A certain circle's area is $x$ square units, and its circumference is $y$ units. The value of $x + y$ is $80\pi$. What is the radius of the circle, in units?
I know that the area of a circle is given by the formula $A = \pi r^2$, where $r$ is the radius. I also know that the circumference of a circle is given by the formula $C = 2\pi r$. So, if the area is $x$ and the circumference is $y$, I can write two equations: $x = \pi r^2$ and $y = 2\pi r$. I can add these two equations together to get $x + y = \pi r^2 + 2\pi r$. But I also know that $x + y = 80\pi$, so I can substitute that in and get $80\pi = \pi r^2 + 2\pi r$. I can simplify this equation by dividing both sides by $\pi$ and get $80 = r^2 + 2r$. I can subtract $80$ from both sides and get $r^2 + 2r - 80 = 0$. This is a quadratic equation that I can factor into $(r + 10)(r - 8) = 0$. This means that either $r + 10 = 0$ or $r - 8 = 0$. Solving for $r$, I get $r = -10$ or $r = 8$. But $r$ cannot be negative, because that would not make sense for a radius. So the only valid solution is $r = 8$. Therefore, the radius of the circle is $8$ units. 