Solve for two variables $x$ and $y$ for the system of equations Suppose I have the following
$ x^2 - \alpha_1 x + y^2 - \beta_1 y = \gamma_1  \\
  x^2 - \alpha_2 x + y^2 - \beta_2 y = \gamma_2 $
where the $\alpha_i$'s, $\beta_i$'s, and $\gamma_i$'s are known. How can I solve for $x$ and $y$? Is there a nice matrix approach to this problem? If so, I would really appreciate a detailed explanation.
Thanks!
 A: There can't be a "matrix" approach to the problem since you're not dealing with a system of linear equations (remember, matrices represent linear transformations). However, the problem at hand is the question of points of intersection of two circles (with centers $(\alpha_i/2, \beta_i/2)$ and radii $\sqrt{\alpha_i^2/4 + \beta_i^2/4 + \gamma_i}.$ Now one can write down the points of intersection using simple geometry. 
A: Here is an easy algebraic solution.
Subtracting the two equations  gives
$$
(\alpha_2  - \alpha_1) x + (\beta_2 - \beta_1) y = \gamma_1  - \gamma_2
$$
which is linear in $x $ and $y$. 
So if you want to solve for $y$, then solve  this linear equation for $x$ and insert the result in any of your equations. Taking the first one, you get:
$$
 x = \frac{(\beta_1 - \beta_2) y + \gamma_1  - \gamma_2}{\alpha_2  - \alpha_1}
$$
and 
$$
 \Big[ \frac{(\beta_1 - \beta_2) y + \gamma_1  - \gamma_2}{\alpha_2  - \alpha_1} \Big]^2 - \alpha_1 \Big[ \frac{(\beta_1 - \beta_2) y + \gamma_1  - \gamma_2}{\alpha_2  - \alpha_1} \Big] + y^2 - \beta_1 y = \gamma_1 
$$
which is a quadratic equation in $y$ which can be directly  solved. [The result is involved in the parameters.]
The same procedure can be applied to solve for $x$.
