**Property Used to Determine the Value**

The property used to determine the value of the expression 6/7 - 1/3 - 3 is the associative property of addition.

**Explanation**

The associative property of addition states that the grouping of numbers does not affect the sum. In other words, when adding multiple numbers together, changing the grouping of the numbers does not change the result.

To apply the associative property to the given expression, we can rearrange the terms so that we group the numbers differently:

(6/7 - 1/3) - 3

Now, we can simplify the expression within the parentheses:

To subtract fractions, we need to find a common denominator. In this case, the least common multiple of 7 and 3 is 21. We can rewrite the fractions with a common denominator:

(18/21 - 7/21) - 3

Now, we can subtract the fractions:

(11/21) - 3

To subtract a fraction from a whole number, we need to convert the whole number into a fraction with the same denominator as the fraction:

(11/21) - (3/1)

Now, we can subtract the fractions:

(11/21) - (63/21) = -52/21

Therefore, the value of the expression 6/7 - 1/3 - 3 is -52/21.

**Conclusion**

By applying the associative property of addition and simplifying the expression, we determined that the value of the given expression is -52/21.