thanks for all the answers. I dont know if Ive ever gotten so many responses from an online forum before.
Earlier this week I read that windows 10 would be free to everyone as an upgrade including those with versions that werent activated. Would that make any difference in being able to have both windows and linux on the same motherboard?
As to being able to install or upgrade Windows 10 on an unactivated/improperly licensed Windows, I don't know for sure if this is an option for all parts of the world, or those where piracy is most rampant. Regardless, Windows 10 will require proper licensing for all features to be available to you, including activation of the prior OS. Some of these recent Windows Updates are checking for these things, you may or may not be presented with a 'to do' list before upgrading to 10. So it's best to take care of this now, and not after 10 is released. It may be that we can assist you, if you're willing to follow steps, to save you money on shop repairs. Is the OS properly working & activated on this PC?
Dual booting isn't going away, regardless of if 10 temporarily knocks it out or not, there's always EasyBCD 2.2 to the rescue, which will allow you to 'recover' your dual boot, and choose from between Windows 10 & Linux. It installs on Windows, and you can easily add your Linux OS back.
Does the eMachine not work at all, even with only Linux installed? If not, it would take a mighty inexpensive repair for it to be worth the troubles, and most shops gets $100+ per hour for labor alone. Assuming you were to get it back & it runs, would you be able to sell it & get your repair bill back? I ask because in general, these PC have little value, it's now an Acer brand (as is Gateway), and many of us here knows the standing of that brand. Most are budget models at best, though Acer does have some more costly models, the quality of these are not much better than their budget models. This is because they pinch pennies too much. Many of these units has motherboard failures, and some of these can be attributed to the ultra low cost PSU's installed.
Even if it were a non-Acer model, we rarely hear about eMachines any longer, so these must have had quality control issues of their own. I just cannot justify advising you to carry it to a shop for repair, when you can get a brand new PC from Walmart beginning at $248 (you reuse your monitor/speakers). For $40 more, you can get a model with 6GB RAM pre-installed, plus Windows 8.1 with Bing (that's a packaged bundle). At these prices & for what shops charges, why spend close to the same at a shop for what is still not going to be reliable long term? And worse yet, no way to sell it at a price to get your money back. If it were a high dollar, high configuration PC, I could see this, not for a budget model that's 5 years old. While I feel bad that you didn't get long service out of the PC, I dropped more cash on an HP AIO model that didn't make it to the 4 year mark. No, it didn't go to a shop, rather, I removed the HDD, which was reused, the PSU, and the rest came face to face with a sledgehammer & then the recycling bin.
Now, as to your Lenovo, does this PC run? I would like to think it does, since you're posting from some computer. Of course you can reuse that 2nd HDD, as long as you delete the Windows partitions from the eMachine first, it can be used for data, or your Linux install, or a combination of both. That may be best, because your Lenovo recovery partition may not properly work with a Linux install on the computer, and if you've already installed one on the HDD, it's likely torched anyway. I hope that at a minimum, you did create your recovery DVD set, which would have consisted of 3 to 5 DVD's (same as eMachine above), so that you can clean install your OS if needed. If not, you may can order these, I was able to for one much older.
Finally, to address your question about RAM. Those slots that are present may have been for those whom ordered a Lenovo with more RAM than yours, an AIO type of MB to meet all needs, however these were likely built the way they are from the factory whom produced the MB, and it would take a delicate operation & a lot of good luck for what you're proposing to work. I'm not even sure that an Lenovo authorized shop would take this repair/upgrade on, would likely steer you into an upgraded board instead, which would be a refurb for repairing PC's & would cost less. You're still going to have the same CPU & all, so for any cash spent, which will be a small fortune, don't expect a lot for your money.
If you're looking for more than 4GB RAM, look for another computer. That's the best advice I can give in that regard. The only alternative, which if successful, would require a new version of Windows also, would be an upgraded motherboard from a place such as Newegg or Amazon. Still you have to install it, and hope that your RAM sticks (all of them) are of the same type & timing, otherwise the best set will revert to the speed of the lowest rated set. And there's still the issue of a new Windows install to deal with, as the Lenovo one would be good no longer (unless you get very, very lucky & reach a super sympathetic Windows rep to help you activate what you have). That very seldom happens.
And even if you purchased these components, could you or a knowledgeable friend install a new motherboard? If not, there's no need to bother, as again shop labor is costly, by the time you purchased components & pay for install, a newer, more powerful computer with up to 8GB RAM preinstalled can be purchased, complete with keyboard & mouse & even a true HDMI port (or Displayport, which can be converted into HDMI or DVI if you don't have a monitor with DP input). You supply your monitor & speakers.
Hope that my post at least gets you headed in the direction you want. Should you have further questions, don't hesitate to ask, that's what we're here for.