Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid(5x+3y)\}$ is an equivalence relation. 
Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z}  :  4 \mid (5x+3y)\}.$$ Show that $R$ is an equivalence relation. 

I'm having a bit of trouble with this exercise in my book and I am trying to study. Can anyone give guidance for this? I know we have to show reflexivity, symmetry, and transitivity, but I don't think what I have on my paper is completely right. I would appreciate other people's opinions on what the solution should be.

Though the other  answers give a good impression of why $x\sim y\iff 4|(5x+3y)$ defines an equivalence relation, it needs to be said that this is not exactly what is asked in the OP. The OP literally asks:

Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z}  :  4 | (5x+3y)\}$ is an equivalence relation. 

Then one should say: $R$ is not an equivalence relation. $R$ is a set. More precisely, $R$ is a subset of the set $\mathbb Z\times \mathbb Z$. Its definition relies heavily on the equivalence relation mentioned in the first line of this answer, but it is not itself an (equivalence) relation.  
This might seem pedantic, but absolute unambiguous clarity is everything to  a mathematician. 
