TITLE: prove that all lines $ax+y=b$ such that coefficients $a, 1, b$ constitute arithmetic sequence have one common point
QUESTION [6 upvotes]: Prove that all lines $ax+y=b$ such that coefficients $a,$$1,$$b$ constitute arithmetic sequence have one common point.
We know that $1-a=b-1$
and solving for b we get $b=$$2-a$
replacing b in the equation $y = 2-a-ax$ but I am not sure where to go from here that will help me prove my claim.

REPLY [2 votes]: We have the equation,
$ax+y=b$                $\tag{1}$   
also we have
$a,1,b$ in A.P.
therefore $1-a=b-1$
$-a+2=b  $    $\tag{2}$
On comparing $(1)$ and $(2)$ we have
The common point $=(-1,2)$