The system of equations $x+2y+2z=5$, $x+2y+3z=6$, $x+2y+\lambda z=\mu$ has infinitely many solutions if

Explanation

Subtract first equation from second and third This gives the augmented matrix $\begin{bmatrix}1&2&2&5\\0&0&1&1\\0&0&\lambda-2&\mu-5\end{bmatrix}$ For infinitely many solutions $\lambda-2=0$ and $\mu-5=0$ Hence $\lambda=2$ and $\mu=5$