Why doesn't Solve work unless I request a solution for both unknowns?

Question

I have two circuit equations

power == voltage*current
voltage == resistance*current

You can plug the second equation into the first and get

power == current^2 * resistance

I'm wondering why this input is returning no solution

Solve[{power == voltage*current, voltage == resistance*current}
  /. {voltage -> 5, resistance -> 10}, current]

If I manually solve for power, then I can get the correct result

Solve[{power == voltage*current, voltage == resistance*current}
  /. {voltage -> 5, resistance -> 10, power -> 5^2/10}, current]

{{current -> 1/2}}

I also get the correct solution if I request the solution for both current and power

Solve[{power == voltage*current, voltage == resistance*current}
   /. {voltage -> 5, resistance -> 10}, {current, power}]

{{current -> 1/2, power -> 5/2}}

I did some more testing and found that if I provide the MaxExtraConditions->All that I get a solution. Is this what I should be using?

{{current -> ConditionalExpression[1/2, power == 5/2]}}

Answer 1

It can't read your mind to understand that power is a free variable rather than a parameter you intend to define later. One way to deal with this is to solve for both current and power. Another way is to simplify by eliminating power:

Eliminate[{power == voltage*current, voltage == resistance*current}, power]
(* voltage == current resistance *)

Then, solve that.