Understanding Adding and Subtracting on both sides of Equation In algebra, I take the number on one side of the equal side and put it on the the other side as the opposite, as in $2+x=10$ becoming $x=10-2$. 
But when I have a negative on one side do I turn it to a positive on the other?
 A: The main point is: $$\text{anything} + \color{red}{\text{stuff}} = \text{something else}$$ if and only if: $$\text{anything} =\text{something else}- \color{red}{\text{stuff}},$$ but $\color{red}{\text{stuff}}$ can be something like $-2$. Say, for example, that $\text{anything} = 10$ and $\text{something else} = x$. So I'm saying that: $$10 - 2 = 10 + \color{red}{(-2)} = x \iff 10 = x - \color{red}{(-2)} = x+2,$$ and so on. (Here I am using that $- \cdot - = +$)
A: Thinking in term of numbers that go from one side to the other is not a good way to understand this key question (in my opinion). We have to think in terms of  the operations that are defined on numbers and of their properties .
I this case the properties we are using are:
1)  $\forall a,b,c \in \mathbb{R} \,:\, a=b \iff a+c=b+c$ (the sum of the same number conserve the identity).
2) $\forall x \in \mathbb{R}$ there exists an opposite $-x$ such that $x+(-x)=0$
So, if we have $a+x=b$, adding the opposite of $a$ to both side we have:
$$
a+x+(-a)=b+(-a) \iff x=b-a
$$
