Say you have two equations with three variables, the first is the equation of the surface of a sphere and the second of a plane. In this case they intersect in a point $(1,0,0)$. The only way I know to find this point is to rewrite the equation of the sphere so you know its center point and intersect a line going through that point at and at an angle of 90 degrees with the plane. Are there other methods to solve this? Without geometry?
Here are two example equations.
$\begin{cases} x^2 + y^2 + z^2 - 6x + 6y - 12z + 5&=&0\\ 2x - 3y + 6 z - 2&=&0\\ \end{cases}$
and the solution
$\begin{cases} x = 1\\ y = 0\\ z = 0\\ \end{cases}$