Linear System: Determine whether it is consistent

Question

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.

Answer 1

To show whether it is consistent, you just have to exhibit a solution.

Let $w=1$ and let the other variables be $0$.

Answer 2

Yes you system is consistent and you have infinitely many solutions.

You may pick $x$ and $z$ such that $x=2z$ and solve for $y$ and $w$ such that $5y+w=1.$

For example $$(x,y,z,w)=(2,1,1,-4)$$ is such a solution.

Answer 3

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 $4-2=2$.