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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.

A particular solution is $x_p=(1,0,-1,0{)}^T$, as direct substitution into all five equations verifies. Add any vector in the null space to obtain every solution:Conversely, the difference of any two solutions is annihilated by $A$, so there are no further solutions. This is the affine solution space of a consistent linear system.

Reducing the augmented matrix givesThere is no contradictory row. Hence the system is consistent, with affine solution spaceThe three planes intersect in this common line. Their normals span a two-dimensional row space, leaving a one-dimensional common direction, the null space. More explicitly, five times the first equation minus eleven times the second plus twice the third is the zero equation, including the right-hand side; the third plane therefore contains the intersection of the first two.