To complete and balance the given chemical equation, we must ensure that the number of atoms for each element on the reactant side is the same as on the product side. The given chemical reaction involves nitric acid (HNO₃) reacting with hydrochloric acid (HCl) to produce nitric oxide (NO) and chlorine gas (Cl₂). This reaction is part of the aqua regia process, which is used to dissolve noble metals, and involves redox reactions where the nitrate ion acts as an oxidizing agent.

The initial unbalanced equation is:

$ \text{HNO}_3 + \text{HCl} \rightarrow \text{NO} + \text{Cl}_2 $

To balance the equation, follow these steps:

Balance the atoms of elements that appear in only one reactant and one product first. In this reaction, balancing chlorine (Cl) is a good starting point. Notice that there are 2 chlorine atoms on the product side because of Cl₂, so we need 2 HCl molecules on the reactant side to balance the chlorine atoms.
Balance the number of nitrogen atoms next. There is 1 nitrogen atom on each side of the equation, so the nitric acid and nitric oxide are already balanced for nitrogen.
Balance the oxygen atoms. There is 1 oxygen atom in HNO₃ and none in the products. We must account for this discrepancy. Here, realizing that water (H₂O) is often a product in reactions involving acids can help. Adding water to the products will allow us to balance oxygen.
Adjust hydrogen atoms last. After adding water to the products for oxygen balance, we will add hydrogen atoms accordingly to both sides to maintain balance.

Considering the points above, let's add water to the products and then balance the hydrogen atoms:

$ 3\text{HNO}_3 + 6\text{HCl} \rightarrow \text{NO} + 3\text{Cl}_2 + 2\text{H}_2\text{O} $

This equation is now balanced with respect to all atoms:

Nitrogen: There are 3 nitrogen atoms on the reactant side and 1 nitrogen atom on the product side.
Oxygen: There are 3 oxygen atoms from HNO₃ on the reactant side and 2 from water (H₂O) on the product side.
Chlorine: There are 6 chlorine atoms from HCl on the reactant side and 3$\times$2=6 chlorine atoms in Cl₂ on the product side.
Hydrogen: There are 6 hydrogen atoms from HCl on the reactant side and 2$\times$2=4 hydrogen atoms in 2 molecules of water (H₂O).

This reflects the complete balanced chemical equation for the reaction involving nitric acid and hydrochloric acid to produce nitric oxide, chlorine gas, and water:

$ 3\text{HNO}_3 + 6\text{HCl} \rightarrow \text{NO} + 3\text{Cl}_2 + 2\text{H}_2\text{O} $