# What justifies algebraic manipulation in equations with only variables?

I recognise my question is at a beginner level but my current level of knowledge of math is up to what any undergraduate engineer would know, so you can give me a more-than-beginner-level explanation.

I'm puzzled by why exactly our rules for algebraic manipulation are deemed valid in equations like the ideal gas law or Ohm's law (to give a few examples) where everything contained in the equation are variables. Why is it valid to proceed with the mentality of "having the same on both sides" here?

For instance if I have $pV=nRT$ which is the ideal gas law, I can say $p=\frac{nRT}{V}$ following the well-known rules for manipulating equations.

I think this type of manipulation differs from solving an equation like $5x=15$ to get $x=3$ in that in that case you are looking for a value that "exists" or "is some how contained inside x" and in the case of an equation like the ideal gas law, the variables inside (except R) can be any positive real number, so a variable is kind of another set of entity which contains all numbers inside the letter; which makes it difficult to grasp how you can, i.e., "divide by V on both sides" if V is not anything at all.

Thanks, and I really hope someone can help out of this confusion because it's really hampering my ability to study at the moment.