Does the following system of linear equations contain infinite solutions?

Solution 1:

Here is a smaller toy example which, I believe, captures the essence of your problem (it is a system of four equations in two unknowns which is clearly inconsistent):

$$\begin{align} x+y&=1\\ x+y&=2\\ x+y&=1\\ x+y&=2\\ \end{align}$$

Adding either the first or the last pair of equations together yields the line $2x+2y=3$. But you can't conclude anything about the solution space from this single equation. Really all you are saying is that if $(x,y)$ solves a pair of original equations, then it solves the new equation, but the converse is not true at all. What you can do with such a combination of two previous equations is substitute it in the original system (this is one of the elementary row operations), but then you still have to solve the system.