Dead video card or logic board?

In November of last year my iMac started acting strange. The screen would simply shut off (or go totally black) and then would come right back on after just a few moments. I took the computer to the nearest Apple certified store in my town which was a Simply Mac store. They told me for $50 that is was the LCD screen that was going out. I didn't have them fix it and I left.

By the time February rolled around, we were lucky to have the screen stay on at all. Fortunately we have a second monitor for it and that served us well. About 3 months ago we were syncing our apple devices to iTunes and the screen became all jumbles and had thousands of mini lines running through it and the machine froze up. I tried to reboot it with no luck. I am assuming that is it the video car that went bad and I am also assuming that the video car is integrated with the logic board. Am I right in thinking this and is there a possible simple fix for this? Or do I need to start shopping for a new logic board?