TITLE: Linear System: Determine whether it is consistent
QUESTION [0 upvotes]: Problem:
$$
\begin{align}
5y+w=1 \\
2x+5y-4z+w=1
\end{align}
$$
What I've done:
-1 times first equation in order to get rid of $5y$ and $w$.
Then, when I add first and second equation, the equation becomes $2x-4z=0$
And I'm stuck right here.
Thanks.

REPLY [1 votes]: The system is consistent because the matrix of the system and the augmented matrix:
$$\begin{bmatrix}0&5&0&1\\2&5&-4&1\end{bmatrix}\enspace\text{ and }\enspace\begin{bmatrix}0&5&0&1&1\\2&5&-4&1&1\end{bmatrix}\enspace \text{ resp. }$$
have the same rank, which is the maximal rank, $2$.
Furthermore, the set of solutions is an affine subspace of $K^4$ ($K$ denotes the base field) of codimension $2$, i.e. of dimension $4-2=2$.
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