Start by writing the balanced chemical equation for this double replacement reaction
#MgCl_(2(aq)) + 2AgNO_(3(aq)) -> 2AgCl_((s)) + Mg(NO_3)_(2(aq))#
The reaction between magnesium chloride and silver nitrate takes place in aqueous solution, which means that you have
#MgCl_(2(aq)) -> Mg_((aq))^(2+) + 2Cl^(-)#
and
#AgNO_(3(aq)) ->Ag_((aq))^(+) + NO_(3(aq))^(-)#
The two products of this double replacement reaction are silver chloride, #AgCl#, which is an insoluble compound, and magnesium nitrate, #Mg(NO_3)_2#, which is a soluble compound, i.e. it will exist as ions in solution.
Therefore, the complete ionic equation will be
#Mg_((aq))^(2+) + 2Cl^(-) + 2Ag_((aq))^(+) + 2NO_(3(aq))^(-) -> 2AgCl_((s)) darr + Mg_((aq))^(2+) + 2NO_(3(aq))#
To get the ney ionic equation, cancel the spectator ions, i.e.ions that are present on both sides of the equation.
#cancel(Mg_((aq))^(2+)) + 2Cl^(-) + 2Ag_((aq))^(+) + cancel(2NO_(3(aq))^(-)) -> 2AgCl_((s)) darr + cancel(Mg_((aq))^(2+)) + cancel(2NO_(3(aq)))#
#bar(2Cl^(-) + 2Ag_((aq))^(+) -> 2AgCl_((s)) darr)#
Simplify the equation by 2 to get
#Cl^(-) + Ag_((aq))^(+) -> AgCl_((s)) darr#
