I have come up with the following system of equations $$a+b=1,$$ and $$a \times b = -1.$$ I know that the solution is $a = \frac{1}{2}(1+\sqrt{5})$ and $b = \frac{1}{2}(1-\sqrt{5})$.
But how to find such a solution? What is such a system of equations called?
With $$b=-\frac{1}{a}$$ we get $$a-\frac{1}{a}=1$$ multiplying by $$a\neq 0$$ we get $$a^2-a-1=0$$ Can you solve this equation?