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.
2026-03-29 08:18:24.1774772304
Do all nonlinear systems of 2 equations in 2 variables have at least one solution?
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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 $2$.