An urn contains 6 yellow balls and $x$ black balls. When two balls are drawn at random, the probability that both are yellow is $\frac{5}{26}$, then the value of $x$ is

Let $Y = 6$ be the number of yellow balls and $X = x$ be the number of black balls, so the total number of balls is $N = 6 + x$.

The probability of drawing two yellow balls is the ratio of combinations: $\frac{\binom{6}{2}}{\binom{6 + x}{2}} = \frac{5}{26}$.

Evaluating the combinations, $\binom{6}{2} = 15$ and $\binom{6 + x}{2} = \frac{(6 + x)(5 + x)}{2}$, so:

$\frac{15}{\frac{(6 + x)(5 + x)}{2}} = \frac{5}{26}$.

Simplifying, $\frac{30}{(6 + x)(5 + x)} = \frac{5}{26}$.

Dividing both sides by 5: $\frac{6}{(6 + x)(5 + x)} = \frac{1}{26}$.

Cross-multiplying: $(6 + x)(5 + x) = 156$.

Solving, $x^2 + 11x + 30 = 156$ and then $x^2 + 11x - 126 = 0$.

Factoring, $(x - 7)(x + 18) = 0$, so $x = 7$ (since $x \geq 0$).

The number of black balls is $\boxed{\text{C}}$.