Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 17 / c / iii

At $\mu =-1$, the first equation gives $x=y$, and the second then gives $z=0$. All nonzero solutions areAt $\mu =6$, the first equation gives $x=-6y$. The second then gives $3z=7y$, and the third is twice the second. Choosing an integer direction vector, all nonzero solutions areEach full null space, including the zero solution, is a line through the origin. Geometrically, the three homogeneous constraint planes have that common intersection; in each case only two of their equations are independent.