# Regarding elimination method [Simultaneous Equations]

I know how to solve a linear system of equations by using the elimination method. But, i somehow do not understand what it means to "add" two equations. What is the significance of just "adding" two equations?

For e.g.:

\begin{align} x + y = 2 \tag{1}\\ x - y = 4 \tag{2} \end{align}

Now, adding equation $(1)$ and $(2)$, we can obtain $2x = 6$, thus $x = 3$. But how does this exactly work? I have so far just blindly followed the method. But can someone please provide the logic behind this?

Thanks!

Adding two equations works like this: $2x=(x+y)+(x-y)=2+4=6$.
Adding the two equations can be seen as combining two steps. From $x+y=2$ we can add $4$ to each side, getting $x+y+4=6$. Then, using the second equation, we can substitute $x-y$ for $4$ on the left, getting $x+y+(x-y)=6$, and then simplify to $2x=6$. When you notice that things will simplify nicely, it is useful. You will get the same place (there is only one solution) by following the usual substitution method: use the second to get $x=4+y$ then substitute into the first to get 4+y+y=2\$ and so on.