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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?

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    $\begingroup$ that system isn't linear $\endgroup$ – Dr. Sonnhard Graubner Aug 25 '15 at 18:31
  • $\begingroup$ @Dr.SonnhardGraubner I think OP understands that it isn't linear, but he rather poses a question why he can't apply same rules for non linear systems. $\endgroup$ – Kaster Aug 25 '15 at 18:33
  • $\begingroup$ In fact we can add both equations if the equations are nonlinear. $\endgroup$ – Ángel Mario Gallegos Aug 25 '15 at 18:34
  • $\begingroup$ you think it and what thinks the OP? $\endgroup$ – Dr. Sonnhard Graubner Aug 25 '15 at 18:34
  • $\begingroup$ @Dr.SonnhardGraubner it's quite obvious, actually, what OP thinks on this matter. Just read what he wrote, namely "Also, why can't we do the same things with nonlinear systems of equations?". $\endgroup$ – Kaster Aug 25 '15 at 18:43
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I think you are confused as to adding subtracting and eliminating variables in linear system of equations is true or why it came to be. So let's look at things in general ( for all systems of linear equation at once ) Let us define a linear equation with 2 unknowns x and y $$Ax + By + c = 0 $$ where A, B and C are just numbers(known).let us define another equation, $$Dx+Ey+f=0$$ If I show you that subtracting simultaneous equations is true or came to be, then I have automatically showed you addition of equations is also true.so I won't be showing you additions just give you insights. Ok let's do this! From the two equations we know that 0=0 which is true, So, we can see that $$Ax + By + c = Dx+Ey+f$$ Rearranging we get: $$(A-D)x+(B-E)y+c-f=0$$ This equation is the one I get when subtracting two linear equations. let's say I multiplied the first equation by R ( a number ) so that $AR-D= 0$ I would have a new equation like so: $$(AR-D)x+(BR-E)y-(cR-f)=0$$ This is the type of equation you get after multiplying an equation by R and subtracting the two equations together to get y. The reason why we can do such things is the statement which I started 0=0 was true the final equation is also true. I have showed you for subtraction I think it is easy now why it is true for addition HINT: it has something to do with R

We can also do this for quadratic equations and any type of equations, the reason why it is not used is because it does not simplify our equation any better than the first equation. So there is no use in doing that.

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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).

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

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