Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 17 / b / iii

For $\lambda \ne \pm 2$, the reduced equation gives $y=(3-\lambda )/(4-{\lambda }^2)$. Back substitution into $x=1-\lambda y$ and $z=1-4y$ gives all solutions:The nonzero determinant proves that there are no other solutions at these parameter values.