why adding or subtracting linear equations finds their intersection point? I was trying to understand the logic behind linear algebra but got stuck at this point. Why does adding or subtracting two linear equations with one another result in their intersection point ? After searching i came across a method which goes like this . Lets say we have two equations x+y=5 and 2x+y=8. Now they do this 
y=5-x
y=8-2x
and then 5-x=8-2x and then find the value of x. I understood the logic behind this method. But what about the other method where they just subtract those equations , and find x value first and then find y value . What is going on there ? What is the logic behind it ?

Keep in mind that both sides of the equations are the same (that's the point of an equation). From basic algebra we know that you can manipulate an equation by adding or subtracting the same value on both sides, so if you have
\begin{align}
x+y &= 5, \tag{1}\\
2x+y &= 8, \tag{2}
\end{align}
You can subtract the left hand side (LHS) of $(1)$ from the LHS of $(2)$, while subtracting the right hand side (RHS) of $(1)$ from the RHS of $(2)$ since the LHS and the RHS of each equation are the exact same thing. It's like you're subtracting $5$ from both sides, since $x+y$ really is equal to $5$. Hence we write $(2)$ as
\begin{align}
2x+y &= 8\\
2x + y - x - y &= 8 - 5\\
x &= 3,
\end{align}
which (since $x+y = 5$) implies $y$ must equal $2$.
