Why do we sum to prove the transitive property?

Question

In the relation $x − y = 4m$, to prove the transitive property, we do the following:

$$(x − y) + (y − z) = 4m + 4n = 4(m + n)$$

Why?

Answer 1

You have that $x$ is related to $y,$ and $y$ is related to $z.$ You want to show $x$ is related to $z.$ So you want to show $x-z=4k$ for some integer $k.$

You already have $x-y=4m,y-z=4n.$ So $$x-z=(x-y)+(y-z)=4k$$ where $k=m+n.$

Answer 2

You start by assuming that $x \sim y$ and $y \sim z$, so that $x-y = 4m$ and $y-z=4n$. To prove $x \sim z$ you need to show that $x-z = 4k$. But by assumption $x-z = x-y+y-z = 4m+4n = 4(m+n)$, so $k=m+n$.