The system of linear equations$x+y+z=4$,$x+2y+3z=7$,$x+4y+\lambda z=\mu$ has a unique solution if:

A system of linear equations $AX=B$ has a unique solution if $|A|\ne 0$.The matrix $A$ is:
$$A=\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 4& \lambda \end{array}\right)$$
Calculate $|A|$:
$$|A|=1(2\lambda -12)-1(\lambda -3)+1(4-2)$$
$$|A|=2\lambda -12-\lambda +3+2=\lambda -7$$
For a unique solution, $|A|\ne 0$:
$$\lambda -7\ne 0⟹\lambda \ne 7$$
The value of $\mu$ does not affect the uniqueness of the solution as long as $|A|\ne 0$.