Suppose i have 3 independent linear equations with 4 variables. If i add another independent linear equation i can find a unique solution.
If i add,instead, in place of the fourth and last independent linear equation, a second order equation with the same 4 variables, it's still possible to solve the system ? If yes, what happens to the solution(s) of the system ?
A system of 3 linear equations in four variables will (most likely) have a line of solutions. (I put most likely since it is possible to already have no solutions).
If you add another (independent) linear equation, to trying to find the intersectionof that line and the hyperplane, that is defined by the fourth linear equation. This will result in a single point, that fullfills als equations.
Equations of higher order will have different sets of solutions. If you use an equation like $x^2+y^2-z=5$ you will have a parabola-like set of points. Finding the intersection of that set with the line will yield other solutions and maybe multiple solutions.