How do you balance #"CaCl"_2 + "Cr"("NO"_3)_3 -> "Ca"("NO"_3)_2 + "CrCl"_3#?
Look for the element that appears the least and balance that first. (Just as when you solve simultaneous linear equation, the variable that appears least often is the easiest to eliminate.) In this case, it is either Ca or Cr. The equation given to us already has both balanced.
Now we try to balance Cl, while making sure that Ca and Cr remains balanced as well. We get:
To get integer coefficients, multiply everywhere by 3.