Consider the relation $R$ where $\left(a_{1}, a_{2}\right) R\left(a_{3}, a_{4}\right) \Leftrightarrow a_{1} a_{2}=a_{3} a_{4}$ where $a_{1}, a_{2}, a_{3}$ and $a_{4}$ are all rational numbers. Check whether the relation is an equivalence relation or not.Calculate the value of $x$ where $(71,69) R(66, x)$. Which of the following is the correct option ?
Choose an answer
A The relation $R$ is not an equivalence relation and the value of $x$ is $x=\frac{22}{1633}$.
B The relation $R$ is an equivalence relation and the value of $x$ is $x=\frac{22}{1633}$.
C The relation $R$ is not an equivalence relation and the value of $x$ is $x=\frac{22}{1633}$.
D The relation $R$ is an equivalence relation and the value of $x$ is $x=\frac{1633}{22}$.

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. \(\boxed{D}\)