Apologies for the oversight. Let's add some specific numbers to the problem:Consider a connected, undirected graph G with $5$ vertices and $8$ edges, each assigned a distinct positive weight. The task is to find the minimum spanning tree of G using the Prim's algorithm. Before each edge is added to the current spanning tree, you must solve a system of linear equations to determine its weight. The coefficients of the variables are related to the properties of the vertices and edges involved. The weights of the edges are as follows:Edge $1$: $7,$ Edge $2$: $12,$ Edge $3$: $5,$ Edge $4$: $8,$ Edge $5$: $10,$ Edge $6$: $6,$ Edge $7$: $9,$ Edge $8$: $11.$Define a system of linear equations for each edge, considering the properties of the vertices and edges. Solve the equations step by step as you apply Prim's algorithm to find the minimum spanning tree.