Question:
Write the identity element for the binary operation \( * \) on the set \( \mathbb{R} \) defined by:
\[ a * b = \frac{3ab}{7}, \quad \forall a,b \in \mathbb{R} \]
Concept:
An identity element \( e \) satisfies:
\[ a * e = a \quad \text{and} \quad e * a = a \]
Solution:
Step 1: Use definition
\[ a * e = \frac{3a e}{7} = a \]
Step 2: Solve for \( e \)
\[ \frac{3ae}{7} = a \]
Multiply both sides by 7:
\[ 3ae = 7a \]
Divide by \( a \) (for \( a \neq 0 \)):
\[ 3e = 7 \Rightarrow e = \frac{7}{3} \]
Step 3: Verify
\[ a * \frac{7}{3} = \frac{3a \cdot \frac{7}{3}}{7} = a \]
Hence verified.
Final Answer:
\[ \boxed{\frac{7}{3}} \]