How do I plot two variables in a line chart using the row number for the X-axis?

I have two variables in column A and B.
I want to chart these as two lines as a function of row number.
This should be stupid simple, but I have tried for hours and not succeeded.

It may not be possible to automatically use the row numbers as the category (X-axis) but otherwise the process is simple. Either rely on a natural row increment to create X-axis values:

… or hardcode the X-axis values as required:

In either case, selecting the entire data set, clicking the Chart toolbar button, and using the settings indicated, will give two series in a line chart.

I tried exactly that at least a dozen times without success. I just tried it here (at home, on a Windows box) and it worked just as you said. I will have to try again tomorrow at work (Debian Linux box) and see what is different.
Thanks very much for taking the time to answer!