Confused about rules for solving systems of linear equations Why are we allowed to add equations together/eliminate variables when solving systems of linear equations? I get that it works to find the solution but I don't understand why it works.
Also, why can't we do the same things with nonlinear systems of equations?
For example, the system
$x^2 -y = 1$
$x+y = 5$
If you add them together and solve for $x$ you get the correct $x$ coordinates for the two points of intersection.
EDIT: I know that this system is nonlinear and I know how to solve it for the points of intersection. My main question is why are the rules of adding equations and eliminating variables valid for systems of linear equations but not nonlinear equations in general?

If (x,y) is a solution to the first equation then the left side is 1, and if (x,y) is a solution to the second equation, then the left side is 5.  When you add the two equations, the result on the left side must equal 6 only if (x,y) satisfies both equations i.e. the point of intersection.  The same idea applies to a pair linear equations.  The solution is usually the (single) point of intersection between the lines represented by the two linear equations (in 2D).
