The equations you gave represent three lines in $\mathbb R^2$. When we have three (or more) lines in a plane, we can have one of the following situations:
- They have infinitely many common points.
- They have one point in common.
- They have no points in common.
The lines $2x=1$ and $y+5=x$ are clearly not parallel (can you see this?). So they intersect at one point, which means we are not dealing with case one. The lines will intersect at the point $(x,y)=\left(\frac12;-4\frac12\right)$.
Now the three lines will have one point in common if $\left(\frac12;-4\frac12\right)$ lies on the third line and no points in common when it doesn't. I will leave it to you to figure out if it is on the third line or not.
After edit from OP: We can solve this even quicker if we just look at the bottom two equations: $$\begin{cases} x=y+5\\ x=y+3\end{cases}$$ These lines are obviously parallel (and don't coincide). You could also argue that this implies $y+5=y+3$, which is a contradiction.