If we are to find the point of intersection of the lines whose joint equation is $2x^2+3xy-2y^2-9x+7y-5=0$ then we can EITHER consider it as a quadratic in $x$ and then apply quadratic formula and get the equations of two straight lines and then their point of intersection. OR, we can differentiate the given equation first w.r.t $x$ and then w.r.t. $y$ and obtain two equations and then their point of intersection.
We get the same point both ways.
While the equations obtained in both ways are different.
Looks like, partial derivative is giving the equations from the same family as the point of intersection is same.
Why is this so?
What exactly is partial derivative doing here?