Does this linear system have no solutions?

Question

Am I correct in saying if I have a linear system where a variable equals two different outputs, then there are no solutions? E.g., $$\begin{cases}x= 4 \\ x=5\end{cases}$$

Answer 1

Yes, because there is no value of $x$ that satisfies both $x=4$ and $x=5$.

Answer 2

For another point of view

Geometrically, the graph of $x = 4$ is a vertical line which intersects the $x$-axis at the point $(4, 0)$. Similarly, the graph of $x = 5$ is (another) vertical line which intersects the $x$-axis at the point $(5, 0)$. Clearly, as your system of equations is represented by two parallel lines which are not coincident, then the system is inconsistent. We conclude that the system of equations does not have a solution.

Here is the WolframAlpha plot for these two lines: