$(a,b)R(c,d)\iff a+2b=c+2d$: Describing the Pieces of a Partition

$(a,b)R(c,d)\iff a+2b=c+2d$: Describing the Pieces of a Partition

Let $S$ be the Cartesian coordinate plane $\mathbb{R}\times \mathbb{R}$
and define the equivalence relation $R$ on $S$ by $(a,b)R(c,d)\iff
a+2b=c+2d$.
$\hspace{1cm}$(a) Find the partition $P$ determined by $R$ by describing
the pieces in $P$.
$\hspace{1cm}$(b) Describe the piece of the partition that contains the
point $(5,3)$.



Could someone explain this to me, please? I suspect that the pieces are
just the sets of points in the lines that are the intsections of $z=c$ and
$z=x+2y$, but I don't think that is what this question is asking. Could
someone explain this visuallyperhaps?