I'm planning on buying a new computer with the following motherboard: ASUS P8Z77-M Pro. Can the integrated graphics be combined with the video card mentioned above? Or are the on-board graphics automatically switched off when I plug in the video card?
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I think you would be able to use them both at the same time but you can't combine them they would be completely independent. That said the mainboard only supports integrated graphics if the CPU you are using has integrated graphics built in. Since you did not state what CPU you are using so it is hard to tell if it has built in graphics.
The only way to do what you suggest is to get a AMD "APU" or the new "FX" series style processor and an ATI graphics card. You could then "CrossFire" them (I've never tried but its been done) I am sure Tom's has a list of what graphics card with each processor, but imho, why would you want to do that? I am not a big fan of the "CrossFire" technology, it will just add alot more headache when dealing with cooling your CPU. just my 2 cents.