Nodal analysis in circuit contain voltage controlled voltage source

Question

I have a circuit below and I need to determine the potential at \$v_1\$ and \$v_2\$. I have 3 questions:

^{simulate this circuit – Schematic created using CircuitLab}

1. How do I use nodal analysis at \$v_2\$, or how do I solve this without nodal analysis?
2. I have tried to do this:

At node \$v_1\$: \$-1+\frac{v_1-v_2}{10}+\frac{v_1}{5}+\frac{v_1-v_2}{15}=0\$

At node \$v_2\$: \$v_2 = 2v_x = 2\frac{v_2-v_1}{10} = \frac{v_2-v_1}{5}\$

$$=> \frac{11}{30}v_1-\frac{1}{6}v_2=1; \frac{1}{5}v_1+\frac{4}{5}v_2=0$$

$$=> v_1=\frac{120}{49}V; v_2=-\frac{30}{49}V$$

This is wrong, but when I ran a simulator and I accidentally set the function in VCVS is \$0.2(v_a-v_b)\$, then I have the same result. Why does this happen?

3. (This question is unrelate to 2 questions above, but just for curious) When I tried to substitute the \$I_2\$ source with a voltage source in a simulator, the simulation stop immediately, but this does not happen with \$I_1\$ source. Why does this happen?

Any help would be appreciated.

Answer 1

You said that \$v_x = \frac{v_2 - v_1}{10}\$

The left-hand side is a voltage, the right-hand side is a current.