How to solve this matrix using gauss-elimination by hand I feel like i am having a brain fart. I have been given this $Ax=b$ system:
$A= \begin{pmatrix} 0.913 & 0.659 \\ 0.780 & 0.563 \end{pmatrix}$
$b= \begin{pmatrix} 0.254 \\ 0.217 \end{pmatrix}$
I know the answer is $x_1 = 1$ and $x_2 = -1$ but for some reason when i try to solve this using Gaussian elimination, the entire bottom row goes to $0$ giving me $1$ equation with $2$ variables.
Can someone get me out of this please? 

Alright I worked it out step by step, so bear with me (I'll use A followed by b notation):
\begin{pmatrix} 0.913 & 0.659 \\ 0.780 & 0.563 \end{pmatrix} \begin{pmatrix} 0.254\\ 0.217 \end{pmatrix}
We multiply the top row by the bottom and vice versa:
\begin{pmatrix} 0.71214 & 0.51402 \\ 0.71214 & 0.514019 \end{pmatrix} \begin{pmatrix} 0.19812\\ 0.198121 \end{pmatrix}
Subtract the bottom from top:
\begin{pmatrix} 0 & 0.000001 \\ 0.71214 & 0.514019 \end{pmatrix} \begin{pmatrix} -0.000001\\ 0.198121 \end{pmatrix}
Divide the top row by 0.000001 to reduce it to:
\begin{pmatrix} 0 & 1 \\ 0.71214 & 0.514019 \end{pmatrix} \begin{pmatrix} -1\\ 0.198121 \end{pmatrix}
Multiply the top row by 0.514019 and subtract it from the bottom to get:
\begin{pmatrix} 0 & 1 \\ 0.71214 & 0\end{pmatrix} \begin{pmatrix} -1\\ 0.71214\end{pmatrix}
Diving the bottom row by 0.71214 we end with:
\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \begin{pmatrix} -1\\ 1\end{pmatrix}
from which we can readily see x1 = 1 and x2 = -1
