Prove a 3x3 system of linear equations with arithmetic progression coefficients has infinitely many solutions How can I prove that a 3x3 system of linear equations of the form:
$\begin{pmatrix}
a&a+b&a+2b\\
c&c+d&c+2d\\
e&e+f&e+2f
\end{pmatrix}
\begin{pmatrix}
x\\ y\\ z
\end{pmatrix}
=\begin{pmatrix}
a+3b\\
c+3d\\
e+3f
\end{pmatrix}$
for $a,b,c,d,e,f \in \mathbb Z$ will always have infinite solutions and will intersect along the line 
$ r=
\begin{pmatrix}
 -2\\3\\0
\end{pmatrix}
+\lambda
\begin{pmatrix}
 1\\-2\\1
\end{pmatrix}$

This follows immediately from a basic theorem of linear algebra. Namely, the solution of a nonhomogeneous linear system can be expressed as the sum of any particular solution plus the general solution of the homogeneous system. Since your exhibited solution has this form it is necessarily a solution, i.e. $\rm\ \ L(x_0) = a_0,\ L(x) = 0\ \Rightarrow\ L(x_0+\lambda\ x)\ =\ L(x_0)+\lambda\ L(x)\ =\ a_0\:.\ \ $ This sort of "shifted" vector space structure is known as an affine space.
