# Do all nonlinear systems of 2 equations in 2 variables have at least one solution?

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.

• 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 Mar 3, 2020 at 15:49
• 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
– set5
Mar 3, 2020 at 15:50
• 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.
– mjw
Mar 3, 2020 at 15:51
• 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
– set5
Mar 3, 2020 at 15:54
– mjw
Mar 3, 2020 at 15:55

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.
\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 $$2$$.