Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Digital systems, linear algebra, introduction to formal proof (resit) / 6 / e / Solution

The second, third and fourth linear equations giveSetting $x_4=t$ yields $x_3=-1-t$, $x_2=1+t$ and $x_1=1-t$. The first linear equation then gives $(1-t)+(1+t)+(-1-t)+t=1$, and the fifth gives $(1-t)+t=1$, so both remaining conditions hold for every real $t$. ConsequentlyThis is a particular solution plus the whole null space, an affine line in a vector space of solutions. The derivation both produces solutions and shows that none have been omitted.