OK, now I see why the circled part is correct by using your method. Your way actually makes more sense to me

But why is the author trying to combine the two equations dx/dt = y, dy/dt = -x ? How is this going to help us to solve the system?

Combining these two, we get dy/dx = -x/y, the general solution is just x^2 + y^2 = k where k is an arbitrary constant. To parametrize this general solution it in terms of t, just put x = a cos(t), y = a sin(t), right? This parametrization

__satisfies__ the equation x^2 + y^2 = k, so it must be a correct parametrization. What is

__wrong__ with this approach? Can you please point out where this line of logic fails?

Thank you!