I am currently studying circuit analysing techniques using Engineering Circuit Analysis 8th Ed. by Hayt, Kemmerly, and Durbin. In Section 4.2, the authors describe a "difficult approach" to handling voltage sources between two non-reference nodes (the red highlighted box in the image below). Instead of using a supernode, the method involves:

 - Assigning an unknown current to the branch containing the voltage source.
 - Applying KCL (Kirchhoff's Current Law) at each of the three non-reference nodes.
 - Applying one KVL (Kirchhoff's Voltage Law) equation.

[![enter image description here][1]][1]

  [1]: https://i.sstatic.net/LR9MC56d.png

I have attempted to set up the system of four equations with four unknowns. I've defined \$i_x\$ as flowing from node 2 to node 3.
 
KCL 1: \$ -8 \text{ A}-\frac{v_1-v_2}{3\ \Omega} -3 \text{ A} -\frac{v_1-v_3}{4\ \Omega} = 0 \text{ A} \$

KCL 2: \$ 3 \text{ A} + \frac{v_1-v_2}{3\ \Omega} - \frac{v_2}{1\ \Omega} -i_x =0 \text{ A}\$

KCL 3: \$ \frac{v_1-v_3}{4\ \Omega} +i_x -\frac{v_3}{5\ \Omega} + 25 \text{ A}= 0 \text{ A}\$

KVL: \$ -v_2 -22 \text{ V} + v_3 = 0  \text{ V} \$


Resulting Matrix Equation:


$$
\begin{bmatrix} -7 & 7 & 0 & 0\ \Omega \\ 1 & -4 & 0 & -3\ \Omega \\ 5 & 0 & -9 & 20\ \Omega \\ 0 & -1 & 1 & 0\ \Omega \end{bmatrix} \cdot \begin{pmatrix} v_1 \\ v_2 \\ v_3 \\ i_x \end{pmatrix} = \begin{pmatrix} 132 \text{ V} \\ -9 \text{ V} \\ -500 \text{ V} \\ 22 \text{ V} \end{pmatrix}
$$

**Could you please verify the coefficients and constants in this matrix equation?**