Two circles whose equations are (x-3)^2 +(y-5)^2 = 25 and (x-7)^2 +(y-5)^2 = 9 intersect in two points. Write the equation of the line passing through these two points of intersection.
--izzy

Just for information, this is not advanced algebra. More like college algebra.

Advanced algebra is on the lines of group theory.

But, for your problem, subtract the two equations. Take note that the y-coordinates for the centers of the circles are the same. The centers are (3,5) and (7,5) with respective radii 5 and 3.

(x-3)^{2}+(y-5)^{2}=25

(x-7)^{2}+(y-5)^{2}=9

When you subtract you get:

(x-3)^{2}-(x-7)^{2}=16

Solve this for x (it's easier than it looks. The x^2's cancel), then sub that into your two circle equations to find the two y values which correspond to it. The points of intersection are in vertical alignment, so you will have the same x value and two different y values.

Here's the graph: