Nodal analysis with voltage source
There are "two" basic techniques for solving nodal analysis circuits with voltage sources (they're really different sides of the same coin):


Use supernodes


Since there's no equation for the current flowing through a voltage source, we can't easily write KCL equations for nodes \$2\$ and \$3\$ in terms of just nodal voltages. However, we can solve one of these equations for \$i_{V}\$ (current through the voltage source) and plug it into the other. We must then introduce an extra equation which states the effect of the voltage source.

For example, take the KCL equations for nodes \$2\$ and \$3\$.

\begin{gather}
i_{5\Omega} + i_{10\Omega} + i_{17\Omega} + i_V = 0\\
0 = i_{17\Omega} + 4A + i_V
\end{gather}
I can solve for \$i_V\$ in the second equation and plug that into the first equation:
\begin{gather}
i_V = -i_{17\Omega} - 4A\\
i_{5\Omega} + i_{10\Omega} + i_{17\Omega} - i_{17\Omega} - 4A = 0\\
i_{5\Omega} + i_{10\Omega} - 4A = 0
\end{gather}

Lastly, we add the equation associated with the voltage source:
\begin{gather}
V_2 - V_3 = 25V
\end{gather}

Including the nodal KCL equation for node \$1\$ as well as all the Ohm's law relations, we now have enough equations to solve for all unknowns. Finding these equation is left as an exercise to the reader.


Modified Nodal Analysis


Modified nodal analysis can be though of as a formalization of the supernodes technique. Similar to the supernodes technique, we have an unknown \$i_V\$, and we still add the same equation for the voltage source \$V_2 - V_3 = 25V\$. However, we formulate everything into a massive matrix and can solve for all the unknowns at once. This process is very systematic and "brain-dead", making it very good for computers. For more information, see these notes on MNA.