Answer: C. $6$

Explanation:

$1$ . Understand the condition for infinitely many solutions:

   - For a system of two linear equations to have infinitely many solutions, the two equations must represent the same line. This means one equation must be a constant multiple of the other.

$2$ . Compare the two equations:

   - Equation 1: $3x + 9y = 10$

   - Equation 2: $rx + 18y = 20$

   - Let's see how the y-terms and the constant terms are related.

   - The y-coefficient in Eq. 2 ( $18$ ) is $2$ times the y-coefficient in Eq. 1 ($9$).

   - The constant term in Eq. 2 ( $20$ ) is $2$ times the constant term in Eq. 1 ($10$).

$3$ . Determine the relationship and solve for r:

   - Since the y-term and the constant are both multiplied by $2$ , the entire second equation must be the first equation multiplied by $2$.

   - $2 \times (3x + 9y = 10)$ gives $6x + 18y = 20$.

   - Comparing this to $rx + 18y = 20$ , we can see that the coefficient of x, r, must be $6$.

$4$ . Conclusion:

   - The value of r must be $6$ for the system to have infinitely many solutions. This corresponds to option C.