Why is there more than one possibility as to the number of solutions to the graph of a system of equations that consists of a parabola and a circle. Be sure to give examples.

Question

There is more than one solution to a system of equations composed of a parabola and a circle.

Answer 1

There is not always more than one solution. The solutions are the points of intersection of the graphs.

For example:

The graph of the parabola y = x² and the graph of the circle x² + (y+2)² = 1 don't intersect at all.

The graph of the parabola y = x² and the graph of the circle x² + (y+1)² = 1 intersect only once.

The graph of the parabola y = x² and the graph of the circle x² + y² = 1 intersect twice.