Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 2 / c

The first two equations give $x_1=1-2x_3$ and $x_2=2-5x_3$. Substitution into the third givesThis equation is inconsistent exactly when its left side is identically zero and its right side is nonzero. Therefore

The elimination in i determines $x_3$ uniquely exactly when $\lambda \ne 5$. The other two coordinates are then also fixed. Explicitly,For $\lambda =5$, the reduced equation is either inconsistent or leaves $x_3$ free, so a unique solution is impossible.

By i, a free choice of $x_3$ is possible precisely when $\lambda =5$ and $\mu =2$. ThusThe complete solution set of a linear system isThe direction vector spans the kernel, and the displayed straight line is the entire solution set of a linear system.