How can find the voltages at 2 nodes with 1 equations and 2 variables?

Question

^{simulate this circuit – Schematic created using CircuitLab}

At node 1, we have:$$0.001+i_1-i_4-i_5=0$$. At node 2, we have: $$i_4+i_5+i_2=0$$.

$$i_1=\frac{0-V_1}{1000}=\frac{-V_x}{1000}$$(Vx is the voltage across R2) $$i_2=\frac{0-V_2}{6000}$$ $$i_3=0.001A$$ $$i_4=i_1+i_3=-i_2-i_5$$ $$i_5=\frac{V_1-V_2}{1000}=\frac{V_x}{1000}$$

Use the equation at node 1, to find an expression for i4 and plug that into the equation at node 2: $$i_1+0.001+i_2=0$$ $$\frac{-V_x}{1000}+0.001-\frac{V_2}{6000}=0$$

But there are V2 and Vx to solve in one equation.

Answer 1

You have too much variables and arrows (not wrong, but unnecessary). I redrawed the circuit:

You obviously missed that the voltage over R4 can be expressed in terms of Vx. This is Kirchoffs voltage law.

I have written the current equations for nodes 1 and 2. There are 2 unknowns and 2 equations.

Hopefully clear now.