problem: The graphs of $y = x^2 - 7x + 7$ and the line $y = -3$ intersect at two points. What is the sum of the $x$-coordinates of these two points?
solution: The intersection points have $x$-coordinates when $x^2-7x+7=-3$, or when $x^2-7x+10=0$. We can factor this into $(x-2)(x-5) = 0$; the intersections have $x$-coordinates 2 and 5, and the sum follows as $2+5 = \boxed{7}$.