  In $1979$, using the theory of ultrafilters, N. Hindman proved that for every
finite coloring of $\mathbb{N}$, there exists a color that contains both
additive and multiplicative $IP$ sets. Later, in $1993$, V. Bergelson and N.
Hindman found an elementary proof. In this article, we prove that the partial
product of these two $IP$ sets also lies in the same color. As an immediate
consequence, we have the equation $a+b=c\cdot d$ partition regular. This was a
conjecture of P. Csikv\'{a}ri, K. Gyarmati, and A. S\'{a}rk\"{o}zy, which was
solved by V. Bergelson and N. Hindman independently. In this article, we
separately give a very short proof of this conjecture. Then we extend these
results to the flavor of Rado systems.
