When solving a system of linear equations algebraically, how do you know when the system has no solution? infinitely many solutions?

You know that the system of equations has no solution because you get something funky when trying to solve for x like:

This happens because the system dont intersect.
Now there are infinite number of solutions when its the same equation (it may look different because it was multiplied by a scalar).
But in general, you will get something like

If you have any questions feel free to ask!