Finding the position of a planet between two other planets of known mass and distance Here is the question:

A planet with mass $m$ and a second with mass $M$ are separated by a
  distance $d$. A third planet with mass $m_3$ happens to be midway
  between $M$ and $m$. Where could the third planet be positioned
  (distance from the larger planet $M$ in meters) so that the net
  gravitational force is zero?

My confusion lies with how to solve for the position of the third planet.  I am given this equation to find the force between two planets at a given distance $r$:
$$F = {GMm}/{r^2}$$
With this I can then set the sum of the two forces to zero:
$$0 = {GMm_3}/r_1^2 + {Gmm_3}/r_2^2$$
My confusion lies in the fact that both $r_1$ and $r_2$ are unknown.  However, we do know that $r_1 + r_2 = d$.  But I am confused with how to solve for either $r_1$ or $r_2$.
 A: Check the signs in your equation. Draw the three planets in a line with the central planet at the origin. What is the direction of the force each of the side planets exert on the middle one?
If you resolve the above problem you can approach solving the equations. Notice that you have as many equations as you have unknows. Therefore each equation lets you (at least in principle) express one unknown as the function of the others which reduces the problem to one with smaller number of unknows (although possibly with a more complicated equations).
A: For a simpler answer: Rewrite the radii in terms of $r+c$ or $r-c$ (depending on which planet's radius will increase vs. decrease) with $c$  the change in the radii. Note that now you have a common variable, use algebra to isolate the new variable $c$. Now that we've found the CHANGE in radius, we can add or subtract this to the radius value $r$ to determine the new final distance depending on which planet's perspective you're talking about. In this case since it's asking for distance from the larger of the two planets, it will be $r+c$.
