Solve a system of quadratic equations in two variables for integral solutions. Let's say I have two equations like:
$x^2 - y^2 = 6$ or
$x^2 + y^2 = 6$
What is the best way to solve these sort of equations for finding only positive integral solutions?

As a starting point, I suggest plotting the solutions to each equation to see where the intersections (if any) between them are. For the pair of equations given in the question, we see that there appear to be two points where the circle $x^2 + y^2 = 6$ (red) intersects with the hyperbola $x^2 - y^2 = 6$ (blue), and the intersection is at approximately $x = \pm 2.4, y=0$.

You can find the solutions analytically by adding and subtracting the two equations:
$$(x^2 + y^2) + (x^2 - y^2) = 6 + 6 \implies x^2 = 6 \implies x = \pm \sqrt{6} \approx 2.449$$
and
$$(x^2 + y^2) - (x^2 - y^2) = 6 - 6 \implies y^2 = 0 \implies y = 0.$$
Therefore, the solutions to the system of equations are $(\pm \sqrt{6}, 0)$.
