Consider the circuit below. The node voltages are already labelled, please use these labels in your analysis.

Oops! I was in such a rush to finish the exams, I forgot to add all the elements. No matter, you can do it yourself. Use resistors $\backslash ($ R$\_\{$x$\}=1\mathrm{.}1\backslash$mathrm$\{$k$\}\backslash$Omega$,$ R$\_\{$w$\}=1\mathrm{.}1\backslash$mathrm$\{$k$\}\backslash$Omega$,$ R$\_\{$z$\}=1\mathrm{.}6\backslash$mathrm$\{$k$\}\backslash$Omega $\backslash ),$ and independent sources $\backslash ($ V$\_\{$A$\}=8\backslash$mathrm$\{$~V$\}\backslash )$ and $\backslash ($ I$\_\{$B$\}=7\backslash$mathrm$\{$~mA$\}\backslash ).$

First, specify the locations of the $5$ missing elements. You may choose any arrangement you wish. It is unlikely, but it may be possible to create an "unsolvable" circuit by poor placement $-$ so don't do that. The polarity of the sources is "top$-$to$-$bottom" and "left$-$to$-$right", as shown in each generic element. If you want to flip the polarity of a source for whatever reason, indicate the element as a negative $($i$.$e$.,$ write $"-$VA$"$ in position $(3)$ if you want to flip the polarity of $\backslash ($ V$\_\{$A$\}\backslash )$ relative to that shown in the diagram above$).$

You do not earn any marks by specifying the location of the elements, but it is necessary to provide this information to obtain any marks for the rest of this problem.

Answer the following questions:

$($a$).$ Identify the supernodes, if any. Write $"$v $0"$ or "ground" if any labelled node forms a supernode with the ground. $[1$ mark$]$

$($b$).$ Calculate the conductances $\backslash ($ G$\_\{$u$\},$ G$\_\{$w$\},$ G$\_\{$x$\},$ G$\_\{$y$\}\backslash ),$ and $\backslash ($ G$\_\{$z$\}\backslash )$ of resistors $\backslash ($ R$\_\{$u$\},$ R$\_\{$w$\},$ R$\_\{$x$\},$ R$\_\{$y$\}\backslash ),$ and $\backslash ($ R$\_\{$z$\}\backslash ),$ respectively. Include the appropriate units in your answers. $[1$ mark$]$

$($c$).$ Write the nodal equations symbolically. Your equations should use the given labelled node voltages, the given labels for the resistors, voltage sources, and current sources, and the expression for the dependent source. You may use $\backslash ($ G$\_\{$n$\}\backslash )$ to represent the conductance of resistor $\backslash ($ R$\_\{$n$\}\backslash )$ if you prefer $($in fact, this is recommended$).$ The only numeric value in these equations should be the scaling factors for any dependent sources. It does not matter in what order you write these equations, nor how you group the terms within the equations, as long as they are all mathematically correct. You may use $\backslash (1/$ R$\_\{$n$\}\backslash )$ or $\backslash ($ R$\_\{$n$\}^\{-1\}\backslash )$ if you don't like writing in terms of conductance. $[4$ marks$]$

$($d$).$ Write the matrix equation $\backslash (\backslash$mathbf$\{$I$\}=\backslash$mathbf$\{$G v$\}\backslash ).$ The matrices $\backslash (\backslash$mathbf$\{$I$\}\backslash )$ and $\backslash (\backslash$mathbf$\{$G$\}\backslash )$ should be expressed numerically. The matrix $\backslash (\backslash$mathbf$\{$v$\}\backslash )$ should remain symbolic in terms of the unknown node voltages, and all unknown node voltages should be listed $($i$.$e$.$ v should be a $4-$element column matrix$).$ You may omit writing the units as long as the solution to the matrix equation provides the node voltages in units of V$.[4$ marks$]$

$($e$).$ Solve the circuit to obtain the node voltages. Include the appropriate units in your answers. $[4$ marks$]$