So, here was my original approach

Let d represent the quantity of dimes.

Let q represent the quantity of quarters.

Jessica: d + q = 16

Whitney: 2d + 1/3q = ?

We know that both people have the same amount of money. Consequently,

d + q = 2d + 1/3q

Given that we have d + q = 16, I transformed the equation like so: q = 16 - d. Next, I swapped instances of q for 16 - d like so:

d + (16-d) = 2d + 1/3(16-d)

This did not yield the results I expected. Looking for a correct approach. Thanks.

You actually have a good start. We can work with it.

You've defined (without explicitly stating it, which is an important thing to do):

d = number of dimes Jessica has

q = number of quarters Jessica has

Then you correctly wrote an equation saying Jessica has 16 coins: d + q = 16.

You also wrote an expression for the number of coins Whitney has: 2d + 1/3q. This depends on the facts

2d = number of dimes Whitney has

(1/3)q = number of quarters Whitney has

These are correct facts. But we don't know anything about the number of coins Whitney has; what we know is that they both have the same AMOUNT OF MONEY -- that is, the same VALUE.

Write an expression for the VALUE of each one's coins, and set those expressions equal. That will give you the second equation you need.