 3z = 9$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, then the value of $$\lambda + \mu$$ is :

$$\Delta = \Delta_x = \Delta_y = \Delta_z = 0$$ (for infinite solutions)

$$\Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda \end{vmatrix}$$

$$\Delta = 1(2\lambda - 9) - 1(\lambda - 3) + 1(3 - 2)$$

$$\Delta = 2\lambda - 9 - \lambda + 3 + 1 = \lambda - 5$$

For infinitely many solutions, we set $$\Delta = 0$$:

$$\lambda - 5 = 0 \implies \lambda = 5$$

$$\Delta_z = \begin{vmatrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & \mu \end{vmatrix}$$

$$\Delta_z = 1(2\mu - 27) - 1(\mu - 9) + 5(3 - 2)$$

$$\Delta_z = 2\mu - 27 - \mu + 9 + 5 = \mu - 13$$

Set $$\Delta_z = 0$$: $$\mu - 13 = 0 \implies \mu = 13$$

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