First show that every solution is of the form \(X0+Y\) where Y is a solution of the homogeneous system \(AY=0\). Let X be one solution. \(AX=b\).

Then \(AX-AX0=b-b=0 \rightarrow A(X-X0)=0.\)

Conclude that \(X-X0=Y\), therefore \(X=X0+Y\)

Now show that any vector of the form \(X0+Y\) is a solution. \(A(X0+Y)=AX0+AY=b+0=b\)