could someone please show me how they got these equilibrium points? not sure if its just simple algebra or you have to do a derivative...how did they get equilibrium points from this answer?
definition (1) --> equalibrium point for a two-dimentional system is an $(x,y)$ point, where $\dfrac{dx}{dt}=0$ and $\dfrac{dy}{dt}=0$
By setting the derivative to zero, you obtain 2 new algebraic equations.
First equation gives $$x = 1-y$$ second equation gives $$x^2+y^2 = 1 \Longrightarrow (1-y)^2 + y^2 = 1$$ Simplify and solve the second equation (2 solutions). Substitute the solutions for $y$ back in the first to find corresponding $x$.
I find the solutions quite clear, I just did it the other way around for the substitution.