Engineering economics question about the purchase price of a share

I don't understand how I am supposed to know how to figure out that I need those particular equations to solve this problem. I think I get what the question is asking but I'm not too sure about how to answer it. The work given seems like random numbers just placed on the screen like, I don't understand why anything is being done the way it is.

I am extremely weak in this material so please illustrate how to approach this problem and why anything is being done the way it is!
Thanks in advance!