Google assigns each Web page a score, called a PageRank, based on the number of pages which link to it. It uses this score to determine the order in which search results are presented. Computing the PageRank for each Web page is done by computing the dominant eigenvector of a (very large) probability matrix. Although this can be computed directly using the Power Method, it can also be reformulated into the solution of a linear system. After introducing the "random surfer" model and defining the PageRank vector, I will develop an algorithm to solve the PageRank problem as a linear system. Finally, we will briefly discuss some possible modifications or alternative strategies and their numerical implications.