It is easy to understand why we don't do simplification in $(\mathbb{R},+, .)$ without having a good look at while solving equation systems since $(\mathbb{R},+, .)$ is a domain. So I also know that multiplying equations with real numbers, or adding one to another is OK. But i can't figure out why my operations don't satisfy the system even if just do some multiplication and addition.
Let x and y be real numbers
$x^2+6y=36 $ $ $ $ $ $ $ $ $ $ $ $ $ (1)
$4x-y^2=49$ $ $ $ $ $ $ $ $ $ $ $ $ (2)
then $y^2-x^2=?$
My Solution:
multiplying (2) with -1
$x^2+6y=36$
$y^2-4x=-49$
then adding them together
$(x-2)^2 +(y+3)^2=0$
$x=2$ and $y=-3$ which are NOT satify the system.
It is easy to figure out that system have no solution in $\mathbb{R}$ since graphs of (1) and (2) are two parabolas wihch don't intersect within any points.
I would be greatful if you explain that confusing matter, thanks in advance.