Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 2 / 15 / b / Solution

Subtracting the third row from the first gives $z=-1$, and the third row gives $y=2-x$. The second row then reduces toFor $t\ne 1$, the unique solution isConsequently $s=1$ gives $\boxed{(x,y,z)=(1,1,-1)}$ for every $t\ne 1$, whereas $s=6$ gives

When $t=1$, the first and second left-hand sides coincide. If $s=1$, they are the same plane, and the remaining independent plane cuts it in the lineIf $s=6$, the two planes are distinct and parallel, so there is no solution. Geometrically, for $t\ne 1$ the three planes intersect at a single point; at $t=1$ the coincident-plane case leaves an entire line, while the parallel-plane case has empty intersection.