Past exam of the mathematical methods I course of the University of Oxford / 2017 / 4 / b / Solution

For $\lambda =1$ and $\mu =1/2$, the third equation gives $x=z-1$. Combining it with $2x-z=-4$ gives $z=-2$ and $x=-3$, after which the second equation gives $y=12$. HenceThe coefficient matrix has determinant $\lambda -2=-1\ne 0$, proving uniqueness. Direct substitution gives the required right-hand sides $1$, $5$ and $-1$.