Here is the output from lspci:
$ lspci | grep VGA 01:00.0 VGA compatible controller: Advanced Micro Devices [AMD] nee ATI RV370 [Radeon X550] 02:00.0 VGA compatible controller: Advanced Micro Devices [AMD] nee ATI Tahiti XT [Radeon HD 7970]
I installed the fglrx proprietary driver, but it only detects the monitors connected the the Radeon HD 797.
Many of the searches I have done seem to suggest that both graphics cards need to be the same, but there weren't many that were resolved that suggested the both cards could be different.
This question seems to suggest it's possible, but that's a lot of steps for something that no one has commented on whether it works.
I have booted into windows with this hardware setup, and Windows is able to use both cards, so I can at least safely say that both cards work.