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We can now reduce the circuit further:

We can now reduce the circuit further by just moving the ground around (remember, only one node can be ground; but any node can be ground):

So, here's the curious thing. You don't need to care about the current in \$V_2\$! In fact, you can analyze everything without even caring about it. You could just set it to any arbitrary value, say \$0\:\text{A}\$, and so long as you don't write equations depending on it you will have enough information elsewhere to solve the circuit.

I didn't directly solve your problem using the methodology your question suggests. But I don't have your textbook or class or the larger context. But I definitely could address the question you asked in comments. So I did.

Here's the curious thing. You don't need to care about the current in \$V_2\$! In fact, you can analyze everything without even caring about it. You could just set it to any arbitrary value, say \$0\:\text{A}\$, and so long as you don't write equations depending on it you will have enough information elsewhere to solve the circuit.

Summary to your question

So, here's the curious thing. You don't need to care about the current in \$V_2\$! In fact, you can analyze everything without even caring about it. You could just set it to any arbitrary value, say \$0\:\text{A}\$, and so long as you don't write equations depending on it you will have enough information elsewhere to solve the circuit.

Sure, it's there. But it's just not important in this case.