I have a cluster of 10 computers running linux connected to each other via LAN and one of them has internet connection. Is it possible to set up mining in which the computer with internet acts as the master computer and gets work done from 9 slave computers and mine bitcoin?
It certainly is, you could for example install the stratum mining proxy (assuming that the pool you are mining for supports stratum). You would install the mining proxy on the computer with network access and then configure the miners to use the proxy instead of directly going to the pool itself.
Of course this is just one of many options. Other options include creating a reverse HTTP proxy on the computer with internet access, as the
getwork protocol uses HTTP underneath. Also possible would be to tunnel a given port directly to the pool using SSH for example. But I think the mining proxy is the easiest option, and allows for legacy mining software to use the more efficient stratum protocol as an added bonus.