If I'm reading your original post correctly, you don't want the first ten terms of each sequence, right? You want the first ten terms that are COMMON to BOTH sequences. If so:
Here's what I did - it's slightly better than listing out every term.
The first sequence's terms can be described by 2 + 7x.
The second sequence's terms can be described by 3 + 11y.
I want to know when these are equal, so:
2 + 7x = 3 + 11y.
I solved for y and got:
.
So I'm looking for an integer value of x that gives me an integer value for 7 (since all the numbers in both sequences are integers). I found the first one by trial and error. x=8 gives y=5. So the point (8,5) is on my line, and gives 58 for both 2 + 7x and 3 + 11y.
So 58 is in both sequences. Now, noting that the slope of our line is 7/11, we can increase our y value by 7 and our x value by 11 (since slope is change in y / change in x) to get the next value of x and y that gives integer coordinates: (19, 12). Plugging x = 19 and y = 12 into the appropriate equations gives 135 in both, so that's the next term.
Keep using slope to "jump" from each integer-coordinate point to the next. Use x and y, plug them into your equations, and you should be good to go:
(8,5), (19,12), (30, 19), (41, 26), ...
Hope this makes sense. I kind of figured it out while writing this, so it's a little... disorganized.