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I know that not all linear systems of 2 equations in 2 variables have a solution, I was wondering if that is the case also with nonlinear systems.

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  • $\begingroup$ What are your thoughts? have you tried playing around with some non-linear systems and seeing what happens? People generally won't answer a question unless they can see the person asking has put in some effort themselves first $\endgroup$
    – lioness99a
    Mar 3, 2020 at 15:49
  • $\begingroup$ yes, I've been solving some nonlinear systems where the equations represent conics and lines and so far all of them have complex solution/s $\endgroup$
    – set5
    Mar 3, 2020 at 15:50
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    $\begingroup$ The answer is no. Here is an example of such a system: $x^2+y^2=1$ and $x^2+y^2=2$. Two concentric circles that do not intersect. There are many more examples. $\endgroup$
    – mjw
    Mar 3, 2020 at 15:51
  • $\begingroup$ thank you for the counterexample. The fact is that in other systems I was observing no intersection/s and there were still complex solution/s $\endgroup$
    – set5
    Mar 3, 2020 at 15:54
  • $\begingroup$ Please give an example. $\endgroup$
    – mjw
    Mar 3, 2020 at 15:55

1 Answer 1

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The answer is no. It should be very easy to come up with many examples. Geometrically, each equation represents a curve in the $xy$-plane. For there to be a solution to the system of equations, the curves must intersect.

Here is a counter-example:

$$\begin{aligned} x^2+y^2&=1\\ x^2+y^2&=2 \end{aligned}$$

In the $xy$-plane, with $x$ and $y$ both real, these are concentric circles that do not intersect.

Even if you allow $x$ and $y$ to be complex, there are no solutions. The sum of the squares of two variables cannot simultaneously be both $1$ and $2$.

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