systems of linear equations intuition I want to know why in a system of linear equations I'm allowed to sum or subtract the equations. I can't get the intuition of why I can do that to solve for the equations. 
 A: Consider
\begin{align*}
5 + 6 &= 11 \quad \quad(1) \\
7 + 12 &= 19 \quad \quad(2) \\
(1) - (2)\implies (5 - 7) + (6 - 12) &= 11 - 19 \\
-2 - 6 & = -8 \quad \text{(true)}
\end{align*}
You see that for equations not containing unknowns, the equality of both sides is not violated upon subtraction / addition.
You are allowed to do the same thing to equations containing unknowns because they adhere to the same rules of arithmetic and get replaced by real numbers after the system has been solved.
A: think of it like this. Suppose you have a=b and c=d where a,b,c,d can be equations based on different variables. So we could say that if a=b then a-b=0 ,since a and b are the same thing. Same for d-c=0. But now we have a-b=0=c-d, thus a-b=d-c. Now we can add b and c on both sides, thus we get a+c=d+b. did that answer your question?  
A: If $a=b$ and $c=d$, then $a-c=b-d$. This is obviously true as you are doing the same computation on both sides of the equal sign. For the same reason, $3a-2b=3c-2d$.
Now if your equalities are complete equations, $ax+by+cz=d$ and $ex+fy+gz=h$, there is no reason you can't do the same
$$(ax+by+cz)-(ex+fy+gz)=d-h,$$which can be rewritten
$$(a-e)x+(b-f)y+(c-g)z=d-h.$$
Also
$$(3a-2e)x+(3b-2f)y+(3c-2g)z=3d-2h.$$
A: The intuition is the following:
If you have an equation, say, $$2x + 3 = 4$$ you know you are allowed to add the same thing to both sides, or subtract the same thing from both sides, or multiply both sides by the same thing, or divide both sides by the same thing.  For example, an equivalent equation where you add $5$ to both sides is: $$2x + 3 + 5 = 4 + 5 $$
So, you are comfortable with doing that.  Well, when we have another equation, say, $$x + 3 = 19$$ what we are saying is that the quantity $x + 3$ is the same as the quantity $19$ (since we wrote $x + 3 = 19$).  Well, if they are the same, then adding $x + 3$ to the left hand side of $2x + 3 = 4$, and adding $19$ to the right hand side, should be allowed because we are allowed to add the same thing to both sides of an equation.  It just so happens that our second equation tells us that $x + 3$ and $19$ are the same thing.
That's why we can say $$2x + 3 + (x + 3) = 4 + 19$$ -- because all we did was add the same thing to both sides of the equation.
So, that's the intuition for adding one equation to another, and why it is allowed.
