Systems of equations for circles

Question

I was given a problem with a system of equations for two circles:$$\begin{cases}x^2+y^2=4\\(x-5)^2+y^2=4\end{cases}$$It’s clear that the two equations have no solution where both are true, but when I substitute the $4$ for $x^2 + y^2$ I get$$(x-5)^2 + y^2 = x^2 + y^2.$$ If I subtract the $y^2$ from each side, expand the $(x-5)^2$ and then subtract $x^2$ from each side I am left with$$-10x + 25 = 0,$$which can be solved as $x = 2.5$. This obviously doesn’t work if you enter it into the previous equations, so why did I get this solution?

Answer 1

Hint:

If $x=2.5$, and $x^2+y^2=4$, what is $y$?

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Essentially, there is, as you can see from the graph, no solution on the real plane. However, the same can't be said about the complex plane!

Answer 2

Remember that an implication is true if the antecedent is false. You are given some equations as axioms. If the equations are inconsistent, eventually you can prove anything. That is why you check solutions in the original equations, because it is easy to assume something or make a transformation that is not a logical equivalence. Here when you subtracted $y^2$ from each side you lost the fact that it had to be greater than or equal to zero. Your solution is fine if $x,y$ can be complex. You find $y=\pm 1.25i$ and all is well. In the reals, not so much.