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 / 2 / e

Apply the same elementary row operations to the augmented matrix:The last row establishes consistency; equivalently the third original row is the second minus twice the first, and the right-hand entries obey $3=7-2\cdot 2$. With $x_2=s$ and $x_4=5t$, the second equation gives $x_3=1+3t$ and the first gives $x_1=3-2s-17t$. All real solutions are thereforeThis affine solution space is one particular solution plus the null space. The two free variables show that there are no additional restrictions.