Step 1 :First, we need to check if the relation R is an equivalence relation. An equivalence relation must satisfy three properties: reflexivity, symmetry, and transitivity.
Step 2 :Reflexivity: For all a, (a, a) ∈ R. This means that \(a*a = a*a\), which is always true for all rational numbers a. So, R is reflexive.
Step 3 :Symmetry: For all a and b, if (a, b) ∈ R then (b, a) ∈ R. This means that if \(a*b = b*a\), then \(b*a = a*b\), which is also always true for all rational numbers a and b. So, R is symmetric.
Step 4 :Transitivity: For all a, b, and c, if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R. This means that if \(a*b = b*c\) and \(b*c = c*a\), then \(a*b = c*a\). This is also always true for all rational numbers a, b, and c. So, R is transitive.
Step 5 :Therefore, the relation R is an equivalence relation.
Step 6 :Next, we need to calculate the value of x where (71,69) R(66, x). This means that \(71*69 = 66*x\). We can solve this equation for x.
Step 7 :\(a1 = 71\)
Step 8 :\(a2 = 69\)
Step 9 :\(a3 = 66\)
Step 10 :\(x = 74.22727272727273\)
Step 11 :Final Answer: The relation R is an equivalence relation and the value of x is \(x=\frac{1633}{22}\). So, the correct option is D. \(\boxed{D}\)