You're on the right track with your initial attempts to balance the equation. Let's break it down step by step:
1. Start by balancing the carbons (C) and hydrogens (H) in the equation. Since there is one carbon atom in both CH4 and HCN, we can balance the carbons by assigning a coefficient of 1 to both CH4 and HCN:
CH4 + NH3 + O2 ⟶ HCN + 1H2O
Now let's balance the hydrogens. There are four hydrogen atoms in CH4 and only one hydrogen atom in HCN. To balance them, we need to add three hydrogen atoms to the product side by assigning a coefficient of 3 to H2O:
CH4 + NH3 + O2 ⟶ HCN + 3H2O
2. Moving on to nitrogen (N), there is one nitrogen atom in both NH3 and HCN. Since the nitrogen is already balanced, no further coefficients are needed for nitrogen.
3. Finally, let's balance the oxygens (O). There are two oxygen atoms in H2O and only one oxygen atom in O2. Since we assigned a coefficient of 3 to H2O, we need to assign a coefficient of 3 to O2:
CH4 + NH3 + 3O2 ⟶ HCN + 3H2O
And now the equation is balanced with the lowest number of coefficients:
CH4 + NH3 + 3O2 ⟶ HCN + 3H2O
Note that the equation is now balanced in terms of atoms but may not necessarily be balanced in terms of charges or oxidation states. Balancing those aspects involves more advanced concepts in chemistry.