Originally Posted by

**losm1** Hello guys,

I am analyzing system of linear equations with coefficients that are part of arithmetic progression.

Example of such system is

$\displaystyle \begin{array}{rcl}x+2y & = & 3 \\ x-2y & = & -5\end{array}$

I should draw conclusions and prove them for 2x2 and 3x3 systems.

__2x2__

By plotting equation lines it is easy to conclude that all such 2x2 systems have unique solution [-1;2].

By looking at a single equation

$\displaystyle ax+(a+k)y = a+2k \Leftrightarrow a(x+y)+ky=a+2k$

I have found that it will evaluate to true if

$\displaystyle ky=2k \Leftrightarrow y=2 \land x+y=1 \Leftrightarrow x=-1$.

Does this holds? I would appreciate very much if someone could provide real (rigor) proof.

__3x3__

I have tried few combinations of coefficients in matlab and got [0;-1;2] solution column vector few times, but I also got [NaN; Inf ; -Inf] in some cases. I do not have skills to prove this algebraically, any help?

Thanks!