The power system shown below has three buses, where buses $1$ and $2$ are connected through a transformer, and buses $2$ and $3$ are connected through a transmission line with the impedance shown. The transformer has a reactance of $20\%$ on $100-$MVA, $23/115-$kV bases, where the $23-$kV side is connected to bus $1.$ The load at bus $2$ is S$2=184\mathrm{.}8+6\mathrm{.}6$j MVA, and the load at bus $3$ is S$3=20$j MVA. The power base for the system is $100$ MVA. Assume that a generator is connected at bus $1,$ so it should be classified as the system slack bus. Complete the following:

a$)$ Determine the per unit system Ybus.

b$)$ Calculate the initial per unit power mismatches. Organize your answers into the $$ vector. Assume starting bus voltage values of $10$ pu buses $2$ and $3$ and $1\mathrm{.}040$ pu for bus $1.$

c$)$ Determine the values of all entries in the jacobian matrix. Organize your answers into the $$ matrix.

d$)$ Use Gauss elimination with back substitution to solve for the $$ vector. You may use your Gauss elimination code from Lab $6$ for this part.

e$)$ Determine $$ for the next iteration.