The idea behind Gaussian elimination is fairly straightforward. I am going to explain just using words and no diagrams, but I still think it should be fairly clear. If you need further help, I can come up with a specific example and go through it step by step.
Suppose you have a 3 by 3 matrix in variables x, y and z, where x is the first column, y the second and z the third.
Choose a variable, say x, and choose one of the rows where the x coefficient is non-zero. Now subtract that row from each of the other rows having non-zero x coefficient, in such a way that you will end up with zero coefficient for x in the other rows.
Now choose a row with non-zero coefficient for y, but you want it to make sure it is different from the row you chose previously. This means that the row will have a zero x coefficient from the previous step. And similar to what you did before, subtract the row from each other row with non-zero y coefficient, in such a way that you end up with a zero y coefficient in the other rows.
If you think about it for a moment you will see that there will be at most one row with non-zero x coefficient and at most one with non-zero y coefficient.
Suppose that at this point that the third row (the one not chosen in the previous steps) has all zeroes in it, so that you can’t continue, as previously, to choose a row for z different from the two already chosen. This means that you are done. z is the only independent variable with both x and y expressible in terms of z.
I hope this makes sense and that you see how to generalize the method. If there are no degenerate rows that go to all zeroes then you end up with something like what @girlofscience showed. In this case the only solution is having all variables equal to 0.