Show that $2x+4y=3$ and $x+2y=4$ are consistent over $GF(5)$ and inconsistent over $\mathbb{R}$

Question

Show that the two equations $$2x+4y=3$$ $$x+2y=4$$ are consistent over $GF_5$ but inconsistent over $\mathbb{R}$.

For $\mathbb{R}$, I can see that $\begin{pmatrix}2x+4y\\2x+4y\end{pmatrix}=\begin{pmatrix}3\\8\end{pmatrix}$ has no solution.
Also, for $GF_5$, I can solve for each value (since the finite set has very few elements) and verify that $\begin{pmatrix}1\\4\end{pmatrix},\begin{pmatrix}0\\2\end{pmatrix},\begin{pmatrix}2\\1\end{pmatrix},\begin{pmatrix}4\\0\end{pmatrix}$ and $\begin{pmatrix}3\\3\end{pmatrix}$ are the only solutions to the system of equations.
Also, obviously it is enough to find even $1$ solution to the system in $GF_5$ to prove its consistency.

Is there a formal proof for the above? My method seems to involve "verification" instead of a "proof", is it enough anyway?

Answer 1

First, checking all possible solution is a valid proof.

Second, you could also have noticed that in $GF(5)$, $2x+4y=3$ and $2x+4y=8$ are the same equation since $3=8$ in characteristic five.