This network shows possible flows between six nodes $($arrows indicate direction$).$

a$)$ Write the LP model to find MAXIMAL flow from node $1$ to node$6$

b$)$ Explain how the model will change if the path from node$3$ to node$6$ were eliminated. Explain all changes.

c$)$ Explain how the model will change if a direct path from node $2$ to node $6$ was added with capacity $=10.$ Explain all changes

d$)$ Explain how many variables and how many constraints this LP model will have when written in standard form.

e$)$ Explain how many variables and how many constraints this LP model will have if node $3$ was eliminated.