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?

$$\\ \\ \begin{cases} { x }^{ 2 }-y=1 \\ x+y=5\quad  \end{cases}\Rightarrow { x }^{ 2 }+x-6=0\Rightarrow \left( x-2 \right) \left( x+3 \right) =0\Rightarrow { x }_{ 1 }=2,{ x }_{ 2 }=-3\Rightarrow \begin{cases} { x }_{ 1 }=2,{ y }_{ 1 }=3 \\ { x }_{ 2 }=-3,y_{ 2 }=2 \end{cases}$$
