Simple questions about current flow and superposition

Question

So I was doing this exercise to prepare (not homework, school hasn't even started yet), and I think I got the math and the logic right, but there are a few things that I didn't understand, and since my teachers are on vacation I was hoping you could help. Below is a picture with the question that I transcribed and all of my work after it, including my questions: what's the best way to find the direction of the current, if it's positive or negative, and a minor uncertainty that I got regarding simple circuit analysis.

Please correct me if you see any errors.

Thank you so much for reading this far, if there is something you don't understand please ask!

Answer 1

+1 for showing all your work, that's excellent. -0.01 for the low contrast -- I did have to squint a bit, but it was worth it.

The intuitive way to figure out the sign of the voltage contributions of each source is to start at the '+' sign on a voltage source, or the point of the arrow on a current source, and trace through your \$V_x\$. If your finger hits '+' and then '-', it's a positive contribution; if your finger hits '-' and then '+', it's negative.

The dreadfully formal way is to label all the nodes or draw in the currents and then do nodal or mesh analysis.

Answer 2

Its hard to read your work, so I'm just going to do the individual sources.

From left to right:

1. Source 1: 15 V over divider with equal resistance: 7.5V. Source + connects to Vx +: positive
2. Source 2: 3 A of current into a 3 Ohm resistor: 9 V. Source current flows from - to + so it's negative.
3. Source 3: 4 V over divider with equal resistance: 2V. Source + connects to Vx - so it's neagtive

In total we have 7.5V-9V-2V = -3.5V.

You can do this more formally with using proper nodes, counting arrows and loops, but in this case it's easier enough to eye ball it. If the the "+" of your source obvisouly connects to the "-" or your target voltage, you count it negative.