price per person = $x

15x + 15% of 15x = 207
15(x + 15% of x) = 207
115x/100 = 207/15
23x/20 = 69/5
x = 69*4/23 = 3*4 = $12


problem with your approach...
i guesst u deducted 15% of $207 from $207, which is wrong

while what u needed to do was deduct 15% of 15x from 207 to get 15x.

The problem with your method is that if you have a number plus 15% to get a new sum you can't just subtract out 15% again to get the original number. (for instance if you have a $100 and you pay 15% interest you have 115 - you can't subtract 15% of 115 to get back to 100).

You can take two approaches in this problem - you can do the algebra BUT in this case picking numbers from the answers and working backwards through the problem works faster. Since the sum is an even number it is probably best to start with an even number near the middle:
You can start with 12 (would be easy to calculate) - 12(15) = 180.

180 (115/100) = 207 - this is the answer.

If you started at 14.00 you would see it was too much - if you apply this approach logically you will only plug in 2 or 3 answers which can be faster and more reliable than algebra for some students.