x + y + z = 5

x + 2y + 2z = 6

x + 3y + $$\lambda $$z = $$\mu $$ have infinite solution

$$\Delta $$ = 0, $$\Delta $$x = $$\Delta $$y = $$\Delta $$z = 0

$$\Delta = \left| {\matrix{ 1 & 1 & 1 \cr 1 & 2 & 2 \cr 1 & 3 & \lambda \cr } } \right| = 0$$

$$ \Rightarrow 1(2\lambda - 6) - 1(\lambda - 2) + 1(3 - 2) = 0$$

$$ \Rightarrow 2\lambda - 6 - \lambda + 2 + 1 = 0$$

$$ \Rightarrow \lambda = 3$$

Now, $$\Delta x = \left| {\matrix{ 5 & 1 & 1 \cr 6 & 2 & 2 \cr \mu & 3 & 3 \cr } } \right| = 0$$,

$$\Delta $$y = $$\left| {\matrix{ 1 & 5 & 1 \cr 1 & 6 & 2 \cr 1 & \mu & 3 \cr } } \right| = 0$$

$$ \Rightarrow \left| {\matrix{ 1 & 5 & 1 \cr 0 & 1 & 1 \cr 0 & {\mu - 5} & 2 \cr } } \right| = 0$$

$$ \Rightarrow \mu = 7$$

$$\Delta z = \left| {\matrix{ 1 & 1 & 5 \cr 1 & 2 & 6 \cr 1 & 3 & \mu \cr } } \right| = \left| {\matrix{ 1 & 1 & 5 \cr 0 & { - 1} & { - 1} \cr 0 & 2 & {\mu - 5} \cr } } \right|$$

$$ \Rightarrow 1(5 - \mu + 2) = 0$$

$$ \Rightarrow \mu = 7$$

So, $$\lambda + \mu = 10$$