1. Use the SHM velocity-displacement relation

For simple harmonic motion, $v^2 = \omega^2\left(A^2 - x^2\right).$

Given in the question: $4v^2 = 50 - x^2$

So, $v^2 = \frac{50 - x^2}{4} = \frac{50}{4} - \frac{x^2}{4}.$

Compare with $v^2 = \omega^2 A^2 - \omega^2 x^2.$

Thus, coefficient of $x^2$ gives $\omega^2 = \frac{1}{4} \implies \omega = \frac{1}{2}\,\text{rad/s}.$

2. Find the time period

We know $T = \frac{2\pi}{\omega}.$

Substitute $\omega = \frac{1}{2}$: $T = \frac{2\pi}{1/2} = 4\pi.$

Using $\pi = \frac{22}{7}$, $T = 4\times \frac{22}{7} = \frac{88}{7}\,\text{s}.$

3. Compare with the given form

The question says the time period is $\frac{x}{7}\,\text{s}.$

So, $\frac{x}{7} = \frac{88}{7}.$

Hence, $x = 88.$

4. Final answer

$\boxed{88}$