How to solve a circuit network involving a resistor parallel to a voltage source using only nodal analysis?
There's actually a fourth node in your diagram, at the junction of E1 and R1. But that one and V2 have known voltages, so the problem reduces to just two unknowns. I get the following equation:

$$
\begin{bmatrix}
  \frac{1}{R1}+\frac{1}{R2}+\frac{1}{R4} &amp;&amp; -\frac{1}{R4} \\
 -\frac{1}{R4} &amp;&amp; \frac{1}{R4}+\frac{1}{R5}+\frac{1}{R6}
\end{bmatrix}
\begin{bmatrix}
 V1 \\
 V3
\end{bmatrix}
  =
\begin{bmatrix}
 \frac{E1}{R1}+\frac{E2}{R2} \\
 0
\end{bmatrix}
$$

This allows you to solve for V3, and once you know that, you can get \$I_x = \frac{V3}{R6}\$.