One way to wring the equations out of the word problem is to start with made-up numbers, and then abstract the letters.
"There are three times as many bicycles as tricycles." So if we have $4$ tricycles, then we have three times as many bicycles, which is $12$. Abstracting, if we have $T$ tricycles, then we have $B=3T$ bicycles.
Same with the wheels. If we have $12$ bikes, then we have twice as many wheels, which is $24$ wheels. So the number of bike wheels is $W_B = 2B$.
Likewise, the number of trike wheels is $W_T = 3T$.
The last piece of information we know is that the total number of wheels is $81$: $W_B + W_T = 81$.
Now, we substitute to solve for either the number of bikes $B$ or the number of trikes $T$:
$$W_B + W_T = 2B + 3T = 2(3T) + 3T = 9T = 81,$$
so the number of trikes is $T=9$. Then, the number of bikes is
$$B = 3T = 3(9) = 27.$$
And we can have some assurance we're right by calculating the number of wheels:
$$W_B + W_T = 2B + 3T = 2(27) + 3(9) = 54 + 27 = 81.$$