Question: 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!

Answer: 
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.
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 $x$.