I'm wondering here, when writing an expression for Vx, I had assumed that it would correct to say that

\(V_{x}=\frac{V_{3}-V_{1}}{R_{1}}\)

Why is this not correct in this case?

 

It's not correct because the units for (V3-V1)/R1 would be ampere not volt. Surely Vx is simply (V3-V1).

 

Ah yes of course... seems sometimes the logical part of my brain is turned off...

 

You need to get in the habit of always, always, ALWAYS checking if the units work out. Always. I can't even begin to count how often I catch mistakes because the units aren't working out. Sometimes is a math goof, sometimes my thinking about the problem when I set it up is flawed, sometimes I go off on a tangent and start solving for something other than what I intended. But the vast majority of these mess up the units.

If you get in that habit (and it truly becomes an ingrained habit in short order), you will look at the above equation and immediately gasp thinking, "Voltage equals Current!? WRONG! WRONG! WRONG!" Far better that happen on line six than on line twenty-six.