Calculate the sum of frequencies (sum f) and the sum of (frequency x mark) (sum fx).

Step 1: Calculate the sum of frequencies ($\sum f$) and the sum of (frequency $\times$ mark) ($\sum fx$).
The given table is:

Marks (x)	5	6	7	8	9	10
Frequency (f)	$3x$	$x+1$	4	7	8	$x$

Sum of frequencies:

$$\sum f = 3x + (x+1) + 4 + 7 + 8 + x$$ $$\sum f = 3x + x + x + 1 + 4 + 7 + 8$$ $$\sum f = 5x + 20$$

Sum of (frequency $\times$ mark):

$$\sum fx = (5 \times 3x) + (6 \times (x+1)) + (7 \times 4) + (8 \times 7) + (9 \times 8) + (10 \times x)$$ $$\sum fx = 15x + 6x + 6 + 28 + 56 + 72 + 10x$$ $$\sum fx = 15x + 6x + 10x + 6 + 28 + 56 + 72$$ $$\sum fx = 31x + 162$$

Step 2: Use the given mean to find the value of $x$.
The mean of the distribution is given as $7\frac{7}{15}$. Convert this to an improper fraction:

$$7\frac{7}{15} = \frac{7 \times 15 + 7}{15} = \frac{105 + 7}{15} = \frac{112}{15}$$

The formula for the mean is $\bar{x} = \frac{\sum fx}{\sum f}$.

$$\frac{31x + 162}{5x + 20} = \frac{112}{15}$$

Cross-multiply:

$$15(31x + 162) = 112(5x + 20)$$ $$465x + 2430 = 560x + 2240$$

Rearrange the terms to solve for $x$:

$$2430 - 2240 = 560x - 465x$$ $$190 = 95x$$ $$x = \frac{190}{95}$$ $$x = 2$$